4. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Polynomial long division is a method for
dividing a polynomial by another polynomials
of a lower degree. It is very similar to dividing
numbers.
5. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Divide using long division.
Example 1: Using Long Division to Divide a
Polynomial
(–y2 + 2y3 + 25) ÷ (y – 3)
Step 1 Write the dividend in standard form, including
terms with a coefficient of 0.
Step 2 Write division in the same way you would
when dividing numbers.
6. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Notice that y times 2y2 is 2y3.
Write 2y2 above 2y3.
Step 3 Divide.
Multiply y – 3 by 2y2. Then
subtract. Bring down the next
term. Divide 5y2 by y.
Multiply y – 3 by 5y. Then
subtract. Bring down the next
term. Divide 15y by y.
Find the remainder.
Multiply y – 3 by 15. Then
subtract.
Example 1 Continued
7. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Step 4 Write the final answer.
Example 1 Continued
8. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Huddle
Divide using long division.
(15x2 + 8x – 12) ÷ (3x + 1)
Step 1 Write the dividend in standard form, including
terms with a coefficient of 0.
Step 2 Write division in the same way you would
when dividing numbers.
9. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Huddle
Step 3 Divide.
Step 4 Write the final answer.
11. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Synthetic division is a shorthand method of
dividing a polynomial by a linear binomial by
using only the coefficients. For synthetic division
to work, the polynomial must be written in
standard form, using 0 and a coefficient for any
missing terms, and the divisor must be in the
form (x – a).
13. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Divide using synthetic division.
Example 2A: Using Synthetic Division to Divide by a
Linear Binomial
(3x2 + 9x – 2) ÷ (x – )
1
3
Step 2 Bring down each coefficient (missing
terms become 0).Then multiply and add for each
column.
Step 3 Write the quotient.
Step 1 Identify a.
14. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Divide using synthetic division.
(3x4 – x3 + 5x – 1) ÷ (x + 2)
Step 1 Find a.
Example 2B: Using Synthetic Division to Divide by a
Linear Binomial
Step 2 Bring down each coefficient (missing
terms become 0).Then multiply and add for each
column.
Step 3 Write the quotient.
17. Holt McDougal Algebra 2
3-3 Dividing Polynomials
You can use synthetic division to evaluate polynomials.
This process is called synthetic substitution. The
process of synthetic substitution is exactly the same as
the process of synthetic division, but the final answer is
interpreted differently, as described by the Remainder
Theorem.
18. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Example 3A: Using Synthetic Substitution
Use synthetic substitution to evaluate the
polynomial for the given value.
P(x) = 2x3 + 5x2 – x + 7 for x = 2.
Write the coefficients of
the dividend. Use a = 2.
P(2) =
Check Substitute 2 for x in P(x) = 2x3 + 5x2 – x + 7.
P(2) = 2(2)3 + 5(2)2 – (2) + 7
P(2) =
19. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Example 3B: Using Synthetic Substitution
Use synthetic substitution to evaluate the
polynomial for the given value.
P(x) = 6x4 – 25x3 – 3x + 5 for x = – .
1
3
Write the coefficients of
the dividend. Use 0 for
the coefficient of x2. Use
a = .
1
3
P( ) =
1
3
20. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Huddle
Use synthetic substitution to evaluate the
polynomial for the given value.
P(x) = x3 + 3x2 + 4 for x = –3.
Write the coefficients of
the dividend. Use 0 for
the coefficient of x2 Use a
= –3.
Check Substitute –3 for x in P(x) = x3 + 3x2 + 4.
P(–3) = (–3)3 + 3(–3)2 + 4
P(–3) =
21. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Mastery
Use synthetic substitution to evaluate the
polynomial for the given value.
P(x) = 5x2 + 9x + 3 for x = .
1
5
22. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Example 4: Geometry Application
Write an expression that represents the area
of the top face of a rectangular prism when the
height is x + 2 and the volume of the prism is
x3 – x2 – 6x.
Substitute.
Use synthetic division.
The volume V is related to the area A and the
height h by the equation V = A h. Rearranging
for A gives A = .
V
h
The area of the face of the
rectangular prism can be
represented by A(x)= _______.
23. Holt McDougal Algebra 2
3-3 Dividing Polynomials
Huddle
Write an expression for the length of a
rectangle with width y – 9 and area y2 – 14y
+ 45.
Substitute.
Use synthetic division.
The area A is related to the width w and the length l
by the equation A = l w.
The length of the rectangle can be represented by
l(x)= _________.
24. Holt McDougal Algebra 2
3-3 Dividing Polynomials
4. Find an expression for the height of a
parallelogram whose area is represented
by 2x3 – x2 – 20x + 3 and whose base is
represented by (x + 3).
Lesson Quiz
2. Divide by using synthetic division.
(x3 – 3x + 5) ÷ (x + 2)
1. Divide by using long division.
(8x3 + 6x2 + 7) ÷ (x + 2)
3. Use synthetic substitution to evaluate
P(x) = x3 + 3x2 – 6 for x = 5 and x = –1.
