Divide Polynomials

Divide a Polynomial by a Monomial

In the last section, you learned how to divide a monomial by a monomial. As you continue to build up your knowledge of polynomials 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.

We’ll state the fraction addition property here just as you learned it and in reverse.

We use the form on the left to add fractions and we use the form on the right to divide a polynomial by a monomial.

For example,	\(\frac{y+2}{5}\)
can be written	\(\frac{y}{5}+\frac{2}{5}.\)

We use this form of fraction addition to divide polynomials by monomials.

Example

Try it.

Find the quotient: \(\frac{7{y}^{2}+21}{7}.\)

Solution

	\(\frac{7{y}^{2}+21}{7}\)
Divide each term of the numerator by the denominator.	\(\frac{7{y}^{2}}{7}+\frac{21}{7}\)
Simplify each fraction.	\({y}^{2}+3\)

Remember that division can be represented as a fraction. When you are asked to divide a polynomial by a monomial and it is not already in fraction form, write a fraction with the polynomial in the numerator and the monomial in the denominator.

Example

Try it.

Find the quotient: \((18{x}^{3}-36{x}^{2})\div 6x.\)

Solution

	\((18{x}^{3}-36{x}^{2})\div 6x\)
Rewrite as a fraction.	\(\frac{18{x}^{3}-36{x}^{2}}{6x}\)
Divide each term of the numerator by the denominator.	\(\frac{18{x}^{3}}{6x}-\frac{36{x}^{2}}{6x}\)
Simplify.	\(3{x}^{2}-6x\)

Example

Try it.

Find the quotient: \(\frac{12{d}^{2}-16d}{-4}.\)

Solution

	\(\frac{12{d}^{2}-16d}{-4}\)
Divide each term of the numerator by the denominator.	\(\frac{12{d}^{2}}{-4}-\frac{16d}{-4}\)
Simplify. Remember, subtracting a negative is like adding a positive!	\(-3{d}^{2}+4d\)

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

Divide a Polynomial by a Binomial

To 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 write the long division
We divide the first two digits, 87, by 25.
We multiply 3 times 25 and write the product under the 87.
Now we subtract 75 from 87.
Then we bring down the third digit of the dividend, 5.
Repeat the process, dividing 25 into 125.

We check division by multiplying the quotient by the divisor.

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

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

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 x2 by x. It may help to ask yourself, "What do I need to multiply x by to get x2?"
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 x2 + 5x from x2 + 9x.
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.
x2 + 9x + 20✓

When the divisor has subtraction sign, we must be extra careful when we multiply the partial quotient and then subtract. It may be safer to show that we change the signs and then add.

Example

Try it.

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

Solution

Write it as a long division problem.
Be sure the dividend is in standard form.
Divide 2x2 by x. Put the answer, 2x, in the quotient over the x term.
Multiply 2x times x − 3. Line up the like terms under the dividend.
Subtract 2x2 − 6x from 2x2 − 5x. Change the signs and then add. Then bring down the last term.
Divide x by x. Put the answer, 1, in the quotient over the constant term.
Multiply 1 times x − 3.
Subtract x − 3 from x − 3 by changing the signs and adding.
To check, multiply (x − 3)(2x + 1).
The result should be 2x2 − 5x − 3.

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

Key Concepts

• Fraction Addition
  • If \(a,b,\ \text{and}\ c\) are numbers where \(c\ne 0\), then
    \(\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\ \text{and}\ \frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\)
• Division of a Polynomial by a Monomial
  • To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Divide Polynomials

In the following exercises, divide each polynomial by the monomial.

Try it.

\(\frac{45y+36}{9}\)

Try it.

\(\frac{30b+75}{5}\)

Solution

\(6b+15\)

Try it.

\(\frac{8{d}^{2}-4d}{2}\)

Try it.

\(\frac{42{x}^{2}-14x}{7}\)

Solution

\(6{x}^{2}-2x\)

Try it.

