Transcript Multiplying and Factoring

Multiplying and Factoring
Section 8-2
Goals
Goal
• To multiply a monomial by
a polynomial.
• To factor a monomial from
a polynomial.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to solve
simple problems.
Level 4 – Use the goals to solve
more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• None
Multiplying Polynomials
To multiply monomials and polynomials,
you will use some of the properties of
exponents that you learned earlier in this
chapter.
Multiplying Monomials
Multiply.
A. (6y3)(3y5)
(6y3)(3y5)
(6 • 3)(y3 • y5)
18y8
Group factors with like bases together.
Multiply.
B. (3mn2) (9m2n)
(3mn2)(9m2n)
(3 • 9)(m • m2)(n2 • n)
27m3n3
Group factors with like bases together.
Multiply.
Multiplying Monomials
Multiply.
(4s2t2)(st)(-12st2)
(4s2t2)(st)(-12st2)
(4 • -12)(s2 • s • s)(t2 • t • t2)
-48s4t5
Group factors with like bases
together.
Multiply.
Remember!
When multiplying powers with the same base,
keep the base and add the exponents.
x2  x3 = x2+3 = x5
Your Turn:
Multiply.
a. (3x3)(6x2)
(3x3)(6x2)
Group factors with like bases together.
(3 • 6)(x3 • x2)
18x5
Multiply.
b. (2r2t)(5t3)
Group factors with like bases together.
(2r2t)(5t3)
(2 • 5)(r2)(t3 • t)
10r2t4
Multiply.
Your Turn:
Multiply.
c.
(3x2y)(2x3z2)(y4z5)
(3x2y)(2x3z2)(y4z5)
(3 • 2)(x2 • x3)(y • y4)(z2 • z5)
6x5y5z7
Group factors with like
bases together.
Multiply.
Multiplying Monomials
and Polynomials
To multiply a polynomial by a monomial,
use the Distributive Property.
Example: Multiplying a
Polynomial by a Monomial
Multiply.
4(3x2 + 4x – 8)
4(3x2 + 4x – 8)
Distribute 4.
(4)3x2 +(4)4x – (4)8
Multiply.
12x2 + 16x – 32
Example: Multiplying a
Polynomial by a Monomial
Multiply.
6pq(2p – q)
(6pq)(2p – q)
Distribute 6pq.
(6pq)2p + (6pq)(–q)
Group like bases together.
(6  2)(p  p)(q) + (–1)(6)(p)(q  q)
12p2q – 6pq2
Multiply.
Your Turn:
Multiply.
a. 2(4x2 + x + 3)
2(4x2 + x + 3)
Distribute 2.
2(4x2) + 2(x) + 2(3)
Multiply.
8x2 + 2x + 6
Your Turn:
Multiply.
b. 3ab(5a2 + b)
3ab(5a2 + b)
Distribute 3ab.
(3ab)(5a2) + (3ab)(b)
(3  5)(a  a2)(b) + (3)(a)(b  b)
15a3b + 3ab2
Group like bases
together.
Multiply.
Your Turn:
Multiply.
c. 5r2s2(r – 3s)
5r2s2(r – 3s)
Distribute 5r2s2.
(5r2s2)(r) – (5r2s2)(3s)
(5)(r2  r)(s2) – (5  3)(r2)(s2  s)
5r3s2 – 15r2s3
Group like bases
together.
Multiply.
Helpful Hint
When multiplying a polynomial by a negative
monomial, be sure to distribute the negative sign.
Your Turn:
Multiply.
d. –5y3(y2 + 6y – 8)
–5y3(y2 + 6y – 8)
–5y5 – 30y4 + 40y3
Multiply each term in
parentheses by –5y3.
Factoring
• Factoring a polynomial reverses the
multiplication process (factoring is
unmultiplying).
• When factoring a monomial from a
polynomial, the first step is to find the
greatest common factor (GCF) of the
polynomial’s terms.
Greatest Common Factor
Factors that are shared by two or more whole numbers are
called common factors. The greatest of these common factors is
called the greatest common factor, or GCF.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 32: 1, 2, 4, 8, 16, 32
Common factors: 1, 2, 4
The greatest of the common factors is 4.
