Dividing Polynomials

Dividing Monomials

We are now familiar with all the properties of exponents and used them to multiply polynomials. Next, we’ll use these properties to divide monomials and polynomials.

Example

Try it.

Find the quotient: \(54{a}^{2}{b}^{3}\div (-6a{b}^{5}).\)

Solution

When we divide monomials with more than one variable, we write one fraction for each variable.

	\(54{a}^{2}{b}^{3}\div (-6a{b}^{5})\)
Rewrite as a fraction.	\(\frac{54{a}^{2}{b}^{3}}{-6a{b}^{5}}\)
Use fraction multiplication.	\(\frac{54}{-6}\cdot \frac{{a}^{2}}{a}\cdot \frac{{b}^{3}}{{b}^{5}}\)
Simplify and use the Quotient Property.	\(-9\cdot a\cdot \frac{1}{{b}^{2}}\)
Multiply.	\(-\frac{9a}{{b}^{2}}\)

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

Example

Try it.

Find the quotient: \(\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}.\)

Solution

Be very careful to simplify \(\frac{14}{21}\) by dividing out a common factor, and to simplify the variables by subtracting their exponents.

	\(\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}\)
Simplify and use the Quotient Property.	\(\frac{2{y}^{6}}{3{x}^{4}}\)

Divide a Polynomial by a Monomial

Now that we know how to divide a monomial by a monomial, the next procedure is to divide a polynomial of two or more terms by a monomial.

The method we’ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we’ll start with an example to review fraction addition. The sum \(\frac{y}{5}+\frac{2}{5}\) simplifies to \(\frac{y+2}{5}.\)

Now we will do this in reverse to split a single fraction into separate fractions. For example, \(\frac{y+2}{5}\) can be written \(\frac{y}{5}+\frac{2}{5}.\)

This is the “reverse” of fraction addition and it states that if a, b, and c are numbers where \(c\ne 0,\) then \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}.\) We will use this to divide polynomials by monomials.

Example

Try it.

Find the quotient: \((18{x}^{3}y-36x{y}^{2})\div (-3xy).\)

Solution

	\((18{x}^{3}y-36x{y}^{2})\div (-3xy)\)
Rewrite as a fraction.	\(\frac{18{x}^{3}y-36x{y}^{2}}{-3xy}\)
Divide each term by the divisor. Be careful with the signs!	\(\frac{18{x}^{3}y}{-3xy}-\frac{36x{y}^{2}}{-3xy}\)
Simplify.	\(-6{x}^{2}+12y\)

Divide Polynomials Using Long Division

Divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So let’s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.

We check division by multiplying the quotient by the divisor.

If we did the division correctly, the product should equal the dividend.

\[\begin{array}{l}35\cdot 25 \\ 875✓\end{array}\]

Now we will divide a trinomial by a binomial. As you read through the example, notice how similar the steps are to the numerical example above.

Example

Try it.

Find the quotient: \(({x}^{2}+9x+20)\div (x+5).\)

Solution

Write it as a long division problem. Be sure the dividend is in standard form.
Divide \({x}^{2}\) by \(x.\) It may help to ask yourself, “What do I need to multiply \(x\) by to get \({x}^{2}\)?”
Put the answer, \(x,\) in the quotient over the \(x\) term. Multiply \(x\) times \(x+5.\) Line up the like terms under the dividend.
Subtract \({x}^{2}+5x\) from \({x}^{2}+9x.\) You may find it easier to change the signs and then add. Then bring down the last term, 20.
Divide \(4x\) by \(x.\) It may help to ask yourself, “What do I need to multiply \(x\) by to get \(4x\)?” Put the answer, \(4\), in the quotient over the constant term.
Multiply 4 times \(x+5.\)
Subtract \(4x+20\) from \(4x+20.\)
Check: Multiply the quotient by the divisor. \(\ (x+4)(x+5)\) You should get the dividend. \(\ {x}^{2}+9x+20✓\)

When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In the next example, we’ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.

