Download M. MALTBY INGERSOLL APRIL 4, 2017 UNIT 2: FACTORS AND

APRIL 4, 2017
UNIT 2: FACTORS AND PRODUCTS
SECTION 3.3:
COMMON FACTORS OF
A POLYNOMIAL
M. MALTBY INGERSOLL
NUMBERS, RELATIONS AND FUNCTIONS 10
1
WHAT'S THE POINT OF TODAY'S LESSON?
We will begin working on the NRF 10 Specific Curriculum Outcome (SCO) "Algebra and Numbers 5" OR "AN5" which states:
"Demonstrate an understanding of common factors and trinomial factoring."
2
What does THAT mean???
SCO AN5 means that we will:
* determine the common factors in the terms of a polynomial and express the polynomial in factored form
* factor a polynomial that is a "difference of squares" and explain why it is a special case of trinomial factoring where b = 0
* identify and explain errors in a polynomial factorization
* factor a polynomial and verify by multiplying the factors
* explain, using examples, the relationship
between multiplication and factoring of
polynomials
* generalize and explain strategies used to
factor a trinomial
* express a polynomial as a product of its
factors
3
WARM­UP:
EXPAND: (5x ­ 7)3
SOLUTION:
(5x ­ 7)3
= (25x2 ­ 70x + 49)(5x ­ 7)
= 125x3 ­ 175x2 ­ 350x2 + 490x + 245x ­ 343
= 125x3 ­ 525x2 + 735x ­ 343
4
HOMEWORK QUESTIONS???
(pages 186 / 187, #6(i, iii, v, vii) TO #11 and #18)
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IN ARITHMETIC:
IN ALGEBRA:
Multiply factors to form
a product:
Expand an expression
to form a product:
(4)(7) = 28
3(2 ­ 5a) = 6 ­ 15a
Factor a number by
writing it as a product
of factors:
Factor a polynomial
by writing it as a product of factors:
28 = (4)(7)
6 ­ 15a = 3(2 ­ 5a)
(EXPANDING and
FACTORING are
inverse processes.)
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MULTIPLYING MONOMIALS BY BINOMIALS:
Last week, we built gradually on what we knew in grade 9. We EXPANDED expressions to form
products.
For example:
3(x + 4)
= 3(x) + 3(4) (We use the DISTRIBUTIVE PROPERTY here.)
= 3x + 12
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FACTORING BINOMIALS:
For example, FACTOR the following binomial:
3x + 12
= (3)(x) + (3)(2)(2)
= 3(x + 4)
(This is a PRODUCT; it is the answer to a multiplication.)
(Prime factorization was used here; GCF = 3.)
(This is the FACTORED form of the product
using the GCF of 3.)
To check that your answer is correct, EXPAND!
3(x + 4)
= 3x + 12
8
FACTORING BINOMIALS:
This is how factoring actually works without
prime factorization:
3x + 12
= 3(x + 4)
(Ask yourself: What is the GCF of 3 and 12? 3!)
(Mentally divide 3x by 3 and 12 by 3.)
9
MULTIPLYING MONOMIALS BY BINOMIALS:
Again, we built gradually on what we knew in
grade 9...
For example:
6x(2x + 3)
= (6)(2)(x)(x) + (6)(3)(x)
= 12x2 + 18x
10
FACTORING BINOMIALS:
Now, let's FACTOR the following binomial:
12x2 + 18x
= (2)(2)(3)(x)(x) + (2)(3)(3)(x)
= 6x(2x + 3)
(GCF = 6x)
Check:
6x(2x + 3)
= 12x2 + 18x
11
FACTORING BINOMIALS:
This is how factoring actually works without
prime factorization:
12x2 + 18x
= 6x(2x + 3)
(Ask yourself: What is the GCF of 12 and 18? 6!
What is the GCF of x2 and x? x! GCF = 6x)
2
(Mentally divide 12x by 6x and 18x by 6x.)
12
FACTORING BINOMIALS:
We have to deal with negative terms when factoring binomials.
For example, FACTOR:
3y2 ­ 9y
= 3y(y ­ 3)
(Ask yourself: What is the GCF of 3 and 9? 3!
What is the GCF of y2 and y? y! GCF = 3y)
2
(Mentally divide 3y by 3y and ­9y by 3y.)
Check:
3y(y ­ 3)
= 3y2 ­ 9y
13
FACTORING BINