To simplify a product of monomials
(4x)(2x)
• Use the Commutative and Associative Properties of Multiplication to group the numerical coefficients and to group like variable;
(4x)(2x) = (4 · 2)(x · x ) =
• Find the product of the numbers
(4 · 2) = 8
• Use the properties of exponents to simplify the variable product.
(x · x) = x 1 · x 1 = x 1+1 = x 2
So your answer is (4x)(2x) = 8x 2

You can also use the Distributive Property for multiplying powers with the same base when multiplying a polynomial by a monomial.
Simplify -4y 2 (5y 4 – 3y 2 + 2) Remember,
Multiply powers with the same base:
3 5 · 3 4 = 3 5 + 4 = 3 9
-4y 2 (5y 4 – 3y 2 + 2) =
-4y 2 (5y 4 ) – 4y 2 (-3y 2 ) – 4y 2 (2) =
Use the Distributive Property
-20y 2 + 4 + 12y 2 + 2 – 8y 2 =
Multiply the coefficients and add the
-20y 6 + 12y 4 – 8y 2 exponents of powers with the same base.

Simplify each product.
a) 4b(5b 2 + b + 6) b) -7h(3h 2 – 8h – 1) c) 2x(x 2 – 6x + 5) d) 4y 2 (9y 3 + 8y 2 – 11)
Remember,
Multiplying powers with the same base.
20b 3 + 4b 2 + 24b
-21h 3 + 56h 2 + 7h
2x 3 -12x 2 + 10x
36y 5 + 32y 4 – 44y 2

Factoring a Monomial from a Polynomial
Find the GCF of the terms of:
4x 3 + 12x 2 – 8x
List the prime factors of each term.
4x 3 = 2 · 2 · x · x x
12x 2 = 2 · 2 · 3 · x · x
8x = 2 · 2 · 2 · x
Factoring a polynomial reverses the multiplication process.
To factor a monomial from a polynomial, first find the greatest
common factor (GCF) of its terms.
The GCF is 2 · 2 · x or 4x .
The GCF is what the terms all have in common!

Find the GCF of the terms of each polynomial.
a) 5v 5 + 10v 3 5v 3 b) 3t 2 – 18 3 c) 4b 3 – 2b 2 – 6b 2b d) 2x 4 + 10x 2 – 6x 2x

Factoring Out a
Monomial
Factor 3x 3 – 12x 2 + 15x
Step 1
Find the GCF
3x 3 = 3 · x · x · x
12x 2 = 2 · 2 · 3 · x · x
15x = 3 · 5 · x
The GCF is 3 · x or 3x
To factor a polynomial completely, you must factor until there are no common factors other than 1.
Step 2
Factor out the GCF
3x 3 – 12x 2 + 15x
= 3x (x 2 ) + 3x (-4x) + 3x (5)
= 3x (x 2 – 4x + 5)

Use the GCF to factor each polynomial.
a) 8x 2 – 12x 4x(2x-3) b) 5d 3 + 10d 5d(d 2 + 2) c) 6m 3 – 12m 2 – 24m 6m(m 2 -2m -4) d) 4x 3 – 8x 2 + 12x 4x(x 2 –2x +3)
Try to factor mentally by scanning the coefficients of each term to find the GCF.
Next, scan for the least power of the variable.

Using the
Distributive
Property
Distribute x + 4
As with the other examples we have seen, we can also use the Distributive
Property to find the product of two binomials.
Now Distribute 2x and 3
Simplify: (2x + 3)(x + 4)
2x 2 +8x +3x +12
2x 2 + 8x + 3x + 12 =
2x 2 + 11x + 12

Multiplying using FOIL
Another way to organize multiplying two binomials is to use FOIL, which stands for,
“ First , Outer , Inner , Last ”. The term FOIL is a memory device for applying the Distributive
Property to the product of two binomials.
Simplify (3x – 5)(2x + 7)
First Outer Inner Last
(3x – 5)(2x + 7)
= 6x 2 + 21x 10x - 35
= 6x 2 + 11x 35
The product is 6x 2 + 11x - 35

Simplify each product.
a) (6h – 7)(2h + 3) b) (5m + 2)(8m – 1) c) (9a – 8)(7a + 4)
12h 2 +4h - 21
40m 2 + 11m - 2 d) (y – 3)(y + 3) e) (2x-1) 2
63a 2 -20a - 32 y 2 + 3y -3y – 9 or y 2 – 9
(2x – 1)(2x – 1)
4x 2 - 2x - 2x + 1
4x 2 - 4x + 1

Applying
Multiplication of
Polynomials.
Find the area of the white region.
Simplify.
2x + 5 x + 2 x
Use the FOIL method to simplify (2x + 5)(3x + 1) area of BIG rectangle =
( 2x + 5 )( 3x + 1 ) area of little rectangle = x ( x + 2 ) area of white region
= area of BIG rectangle – area of black rectangle
(2x + 5)(3x + 1) – x(x + 2) =
6x 2 + 15x + 2x + 5 – x 2 – 2x =
Combine like terms…
6x 2 – x 2 + 15x + 2x – 2x + 5 =
5x 2 + 15x + 5
Use the Distributive Property to simplify –x(x + 2)

Find the area of the shaded region.
Simplify.
Find the area of the white region. Simplify.
5x + 8
5x area of BIG rectangle =
( 5x + 8 )( 6x + 2 ) area of little rectangle =
5x ( x + 6 ) area of white region
= area of BIG rectangle – area of black rectangle
( 5x + 8 )( 6x + 2 ) 5x ( x + 6 )=
30x 2 + 10x + 48x + 16 – 5x 2 –30x=
Combine like terms…
30x 2 + 10x + 48x + 16 – 5x 2 –30x = x + 6
Answer: 25x 2 + 28x + 16