Look back at the dividends in previous examples. The terms were written in descending order of degrees, and there were no missing degrees. The dividend in this example will be \({x}^{4}-{x}^{2}+5x-6.\) It is missing an \({x}^{3}\) term. We will add in \(0{x}^{3}\) as a placeholder.

In the next example, we will divide by \(2a+3.\) As we divide, we will have to consider the constants as well as the variables.

Example

Try it.

Find the quotient: \((8{a}^{3}+27)\div (2a+3).\)

Solution

This time we will show the division all in one step. We need to add two placeholders in order to divide.

To check, multiply \((2a+3)(4{a}^{2}-6a+9).\)

The result should be \(8{a}^{3}+27.\)

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

Divide Polynomials using Synthetic Division

As we have mentioned before, mathematicians like to find patterns to make their work easier. Since long division can be tedious, let’s look back at the long division we did in and look for some patterns. We will use this as a basis for what is called synthetic division. The same problem in the synthetic division format is shown next.

Synthetic division basically just removes unnecessary repeated variables and numbers. Here all the \(x\) and \({x}^{2}\) are removed. as well as the \(\text{-}{x}^{2}\) and \(-4x\) as they are opposite the term above.

The first row of the synthetic division is the coefficients of the dividend. The \(-5\) is the opposite of the 5 in the divisor.

The second row of the synthetic division are the numbers shown in red in the division problem.

The third row of the synthetic division are the numbers shown in blue in the division problem.

Notice the quotient and remainder are shown in the third row.

\[\text{Synthetic division only works when the divisor is of the form}\ x-c.\]

The following example will explain the process.

Example

Try it.

Use synthetic division to find the quotient and remainder when \(2{x}^{3}+3{x}^{2}+x+8\) is divided by \(x+2.\)

Solution

Write the dividend with decreasing powers of \(x.\)
Write the coefficients of the terms as the first row of the synthetic division.
Write the divisor as \(x-c\) and place c in the synthetic division in the divisor box.
Bring down the first coefficient to the third row.
Multiply that coefficient by the divisor and place the result in the second row under the second coefficient.
Add the second column, putting the result in the third row.
Multiply that result by the divisor and place the result in the second row under the third coefficient.
Add the third column, putting the result in the third row.
Multiply that result by the divisor and place the result in the third row under the third coefficient.
Add the final column, putting the result in the third row.
The quotient is \(2{x}^{2}-1x+3\) and the remainder is 2.

The division is complete. The numbers in the third row give us the result. The \(2\ -1\ 3\) are the coefficients of the quotient. The quotient is \(2{x}^{2}-1x+3.\) The 2 in the box in the third row is the remainder.

Check:

\(\ \begin{array}{lll} \\ \\ \text{(quotient)(divisor)}+\text{remainder} & = & \text{dividend} \\ (2{x}^{2}-1x+3)(x+2)+2 & \overset{?}{=} & 2{x}^{3}+3{x}^{2}+x+8 \\ 2{x}^{3}-{x}^{2}+3x+4{x}^{2}-2x+6+2 & \overset{?}{=} & 2{x}^{3}+3{x}^{2}+x+8 \\ 2{x}^{3}+3{x}^{2}+x+8 & = & 2{x}^{3}+3{x}^{2}+x+8✓\end{array}\)

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

Divide Polynomial Functions

Just as polynomials can be divided, polynomial functions can also be divided.

Example

Try it.

