maths.free › Algebra › 8. Rational Expressions and Equations › Simplify Rational Expressions

Simplify Rational Expressions

Determine the values for which a rational expression is undefined

Determine the Values for Which a Rational Expression is Undefined

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.

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

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 values for which the rational expression is undefined:

ⓐ \(\frac{9y}{x}\) ⓑ \(\frac{4b-3}{2b+5}\) ⓒ \(\frac{x+4}{{x}^{2}+5x+6}\)

Solution

The expression will be undefined when the denominator is zero.

ⓐ

	\(\frac{9y}{x}\)
Set the denominator equal to zero. Solve for the variable.	\(x=0\)
	\(\frac{9y}{x}\ \text{is undefined for}\ x=0.\)

ⓑ

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

ⓒ

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

Saying that the rational expression \(\frac{x+4}{{x}^{2}+5x+6}\) is undefined for \(x=-2\ \text{or}\ x=-3\) is similar to writing the phrase “void where prohibited” in contest rules.

Evaluate Rational Expressions

To evaluate a rational expression, we substitute values of the variables into the expression and simplify, just as we have for many other expressions in this book.

Example

Try it.

Evaluate \(\frac{2x+3}{3x-5}\) for each value:

ⓐ \(x=0\) ⓑ \(x=2\) ⓒ \(x=-3\)

Solution

ⓐ



Simplify.

ⓑ



Simplify.

ⓒ



Simplify.

Example

Try it.

Evaluate \(\frac{{x}^{2}+8x+7}{{x}^{2}-4}\) for each value:

ⓐ \(x=0\) ⓑ \(x=2\) ⓒ \(x=-1\)

Solution

ⓐ



Simplify.

ⓑ



Simplify.

This rational expression is undefined for x = 2.

ⓒ



Simplify.

Remember that a fraction is simplified when it has no common factors, other than 1, in its numerator and denominator. When we evaluate a rational expression, we make sure to simplify the resulting fraction.

Example

Try it.

Evaluate \(\frac{{a}^{2}+2ab+{b}^{2}}{3a{b}^{2}}\) for each value:

ⓐ \(a=1,b=2\) ⓑ \(a=-2,b=-1\) ⓒ \(a=\frac{1}{3},b=0\)

Solution

ⓐ

	\(\frac{{a}^{2}+2ab+{b}^{2}}{3a{b}^{2}}\) when \(a=1,b=2\).

Simplify.

ⓑ

	\(\frac{{a}^{2}+2ab+{b}^{2}}{3a{b}^{2}}\) when \(a=-2,b=-1\).

Simplify.

ⓒ

	\(\frac{{a}^{2}+2ab+{b}^{2}}{3a{b}^{2}}\) when \(a=\frac{1}{3},b=0\).

Simplify.

	The expression is undefined.

Simplify Rational Expressions

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

For example:

\(\frac{2}{3}\) is simplified because there are no common factors of 2 and 3.
\(\frac{2x}{3x}\) is not simplified because x is a common factor of 2x and 3x.

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. Every time we write a rational expression, we should make a similar statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.

Let’s start by reviewing how we simplify numerical fractions.

Example

Try it.

Simplify: \(-\frac{36}{63}.\)

Solution


Rewrite the numerator and denominator showing the common factors.
Simplify using the Equivalent Fractions Property.

Notice that the fraction \(-\frac{4}{7}\) is simplified because there are no more common factors.

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, \(x\ne 0\) and \(y\ne 0\).

Example

Try it.

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

Solution


Rewrite the numerator and denominator showing the common factors.
Simplify using the Equivalent Fractions Property.

Did you notice that these are the same steps we took when we divided monomials in Polynomials?

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.

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

Simplify Rational Expressions with Opposite Factors

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. Let’s start with a numerical fraction, say \(\frac{7}{-7}\). We know this fraction simplifies to \(-1\). We also recognize that the numerator and denominator are opposites.

In Foundations, we introduced opposite notation: the opposite of \(a\) is \(\text{-}a\). We remember, too, that \(\text{-}a=-1\cdot a\).