\((16{y}^{2}-20y)\div 4y\)

Try it.

\((55{w}^{2}-10w)\div 5w\)

Solution

\(11w-2\)

Try it.

\((9{n}^{4}+6{n}^{3})\div 3n\)

Try it.

\((8{x}^{3}+6{x}^{2})\div 2x\)

Solution

\(4{x}^{2}+3x\)

Try it.

\(\frac{18{y}^{2}-12y}{-6}\)

Try it.

\(\frac{20{b}^{2}-12b}{-4}\)

Solution

\(-5{b}^{2}+3b\)

Try it.

\(\frac{35{a}^{4}+65{a}^{2}}{-5}\)

Try it.

\(\frac{51{m}^{4}+72{m}^{3}}{-3}\)

Solution

\(-17{m}^{4}-24{m}^{3}\)

Try it.

\(\frac{310{y}^{4}-200{y}^{3}}{5{y}^{2}}\)

Try it.

\(\frac{412{z}^{8}-48{z}^{5}}{4{z}^{3}}\)

Solution

\(103{z}^{5}-12{z}^{2}\)

Try it.

\(\frac{46{x}^{3}+38{x}^{2}}{2{x}^{2}}\)

Try it.

\(\frac{51{y}^{4}+42{y}^{2}}{3{y}^{2}}\)

Solution

\(17{y}^{2}+14\)

Try it.

\((24{p}^{2}-33p)\div (-3p)\)

Try it.

\((35{x}^{4}-21x)\div (-7x)\)

Solution

\(-5{x}^{3}+3\)

Try it.

\((63{m}^{4}-42{m}^{3})\div (-7{m}^{2})\)

Try it.

\((48{y}^{4}-24{y}^{3})\div (-8{y}^{2})\)

Solution

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

Try it.

\((63{a}^{2}{b}^{3}+72a{b}^{4})\div (9ab)\)

Try it.

\((45{x}^{3}{y}^{4}+60x{y}^{2})\div (5xy)\)

Solution

\(9{x}^{2}{y}^{3}+12y\)

Try it.

\(\frac{52{p}^{5}{q}^{4}+36{p}^{4}{q}^{3}-64{p}^{3}{q}^{2}}{4{p}^{2}q}\)

Try it.

\(\frac{49{c}^{2}{d}^{2}-70{c}^{3}{d}^{3}-35{c}^{2}{d}^{4}}{7c{d}^{2}}\)

Solution

\(7c-10{c}^{2}d-5c{d}^{2}\)

Try it.

\(\frac{66{x}^{3}{y}^{2}-110{x}^{2}{y}^{3}-44{x}^{4}{y}^{3}}{11{x}^{2}{y}^{2}}\)

Try it.

\(\frac{72{r}^{5}{s}^{2}+132{r}^{4}{s}^{3}-96{r}^{3}{s}^{5}}{12{r}^{2}{s}^{2}}\)

Solution

\(6{r}^{3}+11{r}^{2}s-8r{s}^{3}\)

Try it.

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

Try it.

\(\frac{12{q}^{2}+3q-1}{3q}\)

Solution

\(4q+1-\frac{1}{3q}\)

Try it.

\(\frac{10{x}^{2}+5x-4}{-5x}\)

Try it.

\(\frac{20{y}^{2}+12y-1}{-4y}\)

Solution

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

Try it.

\(\frac{36{p}^{3}+18{p}^{2}-12p}{6{p}^{2}}\)

Try it.

\(\frac{63{a}^{3}-108{a}^{2}+99a}{9{a}^{2}}\)

Solution

\(7a-12+\frac{11}{a}\)

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

Try it.

\(({y}^{2}+7y+12)\div (y+3)\)

Try it.

\(({d}^{2}+8d+12)\div (d+2)\)

Solution

\(d+6\)

Try it.

\(({x}^{2}-3x-10)\div (x+2)\)

Try it.

