Multiplying and Factoring

1
Multiplying and Factoring

• Module VII, Lesson 2
• Online Algebra
• VHS_at_PWCS

2
Multiplication Review
Try these problems. Before going on with the
lesson make sure that you can get these right.

1. -2(4x)
2. 3a(5b)
3. 4x(-3x)
4. -4(2x 3)

1. -8x
2. 15ab
3. -12x2
4. -8x 12

3
Multiplying

• Problems like -4(2x 3) use the distributive
  property to solve.
• The distributive property says that
• -4(2x 3) -4(2x) - -4(3)
• -8x 12
• We can use this property to multiply any
  polynomial by a monomial.
• 5y(8y3 7y2 3y)
• 5y(8y3) 5y(7y2) 5y(3y)
• 40y4 35y3 15y2

4
Try these.

• 1. 2a(5a3 -7a2 2)
• 2a(5a3) 2a(7a2) 2a(2)
• 10a4 14a3 4a
• 2. -2x(5x 11)
• -2x(5x) -2x(11)
• -10x2 22x
• 3. 5m2(m 7) 2m(5m2 3m 7)
• 5m2(m) 5m2(7) (-2m)(5m2) (-2m)(-3m)
  (-2m)(7)
• 5m3 35m2 -10m3 6m2 -14m
• -5m3 41m2 14m

Now combine all like terms!
5
Factoring

• When two or more numbers are multiplied, each
  number is a factor of the product.
• 18(2) 36
• So 18 and 2 are factors of 36
• Other factors of 36 are
• 1 and 36, 3 and 12, 4 and 9, 6 and 6

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Greatest Common Factor

• The Greatest Common Factor of two or more numbers
  is the largest factor that the numbers have in
  common.
• For example
• The factors of 54 are
• 1, 2, 3, 6, 9, 18, 27, 54
• The factors of 63 are
• 1, 3, 7, 9, 21, 63
• The largest factor they have in common is 9, so 9
  is the GCF of 54 and 63.

7
Find the GCF of each pair of numbers.

• 15 and 50
• The factors of 15 are
• 1, 3, 5, 15
• The factors of 50 are
• 1, 2, 5, 10, 25, 50
• The largest factor that 15 and 50 have in common
  is 5, so 5 is their GCF.

• 88 and 40
• The factors of 88 are
• 1, 2, 4, 8, 11, 22, 44, 88
• The factors of 40 are
• 1, 2, 4, 5, 8, 10, 20, 40
• The largest factor that 88 and 40 have in common
  is 8, so 8 is their GCF.

8
Find the GCF of12ab and 4a2b2

• When variables are involved the GCF will be the
  GCF of the coefficients and the lowest power of
  each variable.
• The factors of 12 are
• 1, 2, 3, 4, 6, 12
• The factors of 4 are
• 1, 2, 4
• The lowest power of a is 1
• The lowest power of b is 1
• Putting all that together the GCF of 12ab and
  4a2b2 is
• 4ab

9
Find the GCF of the following.

1. 50n4 and 40n3
2. 56x2y and 49xy
3. 12mn, 10mn and 15mn

1. 10n3
2. 7xy
3. mn

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Factoring Polynomials

• Factoring a polynomial can be done several ways.
  The first step is always to factor out the GCF.
  To do this we must
• Find the GCF of the terms of the polynomial.
• Divide each term of the polynomial by the GCF.
• Write as multiplication.

11
Factor 11x 44x2y

• Find the GCF of the terms of the polynomial.
• The GCF of the terms is 11x
• Divide each term of the polynomial by the GCF.
• Write as multiplication.
• 11x(1 4xy)

12
Try to factor these.

• 16xy2 24y2z 40y2
• Find the GCF.
• 8y2
• Divide each term by the GCF.
• Write as multiplication.
• 8y2(2x 3z 5)

• 28a2 21a 35
• Find the GCF.
• 7
• Divide each term by the GCF.
• Write as multiplication.
• 7(4a2 3a 5)

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Wrap-Up

• We can use the distributive property to multiply
  a polynomial by a monomial.
• Numbers that are multiplied together are called
  factors.
• Greatest Common Factors are the largest factor
  that 2 or more numbers have in common.
• To factor a polynomial, find the GCF of its terms
  and divide each term by that GCF. Finally write
  as multiplication.