We simplify the fraction \(\frac{a}{\text{-}a}\), whose numerator and denominator are opposites, in this way:

\(\begin{array}{llll} & & & \ \frac{a}{\text{-}a} \\ \text{We could rewrite this.} & & & \ \frac{1\cdot a}{-1\cdot a} \\ \text{Remove the common factors.} & & & \ \frac{1}{-1} \\ \text{Simplify.} & & & \ -1\end{array}\)

So, in the same way, we can simplify the fraction \(\frac{x-3}{\text{-}(x-3)}\):

\(\begin{array}{llll}\text{We could rewrite this.} & & & \ \frac{1\cdot (x-3)}{-1\cdot (x-3)} \\ \text{Remove the common factors.} & & & \ \frac{1}{-1} \\ \text{Simplify.} & & & \ -1\end{array}\)

But the opposite of \(x-3\) could be written differently:

Example

Try it.

Simplify: \(\frac{x-8}{8-x}.\)

Solution

\(\begin{array}{llll} & & & \frac{x-8}{8-x} \\ \text{Recognize that}\ x-8\ \text{and}\ 8-x\ \text{are opposites.} & & & -1\end{array}\)

Example

Try it.

Simplify: \(\frac{14-2x}{{x}^{2}-49}.\)

Solution


Factor the numerator and denominator.
Recognize that \(7-x\ \text{and}\ x-7\ \text{are opposites}\).
Simplify.

Example

Try it.

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

Solution


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

Condensed — the full section is in OpenStax Elementary 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, if possible.
Simplified Rational Expression
- A rational expression is considered simplified if there are no common factors in its numerator and denominator.
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 \(b-a\).
  \(\begin{array}{llllll}\frac{a-b}{b-a}=-1 & & & & & a\ne 0,b\ne 0,\text{a}\ne \text{b}\end{array}\)

Simplify Rational Expressions

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

Try it.

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

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.

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

Evaluate Rational Expressions

In the following exercises, evaluate the rational expression for the given values.

Try it.

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

ⓐ \(x=0\) ⓑ \(x=2\) ⓒ \(x=-1\)

Solution

ⓐ \(0\) ⓑ \(4\) ⓒ \(1\)

Try it.

\(\frac{4y-1}{5y-3}\)

ⓐ \(y=0\) ⓑ \(y=2\) ⓒ \(y=-1\)

Try it.

\(\frac{2p+3}{{p}^{2}+1}\)

ⓐ \(p=0\) ⓑ \(p=1\) ⓒ \(p=-2\)

Solution

ⓐ \(3\) ⓑ \(\frac{5}{2}\) ⓒ \(-\frac{1}{5}\)

Try it.

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

ⓐ \(x=0\) ⓑ \(x=1\) ⓒ \(x=-2\)

Try it.

\(\frac{{y}^{2}+5y+6}{{y}^{2}-1}\)

ⓐ \(y=0\) ⓑ \(y=2\) ⓒ \(y=-2\)

Solution

ⓐ \(-6\) ⓑ \(\frac{20}{3}\) ⓒ \(0\)

Try it.

\(\frac{{z}^{2}+3z-10}{{z}^{2}-1}\)

ⓐ \(z=0\) ⓑ \(z=2\) ⓒ \(z=-2\)

Try it.

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

ⓐ \(a=0\) ⓑ \(a=1\) ⓒ \(a=-2\)

Solution

ⓐ \(-1\) ⓑ \(-\frac{3}{10}\) ⓒ \(0\)

Try it.

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

ⓐ \(b=0\) ⓑ \(b=2\) ⓒ \(b=-2\)

Try it.

\(\frac{{x}^{2}+3xy+2{y}^{2}}{2{x}^{3}y}\)

ⓐ \(x=1,y=-1\)
ⓑ \(x=2,y=1\)
ⓒ \(x=-1,y=-2\)

Solution

ⓐ \(0\) ⓑ \(\frac{3}{4}\) ⓒ \(\frac{15}{4}\)

Try it.

\(\frac{{c}^{2}+cd-2{d}^{2}}{c{d}^{3}}\)

ⓐ \(c=2,d=-1\)
ⓑ \(c=1,d=-1\)
ⓒ \(c=-1,d=2\)

Try it.

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

ⓐ \(m=2,n=1\)
ⓑ \(m=-1,n=-1\)
ⓒ \(m=3,n=2\)

Solution

ⓐ \(0\) ⓑ \(-\frac{3}{5}\) ⓒ \(-\frac{7}{120}\)

Try it.