\(({a}^{2}-2a-35)\div (a+5)\)

Solution

\(a-7\)

Try it.

\(({t}^{2}-12t+36)\div (t-6)\)

Try it.

\(({x}^{2}-14x+49)\div (x-7)\)

Solution

\(x-7\)

Try it.

\((6{m}^{2}-19m-20)\div (m-4)\)

Try it.

\((4{x}^{2}-17x-15)\div (x-5)\)

Solution

\(4x+3\)

Try it.

\(({q}^{2}+2q+20)\div (q+6)\)

Try it.

\(({p}^{2}+11p+16)\div (p+8)\)

Solution

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

Try it.

\(({y}^{2}-3y-15)\div (y-8)\)

Try it.

\(({x}^{2}+2x-30)\div (x-5)\)

Solution

\(x+7+\frac{5}{x-5}\)

Try it.

\((3{b}^{3}+{b}^{2}+2)\div (b+1)\)

Try it.

\((2{n}^{3}-10n+24)\div (n+3)\)

Solution

\(2{n}^{2}-6n+8\)

Try it.

\((2{y}^{3}-6y-36)\div (y-3)\)

Try it.

\((7{q}^{3}-5q-2)\div (q-1)\)

Solution

\(7{q}^{2}+7q+2\)

Try it.

\(({z}^{3}+1)\div (z+1)\)

Try it.

\(({m}^{3}+1000)\div (m+10)\)

Solution

\({m}^{2}-10m+100\)

Try it.

\(({a}^{3}-125)\div (a-5)\)

Try it.

\(({x}^{3}-216)\div (x-6)\)

Solution

\({x}^{2}+6x+36\)

Try it.

\((64{x}^{3}-27)\div (4x-3)\)

Try it.

\((125{y}^{3}-64)\div (5y-4)\)

Solution

\(25{y}^{2}+20x+16\)

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

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

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

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

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

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

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

4. Find the quotient: \(\frac{7{y}^{2}+21}{7}.\)

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   	\(\frac{7{y}^{2}+21}{7}\)
   Divide each term of the numerator by the denominator.	\(\frac{7{y}^{2}}{7}+\frac{21}{7}\)
   Simplify each fraction.	\({y}^{2}+3\)

5. Find the quotient: \(\frac{8{z}^{2}+24}{4}.\)

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   \(2{z}^{2}+6\)

6. Find the quotient: \(\frac{18{z}^{2}-27}{9}.\)

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   \(2{z}^{2}-3\)

7. Find the quotient: \((18{x}^{3}-36{x}^{2})\div 6x.\)

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   	\((18{x}^{3}-36{x}^{2})\div 6x\)
   Rewrite as a fraction.	\(\frac{18{x}^{3}-36{x}^{2}}{6x}\)
   Divide each term of the numerator by the denominator.	\(\frac{18{x}^{3}}{6x}-\frac{36{x}^{2}}{6x}\)
   Simplify.	\(3{x}^{2}-6x\)

8. Find the quotient: \((27{b}^{3}-33{b}^{2})\div 3b.\)

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   \(9{b}^{2}-11b\)

9. Find the quotient: \((25{y}^{3}-55{y}^{2})\div 5y.\)

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   \(5{y}^{2}-11y\)

10. Find the quotient: \(\frac{12{d}^{2}-16d}{-4}.\)

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    	\(\frac{12{d}^{2}-16d}{-4}\)
    Divide each term of the numerator by the denominator.	\(\frac{12{d}^{2}}{-4}-\frac{16d}{-4}\)
    Simplify. Remember, subtracting a negative is like adding a positive!	\(-3{d}^{2}+4d\)

11. Find the quotient: \(\frac{25{y}^{2}-15y}{-5}.\)

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

12. Find the quotient: \(\frac{42{b}^{2}-18b}{-6}.\)

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    \(-7{b}^{2}+3b\)

