Transcript Slide 1

6-4
Factoring Polynomials
Warm Up
Factor each expression.
1. 3x – 6y
2. a2 – b2
Find each product.
4. (a + 1)(a2 + 1)
3. (x – 1)(x + 3)
LEARNING GOALS – LESSON 6.4
6.4.1: Use the Factor Theorem to determine factors of a
polynomial.
6.4.2: Factor the sum and difference of two cubes.
Recall: If a number is divided by any of its factors, the remainder is ____.
Likewise, if a polynomial is divided by any of its factors, the remainder is
_____.
The Remainder Theorem states:
If a polynomial is divided by (x – a), the remainder is the value of the
function at a.
RECALL: If (x – a) is a factor of P(x), then P(___)=___
Example 1: Determining Whether a Linear Binomial is a Factor
Determine whether the given binomial is a factor of P(x).
A. (x + 1); (x2 – 3x + 1)
Find P(___) by synthetic
substitution.
P(____) = ____
B. (x + 2); (3x4 + 6x3 – 5x – 10)
Find P(____) by synthetic
substitution.
P(____) = ____
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Factoring Polynomials
Example 1C: Determining Whether a Linear Binomial is a Factor
Determine whether the given binomial is a factor of P(x).
C. (3x – 6); (3x4 – 6x3 + 6x2 + 3x – 30)
Divide the polynomial by 3,
then find P(2) by synthetic substitution.
P(____) = _____, so (3x – 6)
___________________________
___________________________
P(x) = 3x4 – 6x3 + 6x2 + 3x – 30.
You can also FACTOR a polynomial to find its factors.
Example 2A: Factoring by Grouping
A. Factor: x3 – x2 – 25x + 25.
Group terms.
Factor common monomials from each
group.
Factor out the common binomial (x – 1).
Factor the difference of squares.
Check Use the table feature of your
calculator to compare the original
expression and the factored form.
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Factoring Polynomials
Example 2B: Factoring by Grouping
B. Factor: x3 – 2x2 – 9x + 18.
Example 3A: Factoring the Sum or Difference of Two Cubes
A. Factor the expression. 4x4 + 108x
Factor out the GCF, _____.
Rewrite as the sum of cubes.
Use the rule a3 + b3 =
Example 3B: Factoring the Sum or Difference of Two Cubes
B. Factor the expression. 125d3 – 8
Rewrite as the difference of
cubes.
Use the rule a3 – b3 =
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Factoring Polynomials
Example 3C: Factoring the Sum or Difference of Two Cubes
C. Factor the expression.
8 + z6
Example 4: Geometry Application
The volume of a plastic storage box is modeled by the
function V(x) = x3 + 6x2 + 3x – 10. Identify the values of x
for which V(x) = 0, then use the graph to factor V(x).
V(x) has _____ real zeros at
x =_______________.
If the model is accurate, the
box will have no volume if
x =________________.
One corresponding factor is (x _____).
Use synthetic division to factor
the polynomial.
V(x)= (x – 1)(________________)
Write V(x) as a product.
Factor the quadratic.