For functions \(f(x)={x}^{2}-5x-14\) and \(g(x)=x+2,\) find: ⓐ \((\frac{f}{g})(x)\) ⓑ \((\frac{f}{g})(-4).\)

Solution

ⓐ

Substitute for \(f(x)\) and \(g(x).\)	\(\ (\frac{f}{g})(x)=\frac{{x}^{2}-5x-14}{x+2}\)
Divide the polynomials.	\(\ (\frac{f}{g})(x)=x-7\)

ⓑ In part ⓐ we found \((\frac{f}{g})(x)\) and now are asked to find \((\frac{f}{g})(-4).\)

	\(\ (\frac{f}{g})(x)=x-7\)
To find \((\frac{f}{g})(-4),\) substitute \(x=-4.\)	\((\frac{f}{g})(-4)=-4-7\)
	\((\frac{f}{g})(-4)=-11\)

Use the Remainder and Factor Theorem

Let’s look at the division problems we have just worked that ended up with a remainder. They are summarized in the chart below. If we take the dividend from each division problem and use it to define a function, we get the functions shown in the chart. When the divisor is written as \(x-c,\) the value of the function at \(c,f(c),\) is the same as the remainder from the division problem.

Dividend	Divisor \(x-c\)	Remainder	Function	\(f(c)\)
\({x}^{4}-{x}^{2}+5x-6\)	\(x-(-2)\)	\(-4\)	\(f(x)={x}^{4}-{x}^{2}+5x-6\)	\(-4\)
\(3{x}^{3}-2{x}^{2}-10x+8\)	\(x-2\)	4	\(f(x)=3{x}^{3}-2{x}^{2}-10x+8\)	4
\({x}^{4}-16{x}^{2}+3x+15\)	\(x-(-4)\)	3	\(f(x)={x}^{4}-16{x}^{2}+3x+15\)	3

To see this more generally, we realize we can check a division problem by multiplying the quotient times the divisor and add the remainder. In function notation we could say, to get the dividend \(f(x),\) we multiply the quotient, \(q(x)\) times the divisor, \(x-c,\) and add the remainder, r.

If we evaluate this at \(c,\) we get:

This leads us to the Remainder Theorem.

Example

Try it.

Use the Remainder Theorem to find the remainder when \(f(x)={x}^{3}+3x+19\) is divided by \(x+2.\)

Solution

To use the Remainder Theorem, we must use the divisor in the \(x-c\) form. We can write the divisor \(x+2\) as \(x-(-2).\) So, our \(c\) is \(-2.\)

To find the remainder, we evaluate \(f(c)\) which is \(f(-2).\)

To evaluate \(f(-2),\) substitute \(x=-2.\)
Simplify.
	The remainder is 5 when \(f(x)={x}^{3}+3x+19\) is divided by \(x+2.\)
Check: Use synthetic division to check.
The remainder is 5.

When we divided \(8{a}^{3}+27\) by \(2a+3\) in the result was \(4{a}^{2}-6a+9.\) To check our work, we multiply \(4{a}^{2}-6a+9\) by \(2a+3\) to get \(8{a}^{3}+27\).

\[(4{a}^{2}-6a+9)(2a+3)=8{a}^{3}+27\]

Written this way, we can see that \(4{a}^{2}-6a+9\) and \(2a+3\) are factors of \(8{a}^{3}+27.\) When we did the division, the remainder was zero.

Whenever a divisor, \(x-c,\) divides a polynomial function, \(f(x),\) and resulting in a remainder of zero, we say \(x-c\) is a factor of \(f(x).\)

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

Key Concepts

• Division of a Polynomial by a Monomial
  • To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
• Division of Polynomial Functions
  • For functions \(f(x)\) and \(g(x),\) where \(g(x)\ne 0,\)
    \((\frac{f}{g})(x)=\frac{f(x)}{g(x)}\)
• Remainder Theorem
  • If the polynomial function \(f(x)\) is divided by \(x-c,\) then the remainder is \(f(c).\)
• Factor Theorem: For any polynomial function \(f(x),\)
  • if \(x-c\) is a factor of \(f(x),\) then \(f(c)=0\)
  • if \(f(c)=0,\) then \(x-c\) is a factor of \(f(x)\)

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1. Add: \(\frac{3}{d}+\frac{x}{d}.\)

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   Otkrij odgovor

   \(\frac{3+x}{d}\)

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

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   Otkrij odgovor

   \(6{y}^{2}\)

3. Combine like terms: \(8{a}^{2}+12a+1+3{a}^{2}-5a+4.\)

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   Otkrij odgovor

   \(11{a}^{2}+7a+5\)

4. Find the quotient: \(54{a}^{2}{b}^{3}\div (-6a{b}^{5}).\)

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   Otkrij odgovor

   When we divide monomials with more than one variable, we write one fraction for each variable.
   

