Lesson Objectives

- Demonstrate an understanding of division with fractions
- Learn how to set up the division of a polynomial by a monomial in fractional form
- Learn how to divide a polynomial by a monomial
- Learn how to check the result of a division problem with a polynomial divided by a monomial

## How to Divide a Polynomial by a Monomial

In this lesson, we will learn how to divide a polynomial by a monomial (polynomial with one term). In order to perform this action,
we will think back on operations with fractions. Let's think about the following problem:
$$12 ÷ 3 = 4$$
We can re-write this problem using a fraction bar. A fraction bar represents the division of the numerator by the denominator.
In our example, 12 is being divided by 3, this means in fractional form, 12 is our numerator and 3 is our denominator:
$$\frac{12}{3} = 4$$
Additionally, when we add two fractions with a common denominator, we add the numerators and keep the denominator the same:
$$\frac{12}{4} + \frac{16}{4} = \frac{12 + 16}{4}$$
If we sum the numerators (12 and 16), we get 28. We can then divide our numerator (28) by the denominator (4) to obtain 7:
$$\frac{28}{4} = 7$$
We could have obtained the same result by performing individual divisions and finding the sum of the results:
$$\frac{12}{4} + \frac{16}{4} = 3 + 4 = 7$$

Example 1: Find each quotient

$$(4x^4 + 2x^3 + 32x^2) \div 8x^2$$ Step 1) Let's set up the division problem using a fraction: $$\frac{4x^4 + 2x^3 + 32x^2}{8x^2}$$ Step 2) Split the fraction up into separate division problems: $$\frac{4x^4}{8x^2} + \frac{2x^3}{8x^2} + \frac{32x^2}{8x^2}$$ Step 3) Perform each division: $$\require{cancel}\frac{4x^4}{8x^2} = \frac{1\cancel{4}x^{4 - 2}}{2\cancel{8}} = \frac{x^2}{2}$$ $$\frac{2x^3}{8x^2} = \frac{1\cancel{2}x^{3 - 2}}{4 \cancel{8}} = \frac{x}{4}$$ $$\frac{32x^2}{8x^2} = \frac{4\cancel{32} \cancel{{x^2}}}{\cancel{8} \cancel{x^2}} = 4$$ We can now sum these three parts to report our answer: $$\frac{x^2}{2} + \frac{x}{4} + 4$$ Recall that we can check division with multiplication:

$$\frac{6}{2} = 3$$ Check: $$3 \cdot 2 = 6$$ To check division using multiplication, we multiply our quotient times our divisor. If we got the correct answer, we should get our dividend back. In our example, 6, the numerator is our dividend. 2, the denominator is our divisor. 3, is our quotient. To prove our answer is correct, we multiply 3 by 2 and get 6 back as a result. The same process can be used to check division with polynomials. Let's check our division by multiplying our quotient by the divisor (denominator). We should get our dividend (numerator) back: $$8x^2\left(\frac{x^2}{2} + \frac{x}{4} + 4\right)$$ Use the distributive property: $$8x^2 \cdot \frac{x^2}{2} = \frac{4\cancel{8}x^{2 + 2}}{\cancel{2}} = 4x^4$$ $$8x^2 \cdot \frac{x}{4} = \frac{2\cancel{8}x^{2 + 1}}{\cancel{4}} = 2x^3$$ $$8x^2 \cdot 4 = 32x^2$$ When we sum these amounts, we get our original dividend back: $$4x^4 + 2x^3 + 32x^2$$ Example 2: Find each quotient $$(6x^3 + 12x^2 + 24x) \div 6x^2$$ Step 1) Let's set up the division problem using a fraction: $$\frac{6x^3 + 12x^2 + 24x}{6x^2}$$ Step 2) Split the fraction up into separate division problems: $$\frac{6x^3}{6x^2} + \frac{12x^2}{6x^2} + \frac{24x}{6x^2}$$ Step 3) Perform each division: $$\frac{6x^3}{6x^2} = \frac{\cancel{6}x^{3 - 2}}{\cancel{6}} = x$$ $$\frac{12x^2}{6x^2} = \frac{2\cancel{12}x^{2 - 2}}{\cancel{6}} = 2$$ $$\frac{24x}{6x^2} = \frac{4\cancel{24}x^{1 - 2}}{\cancel{6}} = \frac{4}{x}$$ We can now sum these three parts to report our answer: $$x + 2 + \frac{4}{x}$$ Example 3: Find each quotient $$(2x^6 + 24x^5 + 30x^4) \div 6x^2$$ Step 1) Let's set up the division problem using a fraction: $$\frac{2x^6 + 24x^5 + 30x^4}{6x^2}$$ Step 2) Split the fraction up into separate division problems: $$\frac{2x^6}{6x^2} + \frac{24x^5}{6x^2} + \frac{30x^4}{6x^2}$$ Step 3) Perform each division: $$\frac{2x^6}{6x^2} = \frac{\cancel{2} x^{6 - 2}}{3\cancel{6}} = \frac{x^4}{3}$$ $$\frac{24x^5}{6x^2} = \frac{4\cancel{24} x^{5 - 2}}{\cancel{6}} = 4x^3$$ $$\frac{30x^4}{6x^2} = \frac{5 \cancel{30}x^{4 - 2}}{\cancel{6}} = 5x^2$$ We can now sum these three parts to report our answer: $$\frac{x^4}{3} + 4x^3 + 5x^2$$