\(\frac{2{s}^{2}t}{{s}^{2}-9{t}^{2}}\)

ⓐ \(s=4,t=1\)
ⓑ \(s=-1,t=-1\)
ⓒ \(s=0,t=2\)

Simplify Rational Expressions

In the following exercises, simplify.

Try it.

\(-\frac{4}{52}\)

Solution

\(-\frac{1}{13}\)

Try it.

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

Try it.

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

Solution

\(\frac{8}{9}\)

Try it.

\(\frac{65}{104}\)

Try it.

\(\frac{6a{b}^{2}}{12{a}^{2}b}\)

Solution

\(\frac{b}{2a}\)

Try it.

\(\frac{15xy}{3{x}^{3}{y}^{3}}\)

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.

\(\frac{3a+6}{4a+8}\)

Solution

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

Try it.

\(\frac{5b+5}{6b+6}\)

Try it.

\(\frac{3c-9}{5c-15}\)

Solution

\(\frac{3}{5}\)

Try it.

\(\frac{4d+8}{9d+18}\)

Try it.

\(\frac{7m+63}{5m+45}\)

Solution

\(\frac{7}{5}\)

Try it.

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

Try it.

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

Solution

\(\frac{12}{5}\)

Try it.

\(\frac{6q+210}{5q+175}\)

Try it.

\(\frac{{a}^{2}-a-12}{{a}^{2}-8a+16}\)

Solution

\(\frac{a+3}{a-4}\)

Try it.

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

Try it.

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

Solution

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

Try it.

\(\frac{{v}^{2}+8v+15}{{v}^{2}-v-12}\)

Try it.

\(\frac{{x}^{2}-25}{{x}^{2}+2x-15}\)

Solution

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

Try it.

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

Try it.

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

Solution

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

Try it.

\(\frac{{b}^{2}+9b+18}{{b}^{2}-36}\)

Try it.

\(\frac{{y}^{3}+{y}^{2}+y+1}{{y}^{2}+2y+1}\)

Solution

\(\frac{{y}^{2}+1}{y+1}\)

Try it.

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

Try it.

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

Solution

\(x-2\)

Try it.

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

Try it.

\(\frac{3{a}^{2}+15a}{6{a}^{2}+6a-36}\)

Solution

\(\frac{a(a+5)}{2(a+3)(a-2)}\)

Try it.

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

Try it.

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

Solution

\(\frac{c}{2(c-5)}\)

Try it.

\(\frac{4{d}^{2}-24d}{2{d}^{2}-4d-48}\)

Try it.

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

Solution

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

Try it.

\(\frac{5{n}^{2}+30n+45}{2{n}^{2}-18}\)

Try it.

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

Solution

\(\frac{5(r-1)}{r-7}\)

Try it.

\(\frac{3{s}^{2}+30s+72}{3{s}^{2}-48}\)

Try it.

\(\frac{{t}^{3}-27}{{t}^{2}-9}\)

Solution

\(\frac{{t}^{2}+3t+9}{t+3}\)

Try it.

\(\frac{{v}^{3}-1}{{v}^{2}-1}\)

Try it.

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

Solution

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

Try it.

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

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify each rational expression.

Try it.

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

Solution

\(-1\)

Try it.

\(\frac{b-12}{12-b}\)

Try it.

\(\frac{11-c}{c-11}\)

Solution

\(-1\)

Try it.

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

Try it.

\(\frac{12-2x}{{x}^{2}-36}\)

Solution

\(-\frac{2}{x+6}\)

Try it.

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

Try it.

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

Solution

\(-\frac{4}{8+v}\)

Try it.

\(\frac{7w-21}{9-{w}^{2}}\)

Try it.

\(\frac{{y}^{2}-11y+24}{9-{y}^{2}}\)

Solution

\(-\frac{(y-8)}{3+y}\)

Try it.

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

Try it.

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

Solution

\(-\frac{a+4}{9+a}\)

Try it.

\(\frac{{b}^{2}+b-42}{36-{b}^{2}}\)

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 .