   	\(54{a}^{2}{b}^{3}\div (-6a{b}^{5})\)
   Rewrite as a fraction.	\(\frac{54{a}^{2}{b}^{3}}{-6a{b}^{5}}\)
   Use fraction multiplication.	\(\frac{54}{-6}\cdot \frac{{a}^{2}}{a}\cdot \frac{{b}^{3}}{{b}^{5}}\)
   Simplify and use the Quotient Property.	\(-9\cdot a\cdot \frac{1}{{b}^{2}}\)
   Multiply.	\(-\frac{9a}{{b}^{2}}\)

5. Find the quotient: \(-72{a}^{7}{b}^{3}\div (8{a}^{12}{b}^{4}).\)

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   Otkrij odgovor

   \(-\frac{9}{{a}^{5}b}\)

6. Find the quotient: \(-63{c}^{8}{d}^{3}\div (7{c}^{12}{d}^{2}).\)

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   Otkrij odgovor

   \(\frac{-9d}{{c}^{4}}\)

7. Find the quotient: \(\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}.\)

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   Otkrij odgovor

   Be very careful to simplify \(\frac{14}{21}\) by dividing out a common factor, and to simplify the variables by subtracting their exponents.
   

   	\(\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}\)
   Simplify and use the Quotient Property.	\(\frac{2{y}^{6}}{3{x}^{4}}\)

8. Find the quotient: \(\frac{28{x}^{5}{y}^{14}}{49{x}^{9}{y}^{12}}.\)

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   Otkrij odgovor

   \(\frac{4{y}^{2}}{7{x}^{4}}\)

9. Find the quotient: \(\frac{30{m}^{5}{n}^{11}}{48{m}^{10}{n}^{14}}.\)

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   Otkrij odgovor

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

10. Find the quotient: \((18{x}^{3}y-36x{y}^{2})\div (-3xy).\)

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    Otkrij odgovor

    	\((18{x}^{3}y-36x{y}^{2})\div (-3xy)\)
    Rewrite as a fraction.	\(\frac{18{x}^{3}y-36x{y}^{2}}{-3xy}\)
    Divide each term by the divisor. Be careful with the signs!	\(\frac{18{x}^{3}y}{-3xy}-\frac{36x{y}^{2}}{-3xy}\)
    Simplify.	\(-6{x}^{2}+12y\)

11. Find the quotient: \((32{a}^{2}b-16a{b}^{2})\div (-8ab).\)

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    Otkrij odgovor

    \(-4a+2b\)

12. Find the quotient: \((-48{a}^{8}{b}^{4}-36{a}^{6}{b}^{5})\div (-6{a}^{3}{b}^{3}).\)

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    Otkrij odgovor

    \(8{a}^{5}b+6{a}^{3}{b}^{2}\)

13. Find the quotient: \(({x}^{2}+9x+20)\div (x+5).\)

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    Otkrij odgovor

    Write it as a long division problem. Be sure the dividend is in standard form.
    Divide \({x}^{2}\) by \(x.\) It may help to ask yourself, “What do I need to multiply \(x\) by to get \({x}^{2}\)?”
    Put the answer, \(x,\) in the quotient over the \(x\) term. Multiply \(x\) times \(x+5.\) Line up the like terms under the dividend.
    Subtract \({x}^{2}+5x\) from \({x}^{2}+9x.\) You may find it easier to change the signs and then add. Then bring down the last term, 20.
    Divide \(4x\) by \(x.\) It may help to ask yourself, “What do I need to multiply \(x\) by to get \(4x\)?” Put the answer, \(4\), in the quotient over the constant term.
    Multiply 4 times \(x+5.\)
    Subtract \(4x+20\) from \(4x+20.\)
    Check: Multiply the quotient by the divisor. \(\ (x+4)(x+5)\) You should get the dividend. \(\ {x}^{2}+9x+20✓\)

