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Multiply and Divide Rational Expressions

Determine the values for which a rational expression is undefined

Determine the Values for Which a Rational Expression is Undefined

If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator.

When we work with a numerical fraction, it is easy to avoid dividing by zero because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero.

So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.

Example

Try it.

Determine the value for which each rational expression is undefined:

ⓐ \(\frac{8{a}^{2}b}{3c}\) ⓑ \(\frac{4b-3}{2b+5}\) ⓒ \(\frac{x+4}{{x}^{2}+5x+6}.\)

Solution

The expression will be undefined when the denominator is zero.

ⓐ

	\(\ \frac{8{a}^{2}b}{3c}\)
Set the denominator equal to zero and solve for the variable.	\(\ 3c=0\)
	\(\ c=0\)
	\(\ \frac{8{a}^{2}b}{3c}\ \text{is undefined for}\ c=0.\)

ⓑ

	\(\ \frac{4b-3}{2b+5}\)
Set the denominator equal to zero and solve for the variable.	\(\ \begin{array}{lll}2b+5 & = & 0 \\ 2b & = & -5 \\ b & = & -\frac{5}{2}\end{array}\)
	\(\ \frac{4b-3}{2b+5}\ \text{is undefined for}\ b=-\frac{5}{2}.\)

ⓒ

	\(\frac{x+4}{{x}^{2}+5x+6}\\)
Set the denominator equal to zero and solve for the variable.	\(\begin{array}{l}{x}^{2}+5x+6=0\ \\ (x+2)(x+3)=0\ \\ x+2=0\ \text{or}\ x+3=0\ \\ x=-2\ \text{or}\ x=-3\ \end{array}\)
	\(\frac{x+4}{{x}^{2}+5x+6}\ \text{is undefined for}\ x=-2\ \text{or}\ x=-3.\)

Simplify Rational Expressions

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator. Similarly, a simplified rational expression has no common factors, other than 1, in its numerator and denominator.

For example,

\[\begin{array}{l}\frac{x+2}{x+3}\ \text{is simplified because there are no common factors of}\ x+2\ \text{and}\ x+3. \\ \frac{2x}{3x}\ \text{is not simplified because}\ x\ \text{is a common factor of}\ 2x\ \text{and}\ 3x.\end{array}\]

We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expressions.

Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see \(b\ne 0,c\ne 0\) clearly stated.

To simplify rational expressions, we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property.

Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum.

Removing the x’s from \(\frac{x+5}{x}\) would be like cancelling the 2’s in the fraction \(\frac{2+5}{2}!\)

How to Simplify a Rational Expression

Try it.

Simplify: \(\frac{{x}^{2}+5x+6}{{x}^{2}+8x+12}\).

Solution

Example

Try it.

Simplify: \(\frac{3{a}^{2}-12ab+12{b}^{2}}{6{a}^{2}-24{b}^{2}}\).

Solution

	\(\frac{3{a}^{2}-12ab+12{b}^{2}}{6{a}^{2}-24{b}^{2}}\)
Factor the numerator and denominator, first factoring out the GCF.	\(\frac{3({a}^{2}-4ab+4{b}^{2})}{6({a}^{2}-4{b}^{2})}\)
	\(\frac{3(a-2b)(a-2b)}{6(a+2b)(a-2b)}\)
Remove the common factors of \(a-2b\ \text{and}\ 3.\)	\(\frac{3(a-2b)(a-2b)}{3\cdot 2(a+2b)(a-2b)}\)
	\(\frac{a-2b}{2(a+2b)}\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Multiply Rational Expressions

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.

Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, \(x\ne 0,\)\(x\ne 3,\) and \(x\ne 4.\)

How to Multiply Rational Expressions

Try it.

Simplify: \(\frac{2x}{{x}^{2}-7x+12}\cdot \frac{{x}^{2}-9}{6{x}^{2}}.\)

Solution

Example

Try it.

Multiply: \(\frac{3{a}^{2}-8a-3}{{a}^{2}-25}\cdot \frac{{a}^{2}+10a+25}{3{a}^{2}-14a-5}.\)

Solution

	\(\ \frac{3{a}^{2}-8a-3}{{a}^{2}-25}\cdot \frac{{a}^{2}+10a+25}{3{a}^{2}-14a-5}\)
Factor the numerators and denominators and then multiply.	\(\ \frac{(3a+1)(a-3)(a+5)(a+5)}{(a-5)(a+5)(3a+1)(a-5)}\)
Simplify by dividing out common factors.	\(\ \frac{(3a+1)(a-3)(a+5)(a+5)}{(a-5)(a+5)(3a+1)(a-5)}\)
Simplify.	\(\ \frac{(a-3)(a+5)}{(a-5)(a-5)}\)
Rewrite \((a-5)(a-5)\) using an exponent.	\(\ \frac{(a-3)(a+5)}{{(a-5)}^{2}}\)

Divide Rational Expressions

Just like we did for numerical fractions, to divide rational expressions, we multiply the first fraction by the reciprocal of the second.

Once we rewrite the division as multiplication of the first expression by the reciprocal of the second, we then factor everything and look for common factors.

How to Divide Rational Expressions

Try it.

Divide: \(\frac{{p}^{3}+{q}^{3}}{2{p}^{2}+2pq+2{q}^{2}}\div \frac{{p}^{2}-{q}^{2}}{6}.\)

Solution

Recall from Use the Language of Algebra that a complex fraction is a fraction that contains a fraction in the numerator, the denominator or both. Also, remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.

Example

Try it.

Divide: \(\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}}.\)

Solution

	\(\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}}\)
Rewrite with a division sign.	\(\frac{6{x}^{2}-7x+2}{4x-8}\div \frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}\)
Rewrite as product of first times reciprocal of second.	\(\frac{6{x}^{2}-7x+2}{4x-8}\cdot \frac{{x}^{2}-5x+6}{2{x}^{2}-7x+3}\)
Factor the numerators and the denominators, and then multiply.	\(\frac{(2x-1)(3x-2)(x-2)(x-3)}{4(x-2)(2x-1)(x-3)}\)
Simplify by dividing out common factors.	\(\frac{(2x-1)(3x-2)(x-2)(x-3)}{4(x-2)(2x-1)(x-3)}\)
Simplify.	\(\frac{3x-2}{4}\)

If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then, we factor and multiply.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Multiply and Divide Rational Functions

We started this section stating that a rational expression is an expression of the form \(\frac{p}{q},\) where p and q are polynomials and \(q\ne 0.\) Similarly, we define a rational function as a function of the form \(R(x)=\frac{p(x)}{q(x)}\) where \(p(x)\) and \(q(x)\) are polynomial functions and \(q(x)\) is not zero.