13. Find the quotient: \(\frac{105{y}^{5}+75{y}^{3}}{5{y}^{2}}.\)

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    	\(\frac{105{y}^{5}+75{y}^{3}}{5{y}^{2}}\)
    Separate the terms.	\(\frac{105{y}^{5}}{5{y}^{2}}+\frac{75{y}^{3}}{5{y}^{2}}\)
    Simplify.	\(21{y}^{3}+15y\)

14. Find the quotient: \(\frac{60{d}^{7}+24{d}^{5}}{4{d}^{3}}.\)

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    \(15{d}^{4}+6{d}^{2}\)

15. Find the quotient: \(\frac{216{p}^{7}-48{p}^{5}}{6{p}^{3}}.\)

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    \(36{p}^{4}-8{p}^{2}\)

16. Find the quotient: \((15{x}^{3}y-35x{y}^{2})\div (-5xy).\)

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    	\((15{x}^{3}y-35x{y}^{2})\div (-5xy)\)
    Rewrite as a fraction.	\(\frac{15{x}^{3}y-35x{y}^{2}}{-5xy}\)
    Separate the terms.	\(\frac{15{x}^{3}y}{-5xy}-\frac{35x{y}^{2}}{-5xy}\)
    Simplify.	\(-3{x}^{2}+7y\)

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

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    \(-4a+2b\)

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

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

19. Find the quotient: \(\frac{36{x}^{3}{y}^{2}+27{x}^{2}{y}^{2}-9{x}^{2}{y}^{3}}{9{x}^{2}y}.\)

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    	\(\frac{36{x}^{3}{y}^{2}+27{x}^{2}{y}^{2}-9{x}^{2}{y}^{3}}{9{x}^{2}y}\)
    Separate the terms.	\(\frac{36{x}^{3}{y}^{2}}{9{x}^{2}y}+\frac{27{x}^{2}{y}^{2}}{9{x}^{2}y}-\frac{9{x}^{2}{y}^{3}}{9{x}^{2}y}\)
    Simplify.	\(4xy+3y-{y}^{2}\)

20. Find the quotient: \(\frac{40{x}^{3}{y}^{2}+24{x}^{2}{y}^{2}-16{x}^{2}{y}^{3}}{8{x}^{2}y}.\)

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    \(5xy+3y-2{y}^{2}\)

21. Find the quotient: \(\frac{35{a}^{4}{b}^{2}+14{a}^{4}{b}^{3}-42{a}^{2}{b}^{4}}{7{a}^{2}{b}^{2}}.\)

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

22. Find the quotient: \(\frac{10{x}^{2}+5x-20}{5x}.\)

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    	\(\frac{10{x}^{2}+5x-20}{5x}\)
    Separate the terms.	\(\frac{10{x}^{2}}{5x}+\frac{5x}{5x}-\frac{20}{5x}\)
    Simplify.	\(2x+1-\frac{4}{x}\)

23. Find the quotient: \(\frac{18{c}^{2}+6c-9}{6c}.\)

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    \(3c+1-\frac{3}{2c}\)

24. Find the quotient: \(\frac{10{d}^{2}-5d-2}{5d}.\)

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    \(2d-1-\frac{2}{5d}\)

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

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    Write it as a long division problem.
    Be sure the dividend is in standard form.
    Divide x2 by x. It may help to ask yourself, "What do I need to multiply x by to get x2?"
    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 x2 + 5x from x2 + 9x.
    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.
    x2 + 9x + 20✓

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

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    \(y+7\)

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

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    \(m+5\)

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

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    Write it as a long division problem.
    Be sure the dividend is in standard form.
    Divide 2x2 by x. Put the answer, 2x, in the quotient over the x term.
    Multiply 2x times x − 3. Line up the like terms under the dividend.
    Subtract 2x2 − 6x from 2x2 − 5x. Change the signs and then add. Then bring down the last term.
    Divide x by x. Put the answer, 1, in the quotient over the constant term.
    Multiply 1 times x − 3.
    Subtract x − 3 from x − 3 by changing the signs and adding.
    To check, multiply (x − 3)(2x + 1).
    The result should be 2x2 − 5x − 3.

