Multiplying Monomials and Binomials

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Multiplying Monomials and Binomials
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Polynomials Polynomials
Multiplying a Polynomial by another Polynomial
requires more than one distributing step.
Multiply (2a 7b)(3a 5b)
Distribute 2a(3a 5b) and distribute 7b(3a
5b)
6a2 10ab
21ab 35b2
Then add those products, adding like terms
6a2 10ab 21ab 35b2
6a2 31ab 35b2
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Polynomials Polynomials
An alternative is to stack the polynomials and do
long multiplication.
(2a 7b) x (3a 5b)
(2a 7b)(3a 5b)
(2a 7b) x (3a 5b)
Multiply by 5b, then by 3a
When multiplying by 3a, line up the first term
under 3a.
21ab 35b2
6a2 10ab

Add like terms
6a2 31ab 35b2
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Polynomials Polynomials
Multiply the following polynomials
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Polynomials Polynomials
-x -5
2x2 10x
2x2 9x -5
-15w 10
6w2 -4w
6w2 -19w 10
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Polynomials Polynomials
2a2 a -1
4a4 2a3 -2a2
4a4 2a3 a -1
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There are two techniques you can use for
multiplying polynomials.

• Its all about how you write itHere they are!
• Distributive Property
• FOIL

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1) Multiply. (2x 3)(5x 8)

• Using the distributive property, multiply 2x(5x
  8) 3(5x 8).
• 10x2 16x 15x 24
• Combine like terms.
• 10x2 31x 24
• A shortcut of the distributive property is called
  the FOIL method.

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The FOIL method is ONLY used when you multiply 2
binomials. It is an acronym and tells you which
terms to multiply. 2) Use the FOIL method to
multiply the following binomials(y 3)(y 7).
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(y 3)(y 7). F tells you to multiply the
FIRST terms of each binomial.

• y2

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(y 3)(y 7). O tells you to multiply the
OUTER terms of each binomial.

• y2 7y

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(y 3)(y 7). I tells you to multiply the
INNER terms of each binomial.

• y2 7y 3y

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(y 3)(y 7). L tells you to multiply the
LAST terms of each binomial.

• y2 7y 3y 21
• Combine like terms.
• y2 10y 21

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Remember, FOIL reminds you to multiply the

• First terms
• Outer terms
• Inner terms
• Last terms

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F.O.I.L.
Use the FOIL method to multiply these binomials
1) (3a 4)(2a 1) 2) (x 4)(x - 5) 3) (x
5)(x - 5) 4) (c - 3)(2c - 5) 5) (2w 3)(2w - 3)
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F.O.I.L.
Use the FOIL method to multiply these binomials
1) (3a 4)(2a 1) 6a2 3a 8a 4 6a2
11a 4 2) (x 4)(x - 5) x2 -5x 4x -20
x2 -1x -20 3) (x 5)(x - 5) x2 -5x
5x -25 x2 -25 4) (c - 3)(2c - 5) 2c2
-5c -6c 15 2c2 -11c 15 5) (2w 3)(2w
- 3) 4w2 -6w 6w -9 4w2 -9
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Try These

• (2x 1)(x 4)
• (2xy 4x)(-2y y2)

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5) 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

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3 Examples

• A) (4x 3)(2x 1)
• B) (3k - 2)(2k 1)
• C) (m 5)(3m - 4)

• A) 8x2 4x 6x 3 8x2 10x 3
• B) 6k2 3k - 4k - 2 6k2 - k - 2
• C) 3m2 - 4m 15m - 20 3m2 11m - 20

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Trinomial by a Trinomial3 ? 3 9 Terms
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Trinomial by a Trinomial3 ? 3 9 Terms
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Example Multiplying Vertically
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Example Multiplying Vertically
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Squaring a Binomial
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Raising a Binomial to a Higher Power
First find (x5)2
Now multiply this product by (x5)
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Homework 4.5 Multiplication with Polynomials
1-71 Odd page 278
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Practice Problems

• Simplify each product using the Distributive
  Property.
• 1. (x 2)(x 5) 2. (2x 1)(x 2)
• Simplify each product using FOIL.
• (r 6)(r 4) 4. (4b 2)(b 3)
• Simplify each product. Write in standard form.
• 5. (9y2 2)(y2 y 1)

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Multiplying Special Cases

• The Square of a Binomial
• (a b)2 a2 2ab b2
• (a b)2 a2 2ab b2
• The square of a binomial is the square of the
  first term plus twice the product of the two
  terms plus the square of the last term.

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Squaring a Binomial

• Find (x 7)2.
• Find (4k 3)2.

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The Difference of Squares

• (a b)(a b) a2 b2
• The product of the sum and difference of the same
  two terms is the difference of their squares.

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Multiplying Using FOIL

• Find (t3 6)(t3 6)