The domain of a rational function is all real numbers except for those values that would cause division by zero. We must eliminate any values that make \(q(x)=0.\)

Example

Try it.

Find the domain of \(R(x)=\frac{2{x}^{2}-14x}{4{x}^{2}-16x-48}.\)

Solution

The domain will be all real numbers except those values that make the denominator zero. We will set the denominator equal to zero , solve that equation, and then exclude those values from the domain.

Set the denominator to zero.	\(4{x}^{2}-16x-48=0\\)
Factor, first factor out the GCF.	\(4({x}^{2}-4x-12)=0\\)
	\(4(x-6)(x+2)=0\\)
Use the Zero Product Property.	\(4\ne 0\ x-6=0\ x+2=0\\)
Solve.	\(x=6\ x=-2\)
	The domain of \(R(x)\) is all real numbers where \(x\ne 6\) and \(x\ne \text{-}2.\)

To multiply rational functions, we multiply the resulting rational expressions on the right side of the equation using the same techniques we used to multiply rational expressions.

Example

Try it.

Find \(R(x)=f(x)\cdot g(x)\) where \(f(x)=\frac{2x-6}{{x}^{2}-8x+15}\) and \(g(x)=\frac{{x}^{2}-25}{2x+10}.\)

Solution

	\(\ R(x)=f(x)\cdot g(x)\)
	\(\ R(x)=\frac{2x-6}{{x}^{2}-8x+15}\cdot \frac{{x}^{2}-25}{2x+10}\)
Factor each numerator and denominator.	\(\ R(x)=\frac{2(x-3)}{(x-3)(x-5)}\cdot \frac{(x-5)(x+5)}{2(x+5)}\)
Multiply the numerators and denominators.	\(\ R(x)=\frac{2(x-3)(x-5)(x+5)}{2(x-3)(x-5)(x+5)}\)
Remove common factors.	\(\ R(x)=\frac{2(x-3)(x-5)(x+5)}{2(x-3)(x-5)(x+5)}\)
Simplify.	\(\ R(x)=1\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

Determine the values for which a rational expression is undefined.
1. Set the denominator equal to zero.
2. Solve the equation.
Equivalent Fractions Property
If a, b, and c are numbers where \(b\ne 0,c\ne 0,\) then \(\frac{a}{b}=\frac{a\cdot c}{b\cdot c}\) and \(\frac{a\cdot c}{b\cdot c}=\frac{a}{b}.\)
How to simplify a rational expression.
1. Factor the numerator and denominator completely.
2. Simplify by dividing out common factors.
Opposites in a Rational Expression
The opposite of \(a-b\) is \(b-a.\)
\(\frac{a-b}{b-a}=-1\ a\ne b\)
An expression and its opposite divide to \(-1.\)
Multiplication of Rational Expressions
If p, q, r, and s are polynomials where \(q\ne 0,s\ne 0,\) then
\(\ \frac{p}{q}\cdot \frac{r}{s}=\frac{pr}{qs}\)
How to multiply rational expressions.
1. Factor each numerator and denominator completely.
2. Multiply the numerators and denominators.
3. Simplify by dividing out common factors.
Division of Rational Expressions
If p, q, r, and s are polynomials where \(q\ne 0,r\ne 0,s\ne 0,\) then
\(\ \frac{p}{q}\div \frac{r}{s}=\frac{p}{q}\cdot \frac{s}{r}\)
How to divide rational expressions.
1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
2. Factor the numerators and denominators completely.
3. Multiply the numerators and denominators together.
4. Simplify by dividing out common factors.
How to determine the domain of a rational function.
1. Set the denominator equal to zero.
2. Solve the equation.
3. The domain is all real numbers excluding the values found in Step 2.

Multiply and Divide Rational Expressions

Determine the Values for Which a Rational Expression is Undefined

In the following exercises, determine the values for which the rational expression is undefined.

Try it.

ⓐ \(\frac{2{x}^{2}}{z}\), ⓑ \(\frac{4p-1}{6p-5}\), ⓒ \(\frac{n-3}{{n}^{2}+2n-8}\)

Solution

ⓐ \(z=0\) ⓑ \(p=\frac{5}{6}\)
ⓒ \(n=-4,n=2\)

Try it.

ⓐ \(\frac{10m}{11n}\), ⓑ \(\frac{6y+13}{4y-9}\), ⓒ \(\frac{b-8}{{b}^{2}-36}\)

Try it.

ⓐ \(\frac{4{x}^{2}y}{3y}\), ⓑ \(\frac{3x-2}{2x+1}\), ⓒ \(\frac{u-1}{{u}^{2}-3u-28}\)

Solution

ⓐ \(y=0\), ⓑ \(x=-\frac{1}{2}\), ⓒ \(u=-4,u=7\)

Try it.

ⓐ \(\frac{5p{q}^{2}}{9q}\), ⓑ \(\frac{7a-4}{3a+5}\), ⓒ \(\frac{1}{{x}^{2}-4}\)

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

Try it.

\(-\frac{44}{55}\)

Solution

\(-\frac{4}{5}\)

Try it.

\(\frac{56}{63}\)

Try it.

\(\frac{8{m}^{3}n}{12m{n}^{2}}\)

Solution

\(\frac{2{m}^{2}}{3n}\)

Try it.

\(\frac{36{v}^{3}{w}^{2}}{27v{w}^{3}}\)

Try it.

\(\frac{8n-96}{3n-36}\)

Solution

\(\frac{8}{3}(n\ne 2)\)

Try it.

\(\frac{12p-240}{5p-100}\)

Try it.

\(\frac{{x}^{2}+4x-5}{{x}^{2}-2x+1}\)

Solution

\(\frac{x+5}{x-1}\)

Try it.

\(\frac{{y}^{2}+3y-4}{{y}^{2}-6y+5}\)

Try it.

\(\frac{{a}^{2}-4}{{a}^{2}+6a-16}\)

Solution

\(\frac{a+2}{a+8}\)

Try it.

\(\frac{{y}^{2}-2y-3}{{y}^{2}-9}\)

Try it.

\(\frac{{p}^{3}+3{p}^{2}+4p+12}{{p}^{2}+p-6}\)

Solution

\(\frac{{p}^{2}+4}{p-2}\)

Try it.

\(\frac{{x}^{3}-2{x}^{2}-25x+50}{{x}^{2}-25}\)

Try it.

\(\frac{8{b}^{2}-32b}{2{b}^{2}-6b-80}\)

Solution

\(\frac{4b(b-4)}{(b+5)(b-8)}\)

Try it.

\(\frac{-5{c}^{2}-10c}{-10{c}^{2}+30c+100}\)

Try it.

