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Multiplying Polynomials -Distributive property -FOIL -Box Method
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
To multiply a polynomial by a monomial, use the distributive
property and the rule for multiplying exponential expressions.
Examples: 1. Multiply: 2x(3x2 + 2x – 1).
= 6x3 + 4x2 – 2x
2. Multiply: – 3x2y(5x2 – 2xy + 7y2).
= – 3x2y(5x2 ) – 3x2y(– 2xy) – 3x2y(7y2)
= – 15x4y + 6x3y2 – 21x2y3
= 2x(3x2 ) + 2x(2x) + 2x(–1)
3
Multiply: – 2xy(8x2 +2xy – 5y2).
= – 2xy(8x2 ) – 2xy( 2xy) – 2xy(-5y2)
= – 16x3y – 6x2y2 + 10xy3
Multiplying Polynomials-Try it!
Try it!.
Multiply the polynomial by the monomial.
1) 3(x + 4)
2)
3) 26k(2k 4k 3)
2a(a 5)3x 12
22a 10a
3 212k 24k 18k
Distributive Property
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
To multiply two binomials use a method called FOIL,
which is based on the distributive property. The letters
of FOIL stand for First, Outer, Inner, and Last.
1. Multiply the first terms.
3. Multiply the inner terms.
4. Multiply the last terms.
2. Multiply the outer terms.
5. Combine like terms.
For use with the product of binomials only!
(x 3)(x 5)
First Outer Inner Last
2x
For use with the product of binomials only!
(x 3)(x 5)
First Outer Inner Last
2x 5x
For use with the product of binomials only!
(x 3)(x 5)
First Outer Inner Last
2x 5x 3x
For use with the product of binomials only!
(x 3)(x 5)
First Outer Inner Last2x 5x 3x 15
For use with the product of binomials only!
(x 3)(x 5)
First Outer Inner Last
2x 5x 3x 15
2x 2x 15
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11
Examples: 1. Multiply: (2x + 1)(7x – 5).
= 2x(7x) + 2x(–5) + (1)(7x) + (1)(– 5)
= 14x2 – 10x + 7x – 5
2. Multiply: (5x – 3y)(7x + 6y).
= 35x2 + 30xy – 21yx – 18y2
= 14x2 – 3x – 5
= 35x2 + 9xy – 18y2
= 5x(7x) + 5x(6y) + (– 3y)(7x) + (– 3y)(6y)
First Outer Inner Last
First Outer Inner Last
Try...
(m 3)(m 6)
First Outer Inner Last
2m 6m 3m 18
2m 9m 18
The third method is the Box Method. This method works for every problem!
Here’s how you do it. Multiply (3x – 5)(5x + 2)
Draw a box. Write a polynomial on the top and side of a box. It does not matter which goes where.
This will be modeled in the next problem along with
FOIL.
3x -5
5x
+2
Multiply (3x - 5)(5x + 2)
First terms:Outer terms:Inner terms:Last terms: Combine like terms.
15x2 - 19x – 10
3x -5
5x
+2
15x2
+6x
-25x
-10
You have 3 techniques. Pick the one you like the best!
15x2
+6x-25x-10
Try it! Multiply (7p - 2)(3p - 4)
First terms:Outer terms:Inner terms:Last terms: Combine like terms.
21p2 – 34p + 8
7p -2
3p
-4
21p2
-28p
-6p
+8
21p2
-28p-6p+8
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16
To multiply two polynomials, apply the distributive property.
Example: Multiply: (x – 1)(2x2 + 7x + 3).
= (x – 1)(2x2) + (x – 1)(7x) + (x – 1)(3)
= 2x3 – 2x2 + 7x2 – 7x + 3x – 3
= 2x3 + 5x2 – 4x – 3
Two polynomials can also be multiplied using a vertical format.
– 2x2 – 7x – 3 2x3 + 7x2 + 3x
2x3 + 5x2 – 4x – 3x
Multiply – 1(2x2 + 7x + 3).
Multiply x(2x2 + 7x + 3).
Add the terms in each column.
2x2 + 7x + 3x – 1
Example:
Multiply (2x - 5)(x2 - 5x + 4)You cannot use FOIL because they are not
BOTH binomials. You must use the distributive property.
2x(x2 - 5x + 4) - 5(x2 - 5x + 4)2x3 - 10x2 + 8x - 5x2 + 25x - 20
Group and combine like terms.2x3 - 10x2 - 5x2 + 8x + 25x - 20
2x3 - 15x2 + 33x - 20
x2 -5x +4
2x
-5
Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH
binomials. You must use the distributive property or box method.
2x3
-5x2
-10x2
+25x
+8x
-20
Almost
done!
Go to
the next
slide!
x2 -5x +4
2x
-5
Multiply (2x - 5)(x2 - 5x + 4) Combine like terms!
2x3
-5x2
-10x2
+25x
+8x
-20
2x3 – 15x2 + 33x - 20
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20
Example: The length of a rectangle is (x + 5) ft. The width
is (x – 6) ft. Find the area of the rectangle in terms of
the variable x.
A = L · W = Area
x – 6
x + 5
L = (x + 5) ft
W = (x – 6) ft
A = (x + 5)(x – 6 ) = x2 – 6x + 5x – 30
= x2 – x – 30
The area is (x2 – x – 30) ft2.