Otkrij odgovor

\(\frac{6}{y}\)
Factor: \(6{x}^{2}-7x+2.\)
If you missed this problem, review .

Otkrij odgovor

\(\left(2x-1\right)\left(3x-2\right)\)
Factor: \({n}^{3}+8.\)
If you missed this problem, review .

Otkrij odgovor

\(\left(n+2\right)\left({n}^{2}-2n+4\right)\)

Determine the values for which the rational expression is undefined:

Otkrij odgovor

The expression will be undefined when the denominator is zero.

ⓐ

	\(\frac{9y}{x}\)
Set the denominator equal to zero. Solve for the variable.	\(x=0\)
	\(\frac{9y}{x}\ \text{is undefined for}\ x=0.\)

ⓑ

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

ⓒ

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

Saying that the rational expression \(\frac{x+4}{{x}^{2}+5x+6}\) is undefined for \(x=-2\ \text{or}\ x=-3\) is similar to writing the phrase “void where prohibited” in contest rules.

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

Otkrij odgovor

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

Otkrij odgovor

ⓐ \(q=0\) ⓑ \(y=-\frac{2}{3}\) ⓒ \(m=2,m=-3\)
Evaluate \(\frac{2x+3}{3x-5}\) for each value:
ⓐ \(x=0\) ⓑ \(x=2\) ⓒ \(x=-3\)

Otkrij odgovor

ⓐ

Simplify.
ⓑ

Simplify.

ⓒ

Simplify.
Evaluate \(\frac{y+1}{2y-3}\) for each value:
ⓐ \(y=1\) ⓑ \(y=-3\) ⓒ \(y=0\)

Otkrij odgovor

ⓐ \(-2\) ⓑ \(\frac{2}{9}\) ⓒ \(-\frac{1}{3}\)
Evaluate \(\frac{5x-1}{2x+1}\) for each value:
ⓐ \(x=1\) ⓑ \(x=-1\) ⓒ \(x=0\)

Otkrij odgovor

ⓐ \(\frac{4}{3}\) ⓑ \(6\) ⓒ \(-1\)
Evaluate \(\frac{{x}^{2}+8x+7}{{x}^{2}-4}\) for each value:
ⓐ \(x=0\) ⓑ \(x=2\) ⓒ \(x=-1\)

Otkrij odgovor

ⓐ

Simplify.

ⓑ

Simplify.

This rational expression is undefined for x = 2.
ⓒ

Simplify.
Evaluate \(\frac{{x}^{2}+1}{{x}^{2}-3x+2}\) for each value:
ⓐ \(x=0\) ⓑ \(x=-1\) ⓒ \(x=3\)

Otkrij odgovor

ⓐ \(\frac{1}{2}\) ⓑ \(\frac{1}{3}\) ⓒ 5
Evaluate \(\frac{{x}^{2}+x-6}{{x}^{2}-9}\) for each value:
ⓐ \(x=0\) ⓑ \(x=-2\) ⓒ \(x=1\)

Otkrij odgovor

ⓐ \(\frac{2}{3}\) ⓑ \(\frac{4}{5}\) ⓒ \(\frac{1}{2}\)
Evaluate \(\frac{{a}^{2}+2ab+{b}^{2}}{3a{b}^{2}}\) for each value:
ⓐ \(a=1,b=2\) ⓑ \(a=-2,b=-1\) ⓒ \(a=\frac{1}{3},b=0\)

Otkrij odgovor

ⓐ

\(\frac{{a}^{2}+2ab+{b}^{2}}{3a{b}^{2}}\) when \(a=1,b=2\).
Simplify.

ⓑ

\(\frac{{a}^{2}+2ab+{b}^{2}}{3a{b}^{2}}\) when \(a=-2,b=-1\).
Simplify.

