Multiplying Polynomials Handout
1. Multiplying a Monomial by a Monomial
Multiply the coefficients and add the exponents for variables with the same
base.
Example: 3a
2
·2a
5
= (3·2)a
2+5
= 6a
7
Simplify the following:
1. (8x
3
)(2x
5
)
2. (3xy
3
)(6xz)
3. (-4x
2
)(7y
5
z
3
)
4. (6x
2
)(-3y
3
)(2z
5
)
5. (3a
3
)(-5a
4
)(2a
2
)
6. (4x
2
y
3
)(7x
3
y
5
)
7. (-6x
3
y
4
)(-2x
5
z
2
)
8. (7a
2
b
2
c
4
)(-3a
4
c
2
d
4
)
2. Multiplying a Monomial by a Polynomial
Use the distributive property to remove the parentheses and simplify.
Example:
3a
3
(2a
4
-4a
5
) = 3a
3
·2a
4
– 3a
3
·4a
5
= (3·2)a
3+4
– (3·4)a
3+5
= 6a
7
– 12a
8
Simplify the following:
1. 2(x-7)
2. -6(3h+5)
3. 2x(x+20)
4. 7x(6x+4y)
5. 3n
2
(n
2
-6n)
6. 2x
3
(3k
2
+5k-4)
7. 4y
3
(2xy
2
-x
2
y
2
+y)
8. 9x
2
y(x
2
+xy+y
2
)
1
Multiplying Polynomials Handout
3. Multiply a Binomial by a Binomial using FOIL and the box methods.
FOIL Method
Box Method
FOIL Method: Series of four steps using
the distributive property.
(2x + 7)(3x - 5)
(F) FIRST: 2x·3x = 6x
2
(O) Outer: 2x·(-5) = -10x
(I) Inner: 7·3x = 21x
(L) Last: 7·(-5) = -35
Combine Like Terms:
6x
2
-10x+21x-35 = 6x
2
+11x-35
Box Method: Draw a box with four
squares. Place the binomials on the
outside of the boxes.
3x
-5
2x
+7
Multiply terms:
3x
-5
2x 2x·3x 2x(-5)
+7 7(3x) 7(-5)
3x
-5
2x 6x
2
-10x
+7 21x -35
Write the product as an expression and
combine Like Terms:
6x
2
-10x+21x-35 = 6x
2
+11x-35
Use the FOIL and Box Method to find each product:
1. (a+3)(a-2)
2. (t-3)(2t+3)
3. (2x+3)(2x-3)
4. (3s-2t)(2s-3t)
5. (x-5y)(a+2y)
6. (2y-5)(3y+7)
7. (2b-1)(2b-1)
8. 2(x+y)(x-y)
FIRST
OUTER
INNER
LAST
2
Multiplying Polynomials Handout
4. Multiply a Binomial by a Polynomial.
Distributive Method
Box Method
Distribute the binomial to each term in
the polynomial.
(2x - 7)(3x
2
+ x - 5)
Then use the method to multiply a
monomial by a binomial:
(2x-7)(3x
2
) = 6x
3
– 21x
2
(2x-7)(x) = 2x
2
-7x
(2x-7)(-5) = -10x+35
Combine Like Terms:
6x
3
-21x
2
+2x
2
-7x-10x+35 =
6x
3
-19x
2
-17x+35
Box Method: Draw a box with six
squares. Place the binomials on the
outside of the boxes.
3x
2
+x
-5
2x
-7
Multiply terms:
3x
2
+x
-5
2x 2x·3x
2
2x(x) 2x(-5)
-7 -7(3x
2
) -7(x) -7(-5)
3x
2
+x
-5
2x 6x
3
2x
2
-10x
-7 -21x
2
-7x +35
Write the product as an expression and
combine Like Terms:
6x
3
-21x
2
+2x
2
-7x-10x+35 =
6x
3
-19x
2
-17x+35
Use both methods to find each product.
1. (3x - 2)(2x
2
– x + 2)
2. (-11x + 3) (-10x
2
- 7x - 9)
3. (x - 3) (x
2
+ 3x + 9)
4. (7x + 3) (7x
2
+ 3x + 10)
5. (-x + 1) (4x
2
- x + 8)
6. (-4x - 3) (-x
2
- 2x - 1)
7. (2x+1) (4x
2
- 2x + 1)
8. (-6x
4
+ 5x
2
+ 3x) (x + 4)
3
Multiplying Polynomials Handout KEY
1. Multiplying a Monomial by a Monomial
Multiply the coefficients and add the exponents for variables with the same
base.