\(\frac{3{m}^{2}+30mn+75{n}^{2}}{4{m}^{2}-100{n}^{2}}\)

Solution

\(\frac{3(m+5n)}{4(m-5n)}\)

Try it.

\(\frac{5{r}^{2}+30rs-35{s}^{2}}{{r}^{2}-49{s}^{2}}\)

Try it.

\(\frac{a-5}{5-a}\)

Solution

\(-1\)

Try it.

\(\frac{5-d}{d-5}\)

Try it.

\(\frac{20-5y}{{y}^{2}-16}\)

Solution

\(-\frac{5}{y+4}\)

Try it.

\(\frac{4v-32}{64-{v}^{2}}\)

Try it.

\(\frac{{w}^{3}+216}{{w}^{2}-36}\)

Solution

\(\frac{{w}^{2}-6w+36}{w-6}\)

Try it.

\(\frac{{v}^{3}+125}{{v}^{2}-25}\)

Try it.

\(\frac{{z}^{2}-9z+20}{16-{z}^{2}}\)

Solution

\(-\frac{z-5}{4+z}\)

Try it.

\(\frac{{a}^{2}-5a-36}{81-{a}^{2}}\)

Multiply Rational Expressions

In the following exercises, multiply the rational expressions.

Try it.

\(\frac{12}{16}\cdot \frac{4}{10}\)

Solution

\(\frac{3}{10}\)

Try it.

\(\frac{32}{5}\cdot \frac{16}{24}\)

Try it.

\(\frac{5{x}^{2}{y}^{4}}{12x{y}^{3}}\cdot \frac{6{x}^{2}}{20{y}^{2}}\)

Solution

\(\frac{{x}^{3}}{8y}\)

Try it.

\(\frac{12{a}^{3}b}{{b}^{2}}\cdot \frac{2a{b}^{2}}{9{b}^{3}}\)

Try it.

\(\frac{5{p}^{2}}{{p}^{2}-5p-36}\cdot \frac{{p}^{2}-16}{10p}\)

Solution

\(\frac{p(p-4)}{2(p-9)}\)

Try it.

\(\frac{3{q}^{2}}{{q}^{2}+q-6}\cdot \frac{{q}^{2}-9}{9q}\)

Try it.

\(\frac{2{y}^{2}-10y}{{y}^{2}+10y+25}\cdot \frac{y+5}{6y}\)

Solution

\(\frac{y-5}{3(y+5)}\)

Try it.

\(\frac{{z}^{2}+3z}{{z}^{2}-3z-4}\cdot \frac{z-4}{{z}^{2}}\)

Try it.

\(\frac{28-4b}{3b-3}\cdot \frac{{b}^{2}+8b-9}{{b}^{2}-49}\)

Solution

\(-\frac{4(b+9)}{3(b+7)}\)

Try it.

\(\frac{72m-12{m}^{2}}{8m+32}\cdot \frac{{m}^{2}+10m+24}{{m}^{2}-36}\)

Try it.

\(\frac{3{c}^{2}-16c+5}{{c}^{2}-25}\cdot \frac{{c}^{2}+10c+25}{3{c}^{2}-14c-5}\)

Solution

\(\frac{(3c-1)(c+5)}{(3c+1)(c-5)}\)

Try it.

\(\frac{2{d}^{2}+d-3}{{d}^{2}-16}\cdot \frac{{d}^{2}-8d+16}{2{d}^{2}-9d-18}\)

Try it.

\(\frac{6{m}^{2}-13m+2}{9-{m}^{2}}\cdot \frac{{m}^{2}-6m+9}{6{m}^{2}+23m-4}\)

Solution

\(-\frac{(m-2)(m-3)}{(3+m)(m+4)}\)

Try it.

\(\frac{2{n}^{2}-3n-14}{25-{n}^{2}}\cdot \frac{{n}^{2}-10n+25}{2{n}^{2}-13n+21}\)

Divide Rational Expressions

In the following exercises, divide the rational expressions.

Try it.

\(\frac{v-5}{11-v}\div \frac{{v}^{2}-25}{v-11}\)

Solution

\(-\frac{1}{v+5}\)

Try it.

\(\frac{10+w}{w-8}\div \frac{100-{w}^{2}}{8-w}\)

Try it.

\(\frac{3{s}^{2}}{{s}^{2}-16}\div \frac{{s}^{3}+4{s}^{2}+16s}{{s}^{3}-64}\)

Solution

\(\frac{3s}{s+4}\)

Try it.

\(\frac{{r}^{2}-9}{15}\div \frac{{r}^{3}-27}{5{r}^{2}+15r+45}\)

Try it.

\(\frac{{p}^{3}+{q}^{3}}{3{p}^{2}+3pq+3{q}^{2}}\div \frac{{p}^{2}-{q}^{2}}{12}\)

Solution

\(\frac{4({p}^{2}-pq+{q}^{2})}{(p-q)({p}^{2}+pq+{q}^{2})}\)

Try it.

\(\frac{{v}^{3}-8{w}^{3}}{2{v}^{2}+4vw+8{w}^{2}}\div \frac{{v}^{2}-4{w}^{2}}{4}\)

Try it.

\(\frac{{x}^{2}+3x-10}{4x}\div (2{x}^{2}+20x+50)\)

Solution

\(\frac{x-2}{8x(x+5)}\)

Try it.

\(\frac{2{y}^{2}-10yz-48{z}^{2}}{2y-1}\div (4{y}^{2}-32yz)\)

Try it.

\(\frac{\frac{2{a}^{2}-a-21}{5a+20}}{\frac{{a}^{2}+7a+12}{{a}^{2}+8a+16}}\)

Solution

\(\frac{2a-7}{5}\)

Try it.

\(\frac{\frac{3{b}^{2}+2b-8}{12b+18}}{\frac{3{b}^{2}+2b-8}{2{b}^{2}-7b-15}}\)

Try it.

\(\frac{\frac{12{c}^{2}-12}{2{c}^{2}-3c+1}}{\frac{4c+4}{6{c}^{2}-13c+5}}\)

Solution

\(3(3c-5)\)

Try it.

\(\frac{\frac{4{d}^{2}+7d-2}{35d+10}}{\frac{{d}^{2}-4}{7{d}^{2}-12d-4}}\)

For the following exercises, perform the indicated operations.

Try it.

\(\frac{10{m}^{2}+80m}{3m-9}\cdot \frac{{m}^{2}+4m-21}{{m}^{2}-9m+20}\div \frac{5{m}^{2}+10m}{2m-10}\)

Solution

\(\frac{4(m+8)(m+7)}{3(m-4)(m+2)}\)

Try it.