Example: GCF of Two Numbers
Find the GCF of each pair of numbers.
100 and 60
List the factors.
factors of 100: 1, 2, 4,
5, 10, 20, 25, 50, 100
List all the factors.
factors of 60: 1, 2, 3, 4, 5,
6, 10, 12, 15, 20, 30, 60
Circle the GCF.
The GCF of 100 and 60 is 20.
Your Turn:
Find the GCF of each pair of numbers.
12 and 16
List the factors.
factors of 12: 1, 2, 3, 4, 6, 12
factors of 16: 1, 2, 4, 8, 16
The GCF of 12 and 16 is 4.
List all the factors.
Circle the GCF.
GCF of Monomials
You can also find the GCF of monomials that
include variables. To find the GCF of monomials,
write the prime factorization of each coefficient and
write all powers of variables as products. Then find
the product of the common factors.
Example: GCF of a
Monomial
Find the GCF of each pair of monomials.
15x3 and 9x2
15x3 = 3  5  x  x  x
9x2 = 3  3  x  x
3
x  x = 3x2
Write the factorization of each
coefficient and write powers as
products.
Align the common factors.
Find the product of the common
factors.
The GCF of 3x3 and 6x2 is 3x2.
Example: GCF of a
Monomial
Find the GCF of each pair of monomials.
8x2 and 7y3
8x2 = 2  2  2 
7y3 =
7
xx
Write the factorization of
each coefficient and write
powers as products.
yyy
Align the common factors.
The GCF 8x2 and 7y is 1.
There are no common
factors other than 1.
Helpful Hint
If two terms contain the same variable raised to
different powers, the GCF will contain that variable
raised to the lower power.
Your Turn:
Find the GCF of each pair of monomials.
18g2 and 27g3
18g2 = 2  3  3 
27g3 =
gg
333ggg
33
gg
The GCF of 18g2 and 27g3 is 9g2.
Write the factorization of each
coefficient and write powers as
products.
Align the common factors.
Find the product of the common
factors.
Your Turn:
Find the GCF of each pair of monomials.
Write the
factorization of
each coefficient
and write powers
as products.
16a6 and 9b
16a6 = 2  2  2  2  a  a  a  a  a  a
9b =
The GCF of 16a6 and 7b is 1.
33b
Align the common
factors.
There are no common factors
other than 1.
Your Turn:
Find the GCF of each pair of monomials.
8x and 7x2
8x = 2  2  2  x
7v2 = 7 
xx
The GCF of 8x and 7x2 is x.
Write the prime factorization of each
coefficient and write powers as
products.
Align the common factors.
Factoring out a Monomial
Recall that the Distributive Property states that ab + ac =a(b + c).
The Distributive Property allows you to “factor” out the GCF of
the terms in a polynomial to write a factored form of the
polynomial.
A polynomial is in its factored form when it is written as a
product of monomials and polynomials that cannot be factored
further. The polynomial 2(3x – 4x) is not fully factored because
the terms in the parentheses have a common factor of x.
Example: Factoring the GCF
Factor each polynomial. Check your answer.
2x2 – 4
2x2 = 2  x  x
4=22
Find the GCF.
2
2x2 – (2  2)
The GCF of 2x2 and 4 is 2.
2(x2 – 2)
Check 2(x2 – 2)
2x2 – 4

Write terms as products using the GCF as a
factor.
Use the Distributive Property to factor out the
GCF.
Multiply to check your answer.
The product is the original polynomial.
Writing Math
Aligning common factors can help you find the greatest
common factor of two or more terms.
Example: Factoring the GCF
Factor each polynomial. Check your answer.
8x3 – 4x2 – 16x
8x3 = 2  2  2 
xxx
4x2 = 2  2 
xx
16x = 2  2  2  2  x
22
x = 4x
2x2(4x) – x(4x) – 4(4x)
4x(2x2 – x – 4)
Check
4x(2x2 – x – 4)
8x3 – 4x2 – 16x

Find the GCF.
The GCF of 8x3, 4x2, and 16x is 4x.
Write terms as products using the
GCF as a factor.