Example: 3a
2
·2a
5
= (3·2)a
2+5
= 6a
7
Simplify the following:
1. (8x
3
)(2x
5
) 16x
8
2. (3xy
3
)(6xz) 18x
2
y
3
z
3. (-4x
2
)(7y
5
z
3
) -28x
2
y
5
z
3
4. (6x
2
)(-3y
3
)(2z
5
) -36x
2
y
3
z
5
5. (3a
3
)(-5a
4
)(2a
2
) -30a
9
6. (4x
2
y
3
)(7x
3
y
5
) 28x
5
y
8
7. (-6x
3
y
4
)(-2x
5
z
2
) 12x
8
y
4
z
2
8. (7a
2
b
2
c
4
)(-3a
4
c
2
d
4
) -21z
6
b
2
c
6
d
4
2. Multiplying a Monomial by a Polynomial
Use the distributive property to remove the parentheses and simplify.
Example:
3a
3
(2a
4
-4a
5
) = 3a
3
·2a
4
– 3a
3
·4a
5
= (3·2)a
3+4
– (3·4)a
3+5
= 6a
7
– 12a
8
Simplify the following:
1. 2(x-7) 2x-14
2. -6(3h+5) -18h-30
3. 2x(x+20) 2x
2
+40x
4. 7x(6x+4y) 42x
2
+28xy
5. 3n
2
(n
2
-6n) 3n
4
-18n
3
6. 2x
3
(3k
2
+5k-4) 6x
3
k
2
+10x
3
k-8x
3
7. 4y
3
(2xy
2
-x
2
y
2
+y) 8xy
5
-4x
2
y
5
+4y
4
8. 9x
2
y(x
2
+xy+y
2
)
9x
4
y+9x
3
y
2
+9x
2
y
3
1
Multiplying Polynomials Handout KEY
3. Multiply a Binomial by a Binomial using FOIL and the box methods.
FOIL Method
Box Method
FOIL Method: Series of four steps using
the distributive property.
(2x + 7)(3x - 5)
(F) FIRST: 2x·3x = 6x
2
(O) Outer: 2x·(-5) = -10x
(I) Inner: 7·3x = 21x
(L) Last: 7·(-5) = -35
Combine Like Terms:
6x
2
-10x+21x-35 = 6x
2
+11x-35
Box Method: Draw a box with four
squares. Place the binomials on the
outside of the boxes.
3x
-5
2x
+7
Multiply terms:
3x
-5
2x 2x·3x 2x(-5)
+7 7(3x) 7(-5)
3x
-5
2x 6x
2
-10x
+7 21x -35
Write the product as an expression and
combine Like Terms:
6x
2
-10x+21x-35 = 6x
2
+11x-35
Use the FOIL and Box Method to find each product:
1. (a+3)(a-2) a
2
+a-6
2. (t-3)(2t+3) 2t
2
-3t-9
3. (2x+3)(2x-3) 4x
2
-9
4. (3s-2t)(2s-3t) 6s
2
-13st+6t
2
5. (x-5y)(a+2y) xa+2xy-5ay-10y
2
6. (2y-5)(3y+7) 6y
2
-y-35
7. (2b-1)(2b-1) 4b
2
-4b+1
8. 2(x+y)(x-y) 2x
2
-2y
2
FIRST
OUTER
INNER
LAST
2
Multiplying Polynomials Handout KEY
4. Multiply a Binomial by a Polynomial.
Distributive Method
Box Method
Distribute the binomial to each term in
the polynomial.
(2x - 7)(3x
2
+ x - 5)
Then use the method to multiply a
monomial by a binomial:
(2x-7)(3x
2
) = 6x
3
– 21x
2
(2x-7)(x) = 2x
2
-7x
(2x-7)(-5) = -10x+35
Combine Like Terms:
6x
3
-21x
2
+2x
2
-7x-10x+35 =
6x
3
-19x
2
-17x+35
Box Method: Draw a box with six
squares. Place the binomials on the
outside of the boxes.
3x
2
+x
-5
2x
-7
Multiply terms:
3x
2
+x
-5
2x 2x·3x
2
2x(x) 2x(-5)
-7 -7(3x
2
) -7(x) -7(-5)
3x
2
+x
-5
2x 6x
3
2x
2
-10x
-7 -21x
2
-7x +35
Write the product as an expression and
combine Like Terms:
6x
3
-21x
2
+2x
2
-7x-10x+35 =
6x
3
-19x
2
-17x+35
Use both methods to find each product.
1. (3x - 2)(2x
2
– x + 2) 6x
3
-7x
2
+8x-4
2. (-11x + 3) (-10x
2
- 7x - 9) 110x
3
+47x
2
+78x-27
3. (x - 3) (x
2
+ 3x + 9) x
3
- 27
4. (7x + 3) (7x
2
+ 3x + 10) 49x
3
+42x
2
+79x+30
5. (-x + 1) (4x
2
- x + 8) 4x
3
+5x
2
-9x-8
6. (-4x - 3) (-x
2
- 2x - 1) 4x
3
+11x
2
+10x+3
7. (2x+1) (4x
2
- 2x + 1) 8x
3
+1
8. (-6x
4
+ 5x
2
+ 3x) (x + 4) -6x
5
-24x
4
+5x
3
+23x
2
+12x
3