\(\frac{4{n}^{2}+32n}{3n+2}\cdot \frac{3{n}^{2}-n-2}{{n}^{2}+n-30}\div \frac{108{n}^{2}-24n}{n+6}\)

Try it.

\(\frac{12{p}^{2}+3p}{p+3}\div \frac{{p}^{2}+2p-63}{{p}^{2}-p-12}\cdot \frac{p-7}{9{p}^{3}-9{p}^{2}}\)

Solution

\(\frac{(4p+1)(p-4)}{3p(p+9)(p-1)}\)

Try it.

\(\frac{6q+3}{9{q}^{2}-9q}\div \frac{{q}^{2}+14q+33}{{q}^{2}+4q-5}\cdot \frac{4{q}^{2}+12q}{12q+6}\)

Multiply and Divide Rational Functions

In the following exercises, find the domain of each function.

Try it.

\(R(x)=\frac{{x}^{3}-2{x}^{2}-25x+50}{{x}^{2}-25}\)

Solution

\(x\ne 5\) and \(x\ne \text{-}5\)

Try it.

\(R(x)=\frac{{x}^{3}+3{x}^{2}-4x-12}{{x}^{2}-4}\)

Try it.

\(R(x)=\frac{3{x}^{2}+15x}{6{x}^{2}+6x-36}\)

Solution

\(x\ne 2\) and \(x\ne \text{-}3\)

Try it.

\(R(x)=\frac{8{x}^{2}-32x}{2{x}^{2}-6x-80}\)

For the following exercises, find \(R(x)=f(x)\cdot g(x)\) where \(f(x)\) and \(g(x)\) are given.

Try it.

\(f(x)=\frac{6{x}^{2}-12x}{{x}^{2}+7x-18}\)
\(\ g(x)=\frac{{x}^{2}-81}{3{x}^{2}-27x}\)

Solution

\(R(x)=2\)

Try it.

\(f(x)=\frac{{x}^{2}-2x}{{x}^{2}+6x-16}\)
\(\ g(x)=\frac{{x}^{2}-64}{{x}^{2}-8x}\)

Try it.

\(f(x)=\frac{4x}{{x}^{2}-3x-10}\)
\(\ g(x)=\frac{{x}^{2}-25}{8{x}^{2}}\)

Solution

\(R(x)=\frac{x+5}{2x(x+2)}\)

Try it.

\(f(x)=\frac{2{x}^{2}+8x}{{x}^{2}-9x+20}\)
\(\ g(x)=\frac{x-5}{{x}^{2}}\)

For the following exercises, find \(R(x)=\frac{f(x)}{g(x)}\) where \(f(x)\) and \(g(x)\) are given.

Try it.

\(f(x)=\frac{27{x}^{2}}{3x-21}\)
\(\ g(x)=\frac{3{x}^{2}+18x}{{x}^{2}+13x+42}\)

Solution

\(R(x)=\frac{3x(x+7)}{x-7}\)

Try it.

\(f(x)=\frac{24{x}^{2}}{2x-8}\)
\(\ g(x)=\frac{4{x}^{3}+28{x}^{2}}{{x}^{2}+11x+28}\)

Try it.

\(f(x)=\frac{16{x}^{2}}{4x+36}\)
\(\ g(x)=\frac{4{x}^{2}-24x}{{x}^{2}+4x-45}\)

Solution

\(R(x)=\frac{x(x-5)}{x-6}\)

Try it.

\(f(x)=\frac{24{x}^{2}}{2x-4}\)
\(\ g(x)=\frac{12{x}^{2}+36x}{{x}^{2}-11x+18}\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Multiply Rational Expressions

We’ll do the first example with numerical fractions to remind us of how we multiplied fractions without variables.

Example

Try it.

Multiply: \(\frac{10}{28}\cdot \frac{8}{15}.\)

Solution


Multiply the numerators and denominators.
Look for common factors, and then remove them.
Simplify.

Example

Try it.

Mulitply: \(\frac{2x}{3{y}^{2}}\cdot \frac{6x{y}^{3}}{{x}^{2}y}.\)

Solution


Multiply.
Factor the numerator and denominator completely, and then remove common factors.
Simplify.

How to Multiply Rational Expressions

Try it.

Mulitply: \(\frac{2x}{{x}^{2}-7x+12}\cdot \frac{{x}^{2}-9}{6{x}^{2}}.\)

Solution

Example

Try it.

Multiply: \(\frac{{n}^{2}-7n}{{n}^{2}+2n+1}\cdot \frac{n+1}{2n}.\)

Solution

	\(\frac{{n}^{2}-7n}{{n}^{2}+2n+1}\cdot \frac{n+1}{2n}\)
Factor each numerator and denominator.	\(\frac{n(n-7)}{(n+1)(n+1)}\cdot \frac{n+1}{2n}\)
Multiply the numerators and the denominators.	\(\frac{n(n-7)(n+1)}{(n+1)(n+1)2n}\)
Remove common factors.	\(\frac{n(n-7)(n+1)}{(n+1)(n+1)2n}\)
Simplify.	\(\frac{n-7}{2(n+1)}\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Divide Rational Expressions

To divide rational expressions we multiply the first fraction by the reciprocal of the second, just like we did for numerical fractions.

Remember, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). To find the reciprocal we simply put the numerator in the denominator and the denominator in the numerator. We “flip” the fraction.

How to Divide Rational Expressions

Try it.

Divide: \(\frac{x+9}{6-x}\div \frac{{x}^{2}-81}{x-6}.\)

Solution

Example

Try it.

Divide: \(\frac{3{n}^{2}}{{n}^{2}-4n}\div \frac{9{n}^{2}-45n}{{n}^{2}-7n+10}.\)

Solution


Rewrite the division as the product of the first rational expression and the reciprocal of the second.
Factor the numerators and denominators and then multiply.
Simplify by dividing out common factors.

Remember, first rewrite the division as multiplication of the first expression by the reciprocal of the second. Then factor everything and look for common factors.

Example

Try it.

