Math0301
Division of Polynomials
Objective A
To divide a polynomial by a monomial
To divide a polynomial by a monomial, divide each term in the numerator by the
denominator and write the sum of the quotients.
Simplify:
6 x 3 − 3x 2 + 9 x
3x
6 x 3 3x 2 9 x
=
−
+
3x
3x 3x
= 2x 2 − x + 3
Simplify:
Divide each term of the polynomial by
the monomial.
Simplify each expression.
12 x 2 y − 6 xy + 4 x 2
2 xy
12 x 2 y 6 xy 4 x 2
−
+
2 xy
2 xy 2 xy
2x
= 6x − 3 +
y
=
Objective B
To divide polynomials
The procedure for dividing two polynomials is similar to the one for dividing whole
numbers. The same equation used to check division of whole numbers is used to check
polynomial division.
(Quotient x divisor) + remainder = dividend
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Math0301
( x 2 − 5 x + 8) ÷ ( x − 3)
Simplify:
Step 1
x
x – 3 ) x2 – 5 x + 8
x2 – 3 x
–2x +8
x2
=x
x
Multiply: x(x – 3) = x2 – 3 x
Think: x x 2 =
Subtract: (x2 – 5 x) – (x2 – 3 x) = – 2 x
Step 2
x –2
x–3)x –5x +8
2
x2 – 3 x
–2x +8
–2x +6
− 2x
= −2
x
Multiply: – 2(x – 3) = – 2 x + 6
Think: x − 2 x =
Subtract: (2 x + 8) – (2 x + 6) = 2
The remainder is 2.
2
Check:
Therefore:
(x − 2)(x − 3) + 2 = x 2 − 5 x + 6 + 2 = x 2 − 5 x + 8
( x 2 − 5 x + 8) ÷ ( x − 3) = x − 2 +
2
x −3
If a term is missing from the dividend, a zero can be inserted for that term. This helps
keep like terms in the same column.
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Math0301
(6 x + 26 + 2 x 3 ) ÷ (2 + x)
Simplify:
(2 x 3 + 6 x + 26) ÷ ( x + 2)
2 x2 – 4 x + 14
x + 2 ) 2 x + 0 x2 + 6 x + 26
3
2x3 + 4 x2
Arrange the terms of each polynomial in
descending order.
There is no x² term in (2 x 3 + 6 x + 26) ÷ ( x + 2) .
Insert a zero for the missing term.
– 4 x2 + 6 x
– 4 x2 – 8 x
14 x + 26
14 x + 28
-2
Check: (2 x 2 − 4 x + 14)( x + 2) + (−2) = (2 x 3 + 6 x + 28) + (−2) = (2 x 3 + 6 x + 26)
2
Therefore: (2 x 3 + 6 x + 26) ÷ ( x + 2) = 2 x 2 − 4 x + 14 −
x+2
(8 x 2 + 4 x 3 + x − 4) ÷ (2 x + 3)
Simplify:
(4 x 3 + 8 x 2 + x − 4) ÷ (2 x + 3)
Arrange the terms of each polynomial in
descending order.
2x2 + x - 1
2x + 3 ) 4x + 8x2 + x – 4
3
4x3 + 6x2
2x2 +
x
2x2 + 3x
- 2x - 4
- 2x - 3
- 1
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Math0301
Check: (2x2 + x – 1)(2x + 3) + (–1) = (4x3 + 8x2 + x – 3) + (–1) = 4x3 + 8x2 + x – 4
Therefore: (4x3 + 8x2 + x – 4) ÷ (2x + 3) = 2x2 + x – 1 –
1
2x + 3
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