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7.12: Division of a Polynomial by a Monomial

Difficulty Level: Advanced Created by: CK-12
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Can you divide the polynomial by the monomial? How does this relate to factoring?

4e4+6e310e2÷2e

Watch This

James Sousa: Dividing Polynomials by Monomials

Guidance

Recall that a monomial is an algebraic expression that has only one term. So, for example, x, 8, –2, or 3ac are all monomials because they have only one term. The term can be a number, a variable, or a combination of a number and a variable. A polynomial is an algebraic expression that has more than one term.

When dividing polynomials by monomials, it is often easiest to separately divide each term in the polynomial by the monomial. When simplifying each mini-division problem, don't forget to use exponent rules for the variables. For example,

8x52x3=4x2.

Remember that a fraction is just a division problem!

Example A

What is (14s221s+42)÷(7)?

Solution: This is the same as 14s221s+427. Divide each term of the polynomial numerator by the monomial denominator and simplify.

  • 14s47=2s4
  • 21s7=3s
  • 427=6

Therefore, (14s221s+42)÷(7)=2s43s+6.

Example B

What is 3w318w224w6w?

Solution: Divide each term of the polynomial numerator by the monomial denominator and simplify. Remember to use exponent rules when dividing the variables.

  • 3w36w=w22
  • 18w26w=3w
  • 24w6w=4

Therefore, 3w318w224w6w=w223w4.

Example C

What is (27a4b5+81a3b418a2b3)÷(9a2b)?

Solution: This is the same as 27a4b5+81a3b418a2b39a2b. Divide each term of the polynomial numerator by the monomial denominator and simplify. Remember to use exponent rules when dividing the variables.

  • 27a4b59a2b=3a2b4
  • 81a3b49a2b=9ab3
  • 18a2b39a2b=2b2

Therefore, (27a4b5+81a3b418a2b3)÷(9a2b)=3a2b49ab3+2b2.

Concept Problem Revisited

Can you divide the polynomial by the monomial? How does this relate to factoring?

4e4+6e310e2÷2e

This process is the same as factoring out a 2e from the expression 4e4+6e310e2.

  • 4e42e=2e3
  • 6e32e=3e2
  • 10e22e=5e

Therefore, 4e4+6e310e2÷2e=2e3+3e25e.

Vocabulary

Divisor
A divisor is the expression in the denominator of a fraction.
Monomial
A monomial is an algebraic expression that has only one term. x, 8, –2, or 3ac are all monomials because they have only one term.
Polynomial
A polynomial is an algebraic expression that has more than one term.

Guided Practice

Complete the following division problems.

1. (3a55a4+17a39a2)÷(a)

2. (40n332n7+88n11+8n2)÷(8n2)

3. 16m612m4+4m24m2

Answers:

1. (3a55a4+17a39a2)÷(a)=3a45a3+17a29a

2. (40n332n7+88n11+8n2)÷(8n2)=5n4n5+11n9+1

3. (16m612m4+4m2)(4m2)=4m43m2+1

Practice

Complete the following division problems.

  1. (6a3+30a2+24a)÷6
  2. (15b3+20b2+5b)÷5
  3. (12c4+18c2+6c)÷6c
  4. (60d12+90d11+30d8)÷30d
  5. (33e7+99e3+22e2)÷11e
  6. (8a4+8a2)÷(4a)
  7. (3b4+6b330b2+15b)÷(3b)
  8. (40c1220c1125c930c3)÷5c2
  9. (32d11+16d7+24d464d2)÷8d2
  10. (14e1218e1112e1018e7)÷2e5
  11. (18a109a8+72a7+9a5+3a2)÷3a2
  12. (24b9+42b7+42b6)÷6b3
  13. (24c1242c718c6)÷2c5
  14. (14d12+21d9+42d7)÷7d4
  15. (40e12+30e1010e4+30e3+80e)÷10e2

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Vocabulary

Denominator

The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. \frac{5}{8} has denominator 8.

Dividend

In a division problem, the dividend is the number or expression that is being divided.

divisor

In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression 152 \div 6, 6 is the divisor and 152 is the dividend.

Polynomial long division

Polynomial long division is the standard method of long division, applied to the division of polynomials.

Rational Expression

A rational expression is a fraction with polynomials in the numerator and the denominator.

Rational Root Theorem

The rational root theorem states that for a polynomial, f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0, where a_n, a_{n-1}, \cdots a_0 are integers, the rational roots can be determined from the factors of a_n and a_0. More specifically, if p is a factor of a_0 and q is a factor of a_n, then all the rational factors will have the form \pm \frac{p}{q}.

Remainder Theorem

The remainder theorem states that if f(k) = r, then r is the remainder when dividing f(x) by (x - k).

Synthetic Division

Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.

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Apr 30, 2013
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