Divide: \(\frac{2{x}^{2}+5x-12}{{x}^{2}-16}\div \frac{2{x}^{2}-13x+15}{{x}^{2}-8x+16}.\)

Solution

	\(\frac{2{x}^{2}+5x-12}{{x}^{2}-16}\div \frac{2{x}^{2}-13x+15}{{x}^{2}-8x+16}\)
Rewrite the division as multiplication of the first expression by the reciprocal of the second.	\(\frac{2{x}^{2}+5x-12}{{x}^{2}-16}\cdot \frac{{x}^{2}-8x+16}{2{x}^{2}-13x+15}\)
Factor the numerators and denominators and then multiply.	\(\frac{(2x-3)(x+4)(x-4)(x-4)}{(x-4)(x+4)(2x-3)(x-5)}\)
Simplify by dividing out common factors.	\(\frac{(2x-3)(x+4)(x-4)(x-4)}{(x-4)(x+4)(2x-3)(x-5)}\)
Simplify.	\(\frac{x-4}{x-5}\)

\[\begin{array}{l}\frac{3}{5}\div 4 \\ \frac{3}{5}\div \frac{4}{1} \\ \frac{3}{5}\cdot \frac{1}{4}\end{array}\]

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

Multiplication of Rational Expressions
- If \(p,q,r,s\) are polynomials where \(q\ne 0,s\ne 0\), then \(\frac{p}{q}\cdot \frac{r}{s}=\frac{pr}{qs}\).
- To multiply rational expressions, multiply the numerators and multiply the denominators
Multiply a Rational Expression
1. Factor each numerator and denominator completely.
2. Multiply the numerators and denominators.
3. Simplify by dividing out common factors.
Division of Rational Expressions
- If \(p,q,r,s\) are polynomials where \(q\ne 0,r\ne 0,s\ne 0\), then \(\frac{p}{q}\div \frac{r}{s}=\frac{p}{q}\cdot \frac{s}{r}\).
- To divide rational expressions multiply the first fraction by the reciprocal of the second.
Divide Rational Expressions
1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
2. Factor the numerators and denominators completely.
3. Multiply the numerators and denominators together.
4. Simplify by dividing out common factors.

Multiply and Divide Rational Expressions

Multiply Rational Expressions

In the following exercises, multiply.

Try it.

\(\frac{12}{16}\cdot \frac{4}{10}\)

Solution

\(\frac{3}{10}\)

Try it.

\(\frac{32}{5}\cdot \frac{16}{24}\)

Try it.

\(\frac{18}{10}\cdot \frac{4}{30}\)

Solution

\(\frac{6}{25}\)

Try it.

\(\frac{21}{36}\cdot \frac{45}{24}\)

Try it.

\(\frac{5{x}^{2}{y}^{4}}{12x{y}^{3}}\cdot \frac{6{x}^{2}}{20{y}^{2}}\)

Solution

\(\frac{{x}^{3}}{8y}\)

Try it.

\(\frac{8{w}^{3}y}{9{y}^{2}}\cdot \frac{3y}{4{w}^{4}}\)

Try it.

\(\frac{12{a}^{3}b}{{b}^{2}}\cdot \frac{2a{b}^{2}}{9{b}^{3}}\)

Solution

\(\frac{8{a}^{4}}{3{b}^{2}}\)

Try it.

\(\frac{4m{n}^{2}}{5{n}^{3}}\cdot \frac{m{n}^{3}}{8{m}^{2}{n}^{2}}\)

Try it.

\(\frac{5{p}^{2}}{{p}^{2}-5p-36}\cdot \frac{{p}^{2}-16}{10p}\)

Solution

\(\frac{p(p-4)}{2(p-9)}\)

Try it.

\(\frac{3{q}^{2}}{{q}^{2}+q-6}\cdot \frac{{q}^{2}-9}{9q}\)

Try it.

\(\frac{4r}{{r}^{2}-3r-10}\cdot \frac{{r}^{2}-25}{8{r}^{2}}\)

Solution

\(\frac{r+5}{2r(r+2)}\)

Try it.

\(\frac{s}{{s}^{2}-9s+14}\cdot \frac{{s}^{2}-49}{7{s}^{2}}\)

Try it.

\(\frac{{x}^{2}-7x}{{x}^{2}+6x+9}\cdot \frac{x+3}{4x}\)

Solution

\(\frac{x-7}{4(x+3)}\)

Try it.

\(\frac{2{y}^{2}-10y}{{y}^{2}+10y+25}\cdot \frac{y+5}{6y}\)

Try it.

\(\frac{{z}^{2}+3z}{{z}^{2}-3z-4}\cdot \frac{z-4}{{z}^{2}}\)

Solution

\(\frac{z+3}{z(z+1)}\)

Try it.

\(\frac{2{a}^{2}+8a}{{a}^{2}-9a+20}\cdot \frac{a-5}{{a}^{2}}\)

Try it.

\(\frac{28-4b}{3b-3}\cdot \frac{{b}^{2}+8b-9}{{b}^{2}-49}\)

Solution

\(-\frac{4(b+9)}{3(b+7)}\)

Try it.

\(\frac{18c-2{c}^{2}}{6c+30}\cdot \frac{{c}^{2}+7c+10}{{c}^{2}-81}\)

Try it.

\(\frac{35d-7{d}^{2}}{{d}^{2}+7d}\cdot \frac{{d}^{2}+12d+35}{{d}^{2}-25}\)

Solution

\(-7\)

Try it.

\(\frac{72m-12{m}^{2}}{8m+32}\cdot \frac{{m}^{2}+10m+24}{{m}^{2}-36}\)

Try it.

\(\frac{4n+20}{{n}^{2}+n-20}\cdot \frac{{n}^{2}-16}{4n+16}\)

Solution

\(1\)

Try it.

\(\frac{6{p}^{2}-6p}{{p}^{2}+7p-18}\cdot \frac{{p}^{2}-81}{3{p}^{2}-27p}\)

Try it.

\(\frac{{q}^{2}-2q}{{q}^{2}+6q-16}\cdot \frac{{q}^{2}-64}{{q}^{2}-8q}\)

Solution

\(1\)

Try it.

\(\frac{2{r}^{2}-2r}{{r}^{2}+4r-5}\cdot \frac{{r}^{2}-25}{2{r}^{2}-10r}\)

Divide Rational Expressions

In the following exercises, divide.

Try it.

\(\frac{t-6}{3-t}\div \frac{t-5}{{t}^{2}-9}\)

Solution

\(\frac{\left(6-t\right)\left(t+3\right)}{t-5}\)

Try it.

\(\frac{v-5}{11-v}\div \frac{{v}^{2}-25}{v-11}\)

Try it.

\(\frac{10+w}{w-8}\div \frac{100-{w}^{2}}{8-w}\)

Solution

\(-\frac{1}{10-w}\)

Try it.

\(\frac{7+x}{x-6}\div {\frac{49-x}{x+6}}^{2}\)

Try it.

\(\frac{27{y}^{2}}{3y-21}\div \frac{3{y}^{2}+18}{{y}^{2}+13y+42}\)

Solution

\(\frac{3{y}^{2}(y+6)(y+7)}{(y-7)({y}^{2}+6)}\)

Try it.

\(\frac{24{z}^{2}}{2z-8}\div \frac{4z-28}{{z}^{2}-11z+28}\)

Try it.

