Dividing a Polynomial by a Monomial
You probably know how to divide monomials. For example,
We check by multiplying. Because 2x2 · 3x = 6x3, this answer is correct. Recall
that a ÷ b = c if and only if c · b = a. We call a the dividend, b the divisor, and
c the quotient.We may also refer to a ÷ b and
as quotients.
We can use the distributive property to find that
3x(2x2 + 5x - 4) = 6x3 + 15x2 - 12x.
So if we divide 6x3 + 15x2 - 12x by the monomial 3x, we must get
2x2 + 5x - 4. We can perform this division by dividing 3x into each term of 6x3
+ 15x2 - 12x:
In this case the divisor is 3x, the dividend is 6x3 + 15x2
- 12x, and the quotient is 2x2 + 5x - 4.
Example
Dividing polynomials
Find the quotient.
a) -12x5 ÷ (2x3)
b) ( -20x6 + 8x4 - 4x2) ÷ (4x2)
Solution
a) When dividing x5 by x3, we subtract the exponents:
The quotient is -6x2. Check:
-6x2 · 2x3 = -12x5
b) Divide each term of -20x6 + 8x4 - 4x2 by 4x2:
The quotient is -5x4 + 2x2 - 1. Check:
4x2(-5x4 - 2x2 - 1) = 20x6 + 8x4
- 4x2
Helpful hint
Recall that the order of operations
gives multiplication and
division an equal ranking
and says to do them in order
from left to right. So without
parentheses,
-12x5 ÷ 2x3 actually means
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