ⓒ

\(\frac{{a}^{2}+2ab+{b}^{2}}{3a{b}^{2}}\) when \(a=\frac{1}{3},b=0\).
Simplify.
The expression is undefined.
Evaluate \(\frac{2{a}^{3}b}{{a}^{2}+2ab+{b}^{2}}\) for each value:
ⓐ \(a=-1,b=2\) ⓑ \(a=0,b=-1\) ⓒ \(a=1,b=\frac{1}{2}\)

Otkrij odgovor

ⓐ \(-4\) ⓑ \(0\) ⓒ \(\frac{4}{9}\)
Evaluate \(\frac{{a}^{2}-{b}^{2}}{8a{b}^{3}}\) for each value:
ⓐ \(a=1,b=-1\) ⓑ \(a=\frac{1}{2},b=-1\) ⓒ \(a=-2,b=1\)

Otkrij odgovor

ⓐ \(0\) ⓑ \(\frac{3}{16}\) ⓒ \(-\frac{3}{16}\)
Simplify: \(-\frac{36}{63}.\)

Otkrij odgovor

Rewrite the numerator and denominator showing the common factors.
Simplify using the Equivalent Fractions Property.
Notice that the fraction \(-\frac{4}{7}\) is simplified because there are no more common factors.
Simplify: \(-\frac{45}{81}.\)

Otkrij odgovor

\(-\frac{5}{9}.\)
Simplify: \(-\frac{42}{54}.\)

Otkrij odgovor

\(-\frac{7}{9}\)
Simplify: \(\frac{3xy}{18{x}^{2}{y}^{2}}.\)

Otkrij odgovor

Rewrite the numerator and denominator showing the common factors.
Simplify using the Equivalent Fractions Property.
Did you notice that these are the same steps we took when we divided monomials in Polynomials?
Simplify: \(\frac{4{x}^{2}y}{12x{y}^{2}}.\)

Otkrij odgovor

\(\frac{x}{3y}\)
Simplify: \(\frac{16{x}^{2}y}{2x{y}^{2}}.\)

Otkrij odgovor

\(\frac{8x}{y}\)
Simplify: \(\frac{2x+8}{5x+20}.\)
Simplify: \(\frac{3x-6}{2x-4}.\)

Otkrij odgovor

\(\frac{3}{2}\)
Simplify: \(\frac{7y+35}{5y+25}.\)

Otkrij odgovor

\(\frac{7}{5}\)
Simplify: \(\frac{{x}^{2}+5x+6}{{x}^{2}+8x+12}.\)

Otkrij odgovor

\(\begin{array}{llll} & & & \ \frac{{x}^{2}+5x+6}{{x}^{2}+8x+12} \\ \text{Factor the numerator and denominator.} & & & \ \frac{(x+2)(x+3)}{(x+2)(x+6)} \\ \\ \\ \begin{array}{l}\text{Remove the common factor}\ x+2\ \text{from} \\ \text{the numerator and the denominator.}\end{array} & & & \ \frac{(x+2)(x+3)}{(x+2)(x+6)} \\ & & & \ \frac{x+3}{x+6}\end{array}\)
Can you tell which values of x must be excluded in this example?
Simplify: \(\frac{{x}^{2}-x-2}{{x}^{2}-3x+2}.\)

Otkrij odgovor

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

Otkrij odgovor

\(\frac{x-5}{x-1}\)
Simplify: \(\frac{{y}^{2}+y-42}{{y}^{2}-36}.\)

Otkrij odgovor

\(\begin{array}{llll} & & & \ \frac{{y}^{2}+y-42}{{y}^{2}-36} \\ \text{Factor the numerator and denominator.} & & & \ \frac{(y+7)(y-6)}{(y+6)(y-6)} \\ \\ \\ \begin{array}{l}\text{Remove the common factor}\ y-6\ \text{from} \\ \text{the numerator and the denominator.}\end{array} & & & \ \frac{(y+7)(y-6)}{(y+6)(y-6)} \\ & & & \ \frac{y+7}{y+6}\end{array}\)
Simplify: \(\frac{{x}^{2}+x-6}{{x}^{2}-4}.\)

Otkrij odgovor

\(\frac{x+3}{x+2}\)
Simplify: \(\frac{{x}^{2}+8x+7}{{x}^{2}-49}.\)

Otkrij odgovor

\(\frac{x+1}{x-7}\)
Simplify: \(\frac{{p}^{3}-2{p}^{2}+2p-4}{{p}^{2}-7p+10}.\)