\(\frac{16{a}^{2}}{4a+36}\div \frac{4{a}^{2}-24a}{{a}^{2}+4a-45}\)

Solution

\(\frac{a(a-5)}{a-6}\)

Try it.

\(\frac{24{b}^{2}}{2b-4}\div \frac{12{b}^{2}+36b}{{b}^{2}-11b+18}\)

Try it.

\(\frac{5{c}^{2}+9c-2}{{c}^{2}-4}\div \frac{5{c}^{2}-16c+3}{{c}^{2}+4c+4}\)

Solution

\(\frac{(c+2)(c+2)}{(c-2)(c-3)}\)

Try it.

\(\frac{2{d}^{2}+d-3}{{d}^{2}-16}\div \frac{2{d}^{2}-9d-18}{{d}^{2}-8d+16}\)

Try it.

\(\frac{6{m}^{2}-11m-2}{9-{m}^{2}}\div \frac{6{m}^{2}+25m+4}{{m}^{2}-6m+9}\)

Solution

\(-\frac{(m-2)(m-3)}{(3+m)(m+4)}\)

Try it.

\(\frac{2{n}^{2}-3n-14}{25-{n}^{2}}\div \frac{2{n}^{2}-13n+21}{{n}^{2}-10n+25}\)

Try it.

\(\frac{3{s}^{2}}{{s}^{2}-16}\div \frac{{s}^{3}+4{s}^{2}+16s}{{s}^{3}-64}\)

Solution

\(\frac{3s}{s+4}\)

Try it.

\(\frac{{r}^{2}-9}{15}\div \frac{{r}^{3}-27}{5{r}^{2}+15r+45}\)

Try it.

\(\frac{{p}^{3}+{q}^{3}}{3{p}^{2}+3pq+3{q}^{2}}\div \frac{{p}^{2}-{q}^{2}}{12}\)

Solution

\(\frac{4({p}^{2}-pq+{q}^{2})}{(p-q)({p}^{2}+pq+{q}^{2})}\)

Try it.

\(\frac{{v}^{3}-8{w}^{3}}{2{v}^{2}+4vw+8{w}^{2}}\div \frac{{v}^{2}-4{w}^{2}}{4}\)

Try it.

\(\frac{{t}^{2}-9}{2t}\div ({t}^{2}-6t+9)\)

Solution

\(\frac{t+3}{2t(t-3)}\)

Try it.

\(\frac{{x}^{2}+3x-10}{4x}\div (2{x}^{2}+20x+50)\)

Try it.

\(\frac{2{y}^{2}-10yz-48{z}^{2}}{2y-1}\div (4{y}^{2}-32yz)\)

Solution

\(\frac{y+3z}{2y(2y-1)}\)

Try it.

\(\frac{2{m}^{2}-98{n}^{2}}{2m+6}\div ({m}^{2}-7mn)\)

Try it.

\(\frac{\frac{2{a}^{2}-a-21}{5a+20}}{\frac{{a}^{2}+7a+12}{{a}^{2}+8a+16}}\)

Solution

\(\frac{2a-7}{5}\)

Try it.

\(\frac{\frac{3{b}^{2}+2b-8}{12b+18}}{\frac{3{b}^{2}+2b-8}{2{b}^{2}-7b-15}}\)

Try it.

\(\frac{\frac{12{c}^{2}-12}{2{c}^{2}-3c+1}}{\frac{4c+4}{6{c}^{2}-13c+5}}\)

Solution

\(3(3c-5)\)

Try it.

\(\frac{\frac{4{d}^{2}+7d-2}{35d+10}}{\frac{{d}^{2}-4}{7{d}^{2}-12d-4}}\)

Try it.

\(\frac{10{m}^{2}+80m}{3m-9}\cdot \frac{{m}^{2}+4m-21}{{m}^{2}-9m+20}\)
\(\ \div \frac{5{m}^{2}+10m}{2m-10}\)

Solution

\(\frac{4(m+8)(m+7)}{3(m-4)(m+2)}\)

Try it.

\(\frac{4{n}^{2}+32n}{3n+2}\cdot \frac{3{n}^{2}-n-2}{{n}^{2}+n-30}\)
\(\ \div \frac{108{n}^{2}-24n}{n+6}\)

Try it.

\(\frac{12{p}^{2}+3p}{p+3}\div \frac{{p}^{2}+2p-63}{{p}^{2}-p-12}\)
\(\ \cdot \frac{p-7}{9{p}^{3}-9{p}^{2}}\)

Solution

\(\frac{(4p+1)(p-4)}{3p(p+9)(p-1)}\)

Try it.

\(\frac{6q+3}{9{q}^{2}-9q}\div \frac{{q}^{2}+14q+33}{{q}^{2}+4q-5}\)
\(\ \cdot \frac{4{q}^{2}+12q}{12q+6}\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Simplify: \(\frac{90y}{15{y}^{2}}.\)
If you missed this problem, review .

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\(\frac{6}{y}\)
Multiply: \(\frac{14}{15}\cdot \frac{6}{35}.\)
If you missed this problem, review .

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\(\frac{4}{25}\)
Divide: \(\frac{12}{10}\div \frac{8}{25}.\)
If you missed this problem, review .

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\(\frac{15}{4}\)

Determine the value for which each rational expression is undefined:

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The expression will be undefined when the denominator is zero.

ⓐ

	\(\ \frac{8{a}^{2}b}{3c}\)
Set the denominator equal to zero and solve for the variable.	\(\ 3c=0\)
	\(\ c=0\)
	\(\ \frac{8{a}^{2}b}{3c}\ \text{is undefined for}\ c=0.\)

ⓑ

	\(\ \frac{4b-3}{2b+5}\)
Set the denominator equal to zero and solve for the variable.	\(\ \begin{array}{lll}2b+5 & = & 0 \\ 2b & = & -5 \\ b & = & -\frac{5}{2}\end{array}\)
	\(\ \frac{4b-3}{2b+5}\ \text{is undefined for}\ b=-\frac{5}{2}.\)

ⓒ

	\(\frac{x+4}{{x}^{2}+5x+6}\\)
Set the denominator equal to zero and solve for the variable.	\(\begin{array}{l}{x}^{2}+5x+6=0\ \\ (x+2)(x+3)=0\ \\ x+2=0\ \text{or}\ x+3=0\ \\ x=-2\ \text{or}\ x=-3\ \end{array}\)
	\(\frac{x+4}{{x}^{2}+5x+6}\ \text{is undefined for}\ x=-2\ \text{or}\ x=-3.\)

