6.7 Multiplying a Polynomial by a Monomial

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6.7 Multiplying a Polynomial by a Monomial

• CORD Math
• Mrs. Spitz
• Fall 2006

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Objectives

• After studying this lesson, you should be able
  to
• Multiply a polynomial by a monomial, and
• Simplify expressions involving polynomials

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Assignment

• 6.7 Worksheet

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Application

• The worlds largest swimming pool is the Orthlieb
  Pool in Casablanca, Morocco. It is 30 meters
  longer than 6 times its width. Express the area
  of the swimming pool algebraically.
• To find the area of the swimming pool, multiply
  the length by the width. Let w represent the
  width. Then 6w 30 represents the length.

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What does this look like?
30
6w 30
6w
A 30w
w
A w(6w 30)
w
A 6w2

• This diagram of the swimming pool shows that the
  area is w(6w 30) square meters.

• This diagram of the same swimming pool shows that
  the area is (6w2 30w) square meters.

• Since the areas are equal, w(6w 30) 6w2 30w
• The application above shows how the distributive
  property can be used to multiply a polynomial by
  a monomial.

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Ex. 1 Find 5a(3a2 4)You can multiply either
horizontally or vertically.

• A. Use the distributive property.
• 5a(3a2 4) 5a(3a2) 5a(4)
• 15a3 20a

• B. Multiply each term by 5a.
• 3a2 4
• (x) 5a
• 15a3 20a

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Ex. 2 Find 2m2(5m2 7m 8)

• Use the distributive property.
• 2m2(5m2 7m 8) 2m2(5m2) 2m2(-7m) 2m2(8)
• 10m4 - 14m3 16m2

Ex. 3 Find -3xy(2x2y 3xy2 7y3).
Use the distributive property. -3xy (2x2y 3xy2
7y3) -3xy(2x2y) (-3xy)(3xy2)(-3xy)(-7y3).
- 6x3y2 - 9x2y3 21xy4
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Ex. 4 Find the measure of the area of the
shaded region in simplest terms.
5a2 3a - 2
3a2 - 7a 1
2a
8

• 2a(5a2 3a 2) - 8(3a2 7a 1
• 10a3 6a2 4a - 24a2 56a 8
• 10a3 - 18a2 52a 8
• The measure of the area of the shaded region is
  10a3 - 18a2 52a 8.

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Ex. 5 Many equation contain polynomials that
must be added, subtracted, or multiplied before
the equation is solved.

• Solve x(x 3) 4x 3 8x 4 x (3 x)
• x(x 3) 4x 3 8x 4 x (3 x)
• x2 3x 4x 3 8x 4 3x x2 Multiply
• x2 x 3 x2 11x 4 Combine like terms
• x 3 11x 4 Subtract x2 from each side.
• - 3 10x 4 Subtract x from each side.
• - 7 10x Subtract 4 from each side.

-
x
Check this result.