29. Find the quotient: \((2{x}^{2}-3x-20)\div (x-4).\)

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    \(2x+5\)

30. Find the quotient: \((3{x}^{2}-16x-12)\div (x-6).\)

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    \(3x+2\)

31. Find the quotient: \(({x}^{3}-{x}^{2}+x+4)\div (x+1).\)

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    Write it as a long division problem.
    Be sure the dividend is in standard form.
    Divide x3 by x. Put the answer, x2, in the quotient over the x2 term. Multiply x2 times x + 1. Line up the like terms under the dividend.
    Subtract x3 + x2 from x3 − x2 by changing the signs and adding. Then bring down the next term.
    Divide −2x2 by x. Put the answer, −2x, in the quotient over the x term. Multiply −2x times x + 1. Line up the like terms under the dividend.
    Subtract −2x2 − 2x from −2x2 + x by changing the signs and adding. Then bring down the last term.
    Divide 3x by x. Put the answer, 3, in the quotient over the constant term. Multiply 3 times x + 1. Line up the like terms under the dividend.
    Subtract 3x + 3 from 3x + 4 by changing the signs and adding. Write the remainder as a fraction with the divisor as the denominator.
    To check, multiply \((x+1)({x}^{2}-2x+3+\frac{1}{x+1}).\) The result should be \({x}^{3}-{x}^{2}+x+4\).

32. Find the quotient: \(({x}^{3}+5{x}^{2}+8x+6)\div (x+2).\)

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

33. Find the quotient: \((2{x}^{3}+8{x}^{2}+x-8)\div (x+1).\)

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    \(2{x}^{2}+6x-5-\frac{3}{x+1}\)

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

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    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 x4 by x. Put the answer, x3, in the quotient over the x3 term. Multiply x3 times x + 2. Line up the like terms. Subtract and then bring down the next term.
    Divide −2x3 by x. Put the answer, −2x2, in the quotient over the x2 term. Multiply −2x2 times x + 1. Line up the like terms. Subtract and bring down the next term.
    Divide 3x2 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 −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.
    To check, multiply \((x+2)({x}^{3}-2{x}^{2}+3x-1)\).
    The result should be \({x}^{4}-{x}^{2}+5x-2\).

35. Find the quotient: \(({x}^{3}+3x+14)\div (x+2).\)

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    \({x}^{2}-2x+7\)

36. Find the quotient: \(({x}^{4}-3{x}^{3}-1000)\div (x+5).\)

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    \({x}^{3}-8{x}^{2}+40x-200\)

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

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    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 \(8{a}^{3}+27\).

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

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

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

    Darbs caur to Pamudinājums

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

40. \(\frac{45y+36}{9}\)

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Bezmaksas konts pievieno piezīmes par katru nodarbību, ierakstu par to, ko esat pabeidzis, jūsu atrisinātās problēmas vienā vietā, un pasniedzēju, kuru varat jautāt par šo lapu. Pati matemātika ir pieejama ikvienam, kas ir vai nav parakstījies.

Pierakstīties Pieteikšanās

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Piesitiet visiem simboliem, kas apzīmē pilnu definīciju, attēlu un ko katrs burts tajā nozīmē.

\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.

Kā lietot: Divide Polynomials

1. Divide a polynomial by a monomial
2. Divide a polynomial by a binomial
3. If
4. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Jautājumi, ko cilvēki vaicā

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.

Daļas šīs lapas ir pielāgotas no OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Šeit ir pārliecināts un vēlreiz izskaidrots; kļūdas ir mūsu. Tā kā tās avots ir licencēts nekomerciālai izmantošanai, šī nodarbība ir bezmaksas, to nekas nepārdod un tiek dalīta ar to pašu licenci.

Vairāk Algebra

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