Determine the value for which each rational expression is undefined.
ⓐ \(\frac{3{y}^{2}}{8x}\) ⓑ \(\frac{8n-5}{3n+1}\) ⓒ \(\frac{a+10}{{a}^{2}+4a+3}\)

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ⓐ \(x=0\) ⓑ \(n=-\frac{1}{3}\)
ⓒ \(a=-1,a=-3\)
Determine the value for which each rational expression is undefined.
ⓐ \(\frac{4p}{5q}\) ⓑ \(\frac{y-1}{3y+2}\) ⓒ \(\frac{m-5}{{m}^{2}+m-6}\)

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ⓐ \(q=0\) ⓑ \(y=-\frac{2}{3}\)
ⓒ \(m=2,m=-3\)
Simplify: \(\frac{{x}^{2}+5x+6}{{x}^{2}+8x+12}\).
Simplify: \(\frac{{x}^{2}-x-2}{{x}^{2}-3x+2}.\)

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\(\frac{x+1}{x-1},\)\(x\ne 2,\)\(x\ne 1\)
Simplify: \(\frac{{x}^{2}-3x-10}{{x}^{2}+x-2}.\)

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\(\frac{x-5}{x-1},\)\(x\ne \text{-}2,\)\(x\ne 1\)

Simplify: \(\frac{3{a}^{2}-12ab+12{b}^{2}}{6{a}^{2}-24{b}^{2}}\).

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	\(\frac{3{a}^{2}-12ab+12{b}^{2}}{6{a}^{2}-24{b}^{2}}\)
Factor the numerator and denominator, first factoring out the GCF.	\(\frac{3({a}^{2}-4ab+4{b}^{2})}{6({a}^{2}-4{b}^{2})}\)
	\(\frac{3(a-2b)(a-2b)}{6(a+2b)(a-2b)}\)
Remove the common factors of \(a-2b\ \text{and}\ 3.\)	\(\frac{3(a-2b)(a-2b)}{3\cdot 2(a+2b)(a-2b)}\)
	\(\frac{a-2b}{2(a+2b)}\)

Simplify: \(\frac{2{x}^{2}-12xy+18{y}^{2}}{3{x}^{2}-27{y}^{2}}\).

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\(\frac{2(x-3y)}{3(x+3y)}\)
Simplify: \(\frac{5{x}^{2}-30xy+25{y}^{2}}{2{x}^{2}-50{y}^{2}}\).

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\(\frac{5(x-y)}{2(x+5y)}\)
Simplify: \(\frac{{x}^{2}-4x-32}{64-{x}^{2}}.\)

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Factor the numerator and the denominator.
Recognize the factors that are opposites.
Simplify.
Simplify: \(\frac{{x}^{2}-4x-5}{25-{x}^{2}}.\)

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\(-\frac{x+1}{x+5}\)
Simplify: \(\frac{{x}^{2}+x-2}{1-{x}^{2}}.\)

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\(-\frac{x+2}{x+1}\)
Simplify: \(\frac{2x}{{x}^{2}-7x+12}\cdot \frac{{x}^{2}-9}{6{x}^{2}}.\)
Simplify: \(\frac{5x}{{x}^{2}+5x+6}\cdot \frac{{x}^{2}-4}{10x}.\)

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\(\frac{x-2}{2(x+3)}\)
Simplify: \(\frac{9{x}^{2}}{{x}^{2}+11x+30}\cdot \frac{{x}^{2}-36}{3{x}^{2}}.\)

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\(\frac{3(x-6)}{x+5}\)

Multiply: \(\frac{3{a}^{2}-8a-3}{{a}^{2}-25}\cdot \frac{{a}^{2}+10a+25}{3{a}^{2}-14a-5}.\)

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	\(\ \frac{3{a}^{2}-8a-3}{{a}^{2}-25}\cdot \frac{{a}^{2}+10a+25}{3{a}^{2}-14a-5}\)
Factor the numerators and denominators and then multiply.	\(\ \frac{(3a+1)(a-3)(a+5)(a+5)}{(a-5)(a+5)(3a+1)(a-5)}\)
Simplify by dividing out common factors.	\(\ \frac{(3a+1)(a-3)(a+5)(a+5)}{(a-5)(a+5)(3a+1)(a-5)}\)
Simplify.	\(\ \frac{(a-3)(a+5)}{(a-5)(a-5)}\)
Rewrite \((a-5)(a-5)\) using an exponent.	\(\ \frac{(a-3)(a+5)}{{(a-5)}^{2}}\)

Simplify: \(\frac{2{x}^{2}+5x-12}{{x}^{2}-16}\cdot \frac{{x}^{2}-8x+16}{2{x}^{2}-13x+15}.\)

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\(\frac{x-4}{x-5}\)
Simplify: \(\frac{4{b}^{2}+7b-2}{1-{b}^{2}}\cdot \frac{{b}^{2}-2b+1}{4{b}^{2}+15b-4}.\)

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\(-\frac{(b+2)(b-1)}{(1+b)(b+4)}\)
Divide: \(\frac{{p}^{3}+{q}^{3}}{2{p}^{2}+2pq+2{q}^{2}}\div \frac{{p}^{2}-{q}^{2}}{6}.\)
Simplify: \(\frac{{x}^{3}+8}{3{x}^{2}-6x+12}\div \frac{{x}^{2}-4}{6}.\)

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\(\frac{2}{x-2}\)
Simplify: \(\frac{2{z}^{2}}{{z}^{2}-1}\div \frac{{z}^{3}-{z}^{2}+z}{{z}^{3}+1}.\)

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\(\frac{2z}{z-1}\)

Divide: \(\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}}.\)

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	\(\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}}\)
Rewrite with a division sign.	\(\frac{6{x}^{2}-7x+2}{4x-8}\div \frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}\)
Rewrite as product of first times reciprocal of second.	\(\frac{6{x}^{2}-7x+2}{4x-8}\cdot \frac{{x}^{2}-5x+6}{2{x}^{2}-7x+3}\)
Factor the numerators and the denominators, and then multiply.	\(\frac{(2x-1)(3x-2)(x-2)(x-3)}{4(x-2)(2x-1)(x-3)}\)
Simplify by dividing out common factors.	\(\frac{(2x-1)(3x-2)(x-2)(x-3)}{4(x-2)(2x-1)(x-3)}\)
Simplify.	\(\frac{3x-2}{4}\)

Simplify: \(\frac{\frac{3{x}^{2}+7x+2}{4x+24}}{\frac{3{x}^{2}-14x-5}{{x}^{2}+x-30}}.\)

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\(\frac{x+2}{4}\)
Simplify: \(\frac{\frac{{y}^{2}-36}{2{y}^{2}+11y-6}}{\frac{2{y}^{2}-2y-60}{8y-4}}.\)

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\(\frac{2}{y+5}\)

Perform the indicated operations: \(\frac{3x-6}{4x-4}\cdot \frac{{x}^{2}+2x-3}{{x}^{2}-3x-10}\div \frac{2x+12}{8x+16}.\)

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Rewrite the division as multiplication by the reciprocal.
Factor the numerators and the denominators.
Multiply the fractions. Bringing the constants to the front will help when removing common factors.
Simplify by dividing out common factors.
Simplify.