### Dividing a Polynomial by a Monomial

- Set up the division problem as a fraction
- The dividend or first polynomial becomes the numerator
- The divisor or monomial becomes the denominator

- Split the fraction up into separate division problems
- Divide each term of the first polynomial by the monomial

- Perform each division

Example 1: Find each quotient

$$(4x^4 + 2x^3 + 32x^2) \div 8x^2$$ Step 1) Let's set up the division problem using a fraction: $$\frac{4x^4 + 2x^3 + 32x^2}{8x^2}$$ Step 2) Split the fraction up into separate division problems: $$\frac{4x^4}{8x^2} + \frac{2x^3}{8x^2} + \frac{32x^2}{8x^2}$$ Step 3) Perform each division: $$\require{cancel}\frac{4x^4}{8x^2} = \frac{1\cancel{4}x^{4 - 2}}{2\cancel{8}} = \frac{x^2}{2}$$ $$\frac{2x^3}{8x^2} = \frac{1\cancel{2}x^{3 - 2}}{4 \cancel{8}} = \frac{x}{4}$$ $$\frac{32x^2}{8x^2} = \frac{4\cancel{32} \cancel{{x^2}}}{\cancel{8} \cancel{x^2}} = 4$$ We can now sum these three parts to report our answer: $$\frac{x^2}{2} + \frac{x}{4} + 4$$ Recall that we can check division with multiplication:

$$\frac{6}{2} = 3$$ Check: $$3 \cdot 2 = 6$$ To check division using multiplication, we multiply our quotient times our divisor. If we got the correct answer, we should get our dividend back. In our example, 6, the numerator is our dividend. 2, the denominator is our divisor. 3, is our quotient. To prove our answer is correct, we multiply 3 by 2 and get 6 back as a result. The same process can be used to check division with polynomials. Let's check our division by multiplying our quotient by the divisor (denominator). We should get our dividend (numerator) back: $$8x^2\left(\frac{x^2}{2} + \frac{x}{4} + 4\right)$$ Use the distributive property: $$8x^2 \cdot \frac{x^2}{2} = \frac{4\cancel{8}x^{2 + 2}}{\cancel{2}} = 4x^4$$ $$8x^2 \cdot \frac{x}{4} = \frac{2\cancel{8}x^{2 + 1}}{\cancel{4}} = 2x^3$$ $$8x^2 \cdot 4 = 32x^2$$ When we sum these amounts, we get our original dividend back: $$4x^4 + 2x^3 + 32x^2$$ Example 2: Find each quotient $$(6x^3 + 12x^2 + 24x) \div 6x^2$$ Step 1) Let's set up the division problem using a fraction: $$\frac{6x^3 + 12x^2 + 24x}{6x^2}$$ Step 2) Split the fraction up into separate division problems: $$\frac{6x^3}{6x^2} + \frac{12x^2}{6x^2} + \frac{24x}{6x^2}$$ Step 3) Perform each division: $$\frac{6x^3}{6x^2} = \frac{\cancel{6}x^{3 - 2}}{\cancel{6}} = x$$ $$\frac{12x^2}{6x^2} = \frac{2\cancel{12}x^{2 - 2}}{\cancel{6}} = 2$$ $$\frac{24x}{6x^2} = \frac{4\cancel{24}x^{1 - 2}}{\cancel{6}} = \frac{4}{x}$$ We can now sum these three parts to report our answer: $$x + 2 + \frac{4}{x}$$ Example 3: Find each quotient $$(2x^6 + 24x^5 + 30x^4) \div 6x^2$$ Step 1) Let's set up the division problem using a fraction: $$\frac{2x^6 + 24x^5 + 30x^4}{6x^2}$$ Step 2) Split the fraction up into separate division problems: $$\frac{2x^6}{6x^2} + \frac{24x^5}{6x^2} + \frac{30x^4}{6x^2}$$ Step 3) Perform each division: $$\frac{2x^6}{6x^2} = \frac{\cancel{2} x^{6 - 2}}{3\cancel{6}} = \frac{x^4}{3}$$ $$\frac{24x^5}{6x^2} = \frac{4\cancel{24} x^{5 - 2}}{\cancel{6}} = 4x^3$$ $$\frac{30x^4}{6x^2} = \frac{5 \cancel{30}x^{4 - 2}}{\cancel{6}} = 5x^2$$ We can now sum these three parts to report our answer: $$\frac{x^4}{3} + 4x^3 + 5x^2$$

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