14. Find the quotient: \(({y}^{2}+10y+21)\div (y+3).\)

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    Otkrij odgovor

    \(y+7\)

15. Find the quotient: \(({m}^{2}+9m+20)\div (m+4).\)

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    Otkrij odgovor

    \(m+5\)

16. Find the quotient: \(({x}^{4}-{x}^{2}+5x-6)\div (x+2).\)

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    Otkrij odgovor

    Notice that there is no \({x}^{3}\) term in the dividend. We will add \(0{x}^{3}\) as a placeholder.

    Write it as a long division problem. Be sure the dividend is in standard form with placeholders for missing terms.
    Divide \({x}^{4}\) by \(x.\) Put the answer, \({x}^{3},\) in the quotient over the \({x}^{3}\) term. Multiply \({x}^{3}\) times \(x+2.\) Line up the like terms. Subtract and then bring down the next term.
    Divide \(-2{x}^{3}\) by \(x.\) Put the answer, \(-2{x}^{2},\) in the quotient over the \({x}^{2}\) term. Multiply \(-2{x}^{2}\) times \(x+1.\) Line up the like terms Subtract and bring down the next term.
    Divide \(3{x}^{2}\) by \(x.\) Put the answer, \(3x,\) in the quotient over the \(x\) term. Multiply \(3x\) times \(x+1.\) Line up the like terms. Subtract and bring down the next term.
    Divide \(\text{-}x\) by \(x.\) Put the answer, \(-1,\) in the quotient over the constant term. Multiply \(-1\) times \(x+1.\) Line up the like terms. Change the signs, add. Write the remainder as a fraction with the divisor as the denominator.
    To check, multiply \((x+2)({x}^{3}-2{x}^{2}+3x-1-\frac{4}{x+2})\). The result should be \({x}^{4}-{x}^{2}+5x-6.\)

17. Find the quotient: \(({x}^{4}-7{x}^{2}+7x+6)\div (x+3).\)

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    Otkrij odgovor

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

18. Find the quotient: \(({x}^{4}-11{x}^{2}-7x-6)\div (x+3).\)

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    Otkrij odgovor

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

19. Find the quotient: \((8{a}^{3}+27)\div (2a+3).\)

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    Otkrij odgovor

    This time we will show the division all in one step. We need to add two placeholders in order to divide.

    To check, multiply \((2a+3)(4{a}^{2}-6a+9).\)

    The result should be \(8{a}^{3}+27.\)

20. Find the quotient: \(({x}^{3}-64)\div (x-4).\)

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    Otkrij odgovor

    \({x}^{2}+4x+16\)

21. Find the quotient: \((125{x}^{3}-8)\div (5x-2).\)

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    Otkrij odgovor

    \(25{x}^{2}+10x+4\)

22. Use synthetic division to find the quotient and remainder when \(2{x}^{3}+3{x}^{2}+x+8\) is divided by \(x+2.\)

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    Otkrij odgovor

    Write the dividend with decreasing powers of \(x.\)
    Write the coefficients of the terms as the first row of the synthetic division.
    Write the divisor as \(x-c\) and place c in the synthetic division in the divisor box.
    Bring down the first coefficient to the third row.
    Multiply that coefficient by the divisor and place the result in the second row under the second coefficient.
    Add the second column, putting the result in the third row.
    Multiply that result by the divisor and place the result in the second row under the third coefficient.
    Add the third column, putting the result in the third row.
    Multiply that result by the divisor and place the result in the third row under the third coefficient.
    Add the final column, putting the result in the third row.
    The quotient is \(2{x}^{2}-1x+3\) and the remainder is 2.