Otkrij odgovor

\(\begin{array}{llll} & & & \ \frac{{p}^{3}-2{p}^{2}+2p-4}{{p}^{2}-7p+10} \\ \\ \\ \begin{array}{l}\text{Factor the numerator and denominator,} \\ \text{using grouping to factor the numerator.}\end{array} & & & \ \frac{{p}^{2}(p-2)+2(p-2)}{(p-5)(p-2)} \\ & & & \ \frac{({p}^{2}+2)(p-2)}{(p-5)(p-2)} \\ \\ \\ \begin{array}{l}\text{Remove the common factor of}\ p-2 \\ \text{from the numerator and the denominator.}\end{array} & & & \ \frac{({p}^{2}+2)(p-2)}{(p-5)(p-2)} \\ & & & \ \frac{{p}^{2}+2}{p-5}\end{array}\)
Simplify: \(\frac{{y}^{3}-3{y}^{2}+y-3}{{y}^{2}-y-6}.\)

Otkrij odgovor

\(\frac{{y}^{2}+1}{y+2}\)
Simplify: \(\frac{{p}^{3}-{p}^{2}+2p-2}{{p}^{2}+4p-5}.\)

Otkrij odgovor

\(\frac{{p}^{2}+2}{p+5}\)
Simplify: \(\frac{2{n}^{2}-14n}{4{n}^{2}-16n-48}.\)

Otkrij odgovor

\(\begin{array}{llll} & & & \ \frac{2{n}^{2}-14n}{4{n}^{2}-16n-48} \\ \\ \\ \begin{array}{l}\text{Factor the numerator and denominator,} \\ \text{first factoring out the GCF.}\end{array} & & & \ \frac{2n(n-7)}{4({n}^{2}-4n-12)} \\ & & & \ \frac{2n(n-7)}{4(n-6)(n+2)} \\ \\ \\ \text{Remove the common factor, 2.} & & & \ \frac{2n(n-7)}{2\cdot 2(n-6)(n+2)} \\ & & & \ \frac{n(n-7)}{2(n-6)(n+2)}\end{array}\)
Simplify: \(\frac{2{n}^{2}-10n}{4{n}^{2}-16n-20}.\)

Otkrij odgovor

\(\frac{n}{2(n+1)}\)
Simplify: \(\frac{4{x}^{2}-16x}{8{x}^{2}-16x-64}.\)

Otkrij odgovor

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

Otkrij odgovor

\(\begin{array}{llll} & & & \ \frac{3{b}^{2}-12b+12}{6{b}^{2}-24} \\ \\ \\ \begin{array}{l}\text{Factor the numerator and denominator,} \\ \text{first factoring out the GCF.}\end{array} & & & \ \frac{3({b}^{2}-4b+4)}{6({b}^{2}-4)} \\ & & & \ \frac{3(b-2)(b-2)}{6(b+2)(b-2)} \\ \\ \\ \text{Remove the common factors of}\ b-2\ \text{and}\ 3. & & & \ \frac{3(b-2)(b-2)}{3\cdot 2(b+2)(b-2)} \\ & & & \ \frac{b-2}{2(b+2)}\end{array}\)
Simplify: \(\frac{2{x}^{2}-12x+18}{3{x}^{2}-27}.\)

Otkrij odgovor

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

Otkrij odgovor

\(\frac{5(y-1)}{2(y+5)}\)
Simplify: \(\frac{{m}^{3}+8}{{m}^{2}-4}.\)

Otkrij odgovor

\(\begin{array}{llll} & & & \ \frac{{m}^{3}+8}{{m}^{2}-4} \\ \begin{array}{l}\text{Factor the numerator and denominator,} \\ \text{using the formulas for sum of cubes and} \\ \text{difference of squares.}\end{array} & & & \ \frac{(m+2)({m}^{2}-2m+4)}{(m+2)(m-2)} \\ \text{Remove the common factor of}\ m+2. & & & \ \frac{(m+2)({m}^{2}-2m+4)}{(m+2)(m-2)} \\ & & & \ \frac{{m}^{2}-2m+4}{m-2}\end{array}\)

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: Simplify Rational Expressions

Determine the values for which a rational expression is undefined
Evaluate rational expressions
Simplify rational expressions
Simplify rational expressions with opposite factors
Set the denominator equal to zero.
Solve the equation in the set of reals, if possible.
Factor the numerator and denominator completely.
Simplify by dividing out common factors.

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.

Pokušaj i ti

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

Više u Algebra

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