Use the Distributive Property to factor
out the GCF.
Multiply to check your answer.
The product is the original polynomials.
Example: Factoring the GCF
Factor each polynomial.
–14x – 12x2
– 1(14x + 12x2)
14x = 2 
7x
12x2 = 2  2  3 
xx
2
–1[7(2x) + 6x(2x)]
–1[2x(7 + 6x)]
–2x(7 + 6x)
x = 2x
Both coefficients are negative.
Factor out –1.
Find the GCF.
The GCF of 14x and 12x2 is 2x.
Write each term as a product using
the GCF.
Use the Distributive Property to
factor out the GCF.
Caution!
When you factor out –1 as the first step, be sure to include it
in all the other steps as well.
Example: Factoring the GCF
Factor each polynomial.
3x3 + 2x2 – 10
xxx
3x3 = 3
2x2 =
10 =
2
Find the GCF.
xx
25
3x3 + 2x2 – 10
There are no common factors
other than 1.
The polynomial cannot be factored further.
Your Turn:
Factor each polynomial. Check your answer.
5b + 9b3
5b = 5 
b
Find the GCF.
9b = 3  3  b  b  b
b
5(b) + 9b2(b)
b(5 + 9b2)
Check b(5 + 9b2)
5b + 9b3

The GCF of 5b and 9b3 is b.
Write terms as products using the
GCF as a factor.
Use the Distributive Property to factor
out the GCF.
Multiply to check your answer.
The product is the original
polynomial.
Your Turn:
Factor each polynomial.
9d2 – 82
9d2 = 3  3  d  d
82 =
9d2
–
82
Find the GCF.
222222
There are no common factors
other than 1.
The polynomial cannot be factored further.
Example: Factoring the GCF
Factor each polynomial.
–18y3 – 7y2
– 1(18y3 + 7y2)
18y3 = 2  3  3  y  y  y
7y2 = 7 
–1[18y(y2)
+
7(y2)]
–1[y2(18y + 7)]
–y2(18y + 7)
Both coefficients are negative.
Factor out –1.
yy
Find the GCF.
y  y = y2
The GCF of 18y3 and 7y2 is y2.
Write each term as a product using
the GCF.
Use the Distributive Property to
factor out the GCF..
Your Turn:
Factor each polynomial.
8x4 + 4x3 – 2x2
8x4 = 2  2  2  x  x  x  x
4x3 = 2  2  x  x  x
2x2 = 2 
xx
2
x  x = 2x2
4x2(2x2) + 2x(2x2) –1(2x2)
2x2(4x2 + 2x – 1)
Check 2x2(4x2 + 2x – 1)
8x4 + 4x3 – 2x2
Find the GCF.
The GCF of 8x4, 4x3 and –2x2 is 2x2.
Write terms as products using the
GCF as a factor.
Use the Distributive Property to factor
out the GCF.
Multiply to check your answer.
The product is the original polynomial.
To write expressions for the length and width of a rectangle
with area expressed by a polynomial, you need to write the
polynomial as a product. You can write a polynomial as a
product by factoring it.
Example: Application
The area of a court for the game squash is 9x2 + 6x m2.
Factor this polynomial to find possible expressions for
the dimensions of the squash court.
A = 9x2 + 6x
The GCF of 9x2 and 6x is 3x.
= 3x(3x) + 2(3x)
Write each term as a product using the
GCF as a factor.
= 3x(3x + 2)
Use the Distributive Property to factor
out the GCF.
Possible expressions for the dimensions of the squash court are
3x m and (3x + 2) m.
Your Turn:
What if…? The area of the solar panel on another
calculator is (2x2 + 4x) cm2. Factor this polynomial
to find possible expressions for the dimensions of the
solar panel.
A = 2x2 + 4x
The GCF of 2x2 and 4x is 2x.
= x(2x) + 2(2x)
Write each term as a product using the
GCF as a factor.
= 2x(x + 2)
Use the Distributive Property to factor
out the GCF.
Possible expressions for the dimensions of the solar panel are 2x cm,
and (x + 2) cm.
Assignment
• 8-2 Exercises Pg. 493 - 494: #10 – 40 even