    The division is complete. The numbers in the third row give us the result. The \(2\ -1\ 3\) are the coefficients of the quotient. The quotient is \(2{x}^{2}-1x+3.\) The 2 in the box in the third row is the remainder.

    Check:

    \(\ \begin{array}{lll} \\ \\ \text{(quotient)(divisor)}+\text{remainder} & = & \text{dividend} \\ (2{x}^{2}-1x+3)(x+2)+2 & \overset{?}{=} & 2{x}^{3}+3{x}^{2}+x+8 \\ 2{x}^{3}-{x}^{2}+3x+4{x}^{2}-2x+6+2 & \overset{?}{=} & 2{x}^{3}+3{x}^{2}+x+8 \\ 2{x}^{3}+3{x}^{2}+x+8 & = & 2{x}^{3}+3{x}^{2}+x+8✓\end{array}\)

23. Use synthetic division to find the quotient and remainder when \(3{x}^{3}+10{x}^{2}+6x-2\) is divided by \(x+2.\)

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    Otkrij odgovor

    \(3{x}^{2}+4x-2\text{;}\ 2\)

24. Use synthetic division to find the quotient and remainder when \(4{x}^{3}+5{x}^{2}-5x+3\) is divided by \(x+2.\)

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    Otkrij odgovor

    \(4{x}^{2}-3x+1\text{;}\ 1\)

25. Use synthetic division to find the quotient and remainder when \({x}^{4}-16{x}^{2}+3x+12\) is divided by \(x+4.\)

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    Otkrij odgovor

    The polynomial \({x}^{4}-16{x}^{2}+3x+12\) has its term in order with descending degree but we notice there is no \({x}^{3}\) term. We will add a 0 as a placeholder for the \({x}^{3}\) term. In \(x-c\) form, the divisor is \(x-(-4).\)

    We divided a 4^th degree polynomial by a 1^st degree polynomial so the quotient will be a 3^rd degree polynomial.

    Reading from the third row, the quotient has the coefficients \(1\ -4\ 0\ 3,\) which is \({x}^{3}-4{x}^{2}+3.\) The remainder
    is 0.

26. Use synthetic division to find the quotient and remainder when \({x}^{4}-16{x}^{2}+5x+20\) is divided by \(x+4.\)

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    Otkrij odgovor

    \({x}^{3}-4{x}^{2}+5\text{;}\ 0\)

27. Use synthetic division to find the quotient and remainder when \({x}^{4}-9{x}^{2}+2x+6\) is divided by \(x+3.\)

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    Otkrij odgovor

    \({x}^{3}-3{x}^{2}+2\text{;}\ 0\)

28. For functions \(f(x)={x}^{2}-5x-14\) and \(g(x)=x+2,\) find: ⓐ \((\frac{f}{g})(x)\) ⓑ \((\frac{f}{g})(-4).\)

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    Otkrij odgovor

    ⓐ
    
    

    Substitute for \(f(x)\) and \(g(x).\)	\(\ (\frac{f}{g})(x)=\frac{{x}^{2}-5x-14}{x+2}\)
    Divide the polynomials.	\(\ (\frac{f}{g})(x)=x-7\)

    
    
    ⓑ In part ⓐ we found \((\frac{f}{g})(x)\) and now are asked to find \((\frac{f}{g})(-4).\)
    

    	\(\ (\frac{f}{g})(x)=x-7\)
    To find \((\frac{f}{g})(-4),\) substitute \(x=-4.\)	\((\frac{f}{g})(-4)=-4-7\)
    	\((\frac{f}{g})(-4)=-11\)

29. For functions \(f(x)={x}^{2}-5x-24\) and \(g(x)=x+3,\) find ⓐ \((\frac{f}{g})(x)\) ⓑ \((\frac{f}{g})(-3).\)

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    Otkrij odgovor

    ⓐ \((\frac{f}{g})(x)=x-8\)
    ⓑ \((\frac{f}{g})(-3)=-11\)