Perform the indicated operations: \(\frac{4m+4}{3m-15}\cdot \frac{{m}^{2}-3m-10}{{m}^{2}-4m-32}\div \frac{12m-36}{6m-48}.\)

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\(\frac{2(m+1)(m+2)}{3(m+4)(m-3)}\)
Perform the indicated operations: \(\frac{2{n}^{2}+10n}{n-1}\div \frac{{n}^{2}+10n+24}{{n}^{2}+8n-9}\cdot \frac{n+4}{8{n}^{2}+12n}.\)

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\(\frac{(n+5)(n+9)}{2(n+6)(2n+3)}\)

Find the domain of \(R(x)=\frac{2{x}^{2}-14x}{4{x}^{2}-16x-48}.\)

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The domain will be all real numbers except those values that make the denominator zero. We will set the denominator equal to zero , solve that equation, and then exclude those values from the domain.

Set the denominator to zero.	\(4{x}^{2}-16x-48=0\\)
Factor, first factor out the GCF.	\(4({x}^{2}-4x-12)=0\\)
	\(4(x-6)(x+2)=0\\)
Use the Zero Product Property.	\(4\ne 0\ x-6=0\ x+2=0\\)
Solve.	\(x=6\ x=-2\)
	The domain of \(R(x)\) is all real numbers where \(x\ne 6\) and \(x\ne \text{-}2.\)

Find the domain of \(R(x)=\frac{2{x}^{2}-10x}{4{x}^{2}-16x-20}.\)

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The domain of \(R(x)\) is all real numbers where \(x\ne 5\) and \(x\ne \text{-}1.\)
Find the domain of \(R(x)=\frac{4{x}^{2}-16x}{8{x}^{2}-16x-64}.\)

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The domain of \(R(x)\) is all real numbers where \(x\ne 4\) and \(x\ne \text{-}2.\)

Find \(R(x)=f(x)\cdot g(x)\) where \(f(x)=\frac{2x-6}{{x}^{2}-8x+15}\) and \(g(x)=\frac{{x}^{2}-25}{2x+10}.\)

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	\(\ R(x)=f(x)\cdot g(x)\)
	\(\ R(x)=\frac{2x-6}{{x}^{2}-8x+15}\cdot \frac{{x}^{2}-25}{2x+10}\)
Factor each numerator and denominator.	\(\ R(x)=\frac{2(x-3)}{(x-3)(x-5)}\cdot \frac{(x-5)(x+5)}{2(x+5)}\)
Multiply the numerators and denominators.	\(\ R(x)=\frac{2(x-3)(x-5)(x+5)}{2(x-3)(x-5)(x+5)}\)
Remove common factors.	\(\ R(x)=\frac{2(x-3)(x-5)(x+5)}{2(x-3)(x-5)(x+5)}\)
Simplify.	\(\ R(x)=1\)

Find \(R(x)=f(x)\cdot g(x)\) where \(f(x)=\frac{3x-21}{{x}^{2}-9x+14}\) and \(g(x)=\frac{2{x}^{2}-8}{3x+6}.\)

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\(R(x)=2\)
Find \(R(x)=f(x)\cdot g(x)\) where \(f(x)=\frac{{x}^{2}-x}{3{x}^{2}+27x-30}\) and \(g(x)=\frac{{x}^{2}-100}{{x}^{2}-10x}.\)

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\(R(x)=\frac{1}{3}\)

Find \(R(x)=\frac{f(x)}{g(x)}\) where \(f(x)=\frac{3{x}^{2}}{{x}^{2}-4x}\) and \(g(x)=\frac{9{x}^{2}-45x}{{x}^{2}-7x+10}.\)

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	\(\ R(x)=\frac{f(x)}{g(x)}\)
Substitute in the functions \(f(x),\ g(x).\)	\(\ R(x)=\frac{\frac{3{x}^{2}}{{x}^{2}-4x}}{\frac{9{x}^{2}-45x}{{x}^{2}-7x+10}}\)
Rewrite the division as the product of \(f(x)\) and the reciprocal of \(g(x).\)	\(\ R(x)=\frac{3{x}^{2}}{{x}^{2}-4x}\cdot \frac{{x}^{2}-7x+10}{9{x}^{2}-45x}\)
Factor the numerators and denominators and then multiply.	\(\ R(x)=\frac{3\cdot x\cdot x\cdot (x-5)(x-2)}{x(x-4)\cdot 3\cdot 3\cdot x\cdot (x-5)}\)
Simplify by dividing out common factors.	\(\ R(x)=\frac{3\cdot x\cdot x(x-5)(x-2)}{x(x-4)\cdot 3\cdot 3\cdot x(x-5)}\)
	\(\ R(x)=\frac{x-2}{3(x-4)}\)

Find \(R(x)=\frac{f(x)}{g(x)}\) where \(f(x)=\frac{2{x}^{2}}{{x}^{2}-8x}\) and \(g(x)=\frac{8{x}^{2}+24x}{{x}^{2}+x-6}.\)

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\(R(x)=\frac{x-2}{4(x-8)}\)
Find \(R(x)=\frac{f(x)}{g(x)}\) where \(f(x)=\frac{15{x}^{2}}{3{x}^{2}+33x}\) and \(g(x)=\frac{5x-5}{{x}^{2}+9x-22}.\)

Cavabı göstər

\(R(x)=\frac{x(x-2)}{x-1}\)
ⓐ \(\frac{2{x}^{2}}{z}\), ⓑ \(\frac{4p-1}{6p-5}\), ⓒ \(\frac{n-3}{{n}^{2}+2n-8}\)

Cavabı göstər

ⓐ \(z=0\) ⓑ \(p=\frac{5}{6}\)
ⓒ \(n=-4,n=2\)

Symbols used here

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Multiply and Divide Rational Expressions

Determine the values for which a rational expression is undefined
Simplify rational expressions
Multiply rational expressions
Divide rational expressions
Multiply and divide rational functions
Set the denominator equal to zero.
Solve the equation.
Factor the numerator and denominator completely.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Özün sına

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Daha çox Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value


Factor the numerator and the denominator.
Recognize the factors that are opposites.
Simplify.