30. For functions \(f(x)={x}^{2}-5x-36\) and \(g(x)=x+4,\) find ⓐ \((\frac{f}{g})(x)\) ⓑ \((\frac{f}{g})(-5).\)

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    Otkrij odgovor

    ⓐ \((\frac{f}{g})(x)=x-9\)
    ⓑ undefined

31. Use the Remainder Theorem to find the remainder when \(f(x)={x}^{3}+3x+19\) is divided by \(x+2.\)

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    Otkrij odgovor

    To use the Remainder Theorem, we must use the divisor in the \(x-c\) form. We can write the divisor \(x+2\) as \(x-(-2).\) So, our \(c\) is \(-2.\)

    To find the remainder, we evaluate \(f(c)\) which is \(f(-2).\)

    To evaluate \(f(-2),\) substitute \(x=-2.\)
    Simplify.
    	The remainder is 5 when \(f(x)={x}^{3}+3x+19\) is divided by \(x+2.\)
    Check: Use synthetic division to check.
    The remainder is 5.

32. Use the Remainder Theorem to find the remainder when \(f(x)={x}^{3}+4x+15\) is divided by \(x+2.\)

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    Otkrij odgovor

    \(-1\)

33. Use the Remainder Theorem to find the remainder when \(f(x)={x}^{3}-7x+12\) is divided by \(x+3.\)

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    Otkrij odgovor

    \(6\)

34. Use the Remainder Theorem to determine if \(x-4\) is a factor of \(f(x)={x}^{3}-64.\)

    Prođi kroz to. Savjet

    Otkrij odgovor

    The Factor Theorem tells us that \(x-4\) is a factor of \(f(x)={x}^{3}-64\) if \(f(4)=0.\)
    

    	\(f(x)={x}^{3}-64\)
    To evaluate \(f(4)\) substitute \(x=4.\)	\(f(4)={4}^{3}-64\)
    Simplify.	\(f(4)=64-64\)
    Subtract.	\(f(4)=0\)

    Since \(f(4)=0,\) \(x-4\) is a factor of \(f(x)={x}^{3}-64.\)

35. Use the Factor Theorem to determine if \(x-5\) is a factor of \(f(x)={x}^{3}-125.\)

    Prođi kroz to. Savjet

    Otkrij odgovor

    yes

36. Use the Factor Theorem to determine if \(x-6\) is a factor of \(f(x)={x}^{3}-216.\)

    Prođi kroz to. Savjet

    Otkrij odgovor

    yes

37. \(15{r}^{4}{s}^{9}\div (15{r}^{4}{s}^{9})\)

    Prođi kroz to. Savjet

38. \(20{m}^{8}{n}^{4}\div (30{m}^{5}{n}^{9})\)

    Prođi kroz to. Savjet

    Otkrij odgovor

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

39. \(\frac{18{a}^{4}{b}^{8}}{-27{a}^{9}{b}^{5}}\)

    Prođi kroz to. Savjet

40. \(\frac{45{x}^{5}{y}^{9}}{-60{x}^{8}{y}^{6}}\)

    Prođi kroz to. Savjet

    Otkrij odgovor

    \(\frac{-3{y}^{3}}{4{x}^{3}}\)

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Simboli koji se koriste ovdje

Dodirnite bilo koji simbol za punu definiciju, sliku, i što znači svako slovo u njoj.

\neqnot equal
The two sides are different.\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\sqrt{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.iimaginary unit
i² = −1.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

Kako: Dividing Polynomials

1. Dividing monomials
2. Dividing a polynomial by a monomial
3. Dividing polynomials using long division
4. Dividing polynomials using synthetic division
5. Dividing polynomial functions
6. Use the remainder and factor theorems
7. if
8. if

Pitanja koja ljudi postavljaju

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.

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Više u Algebra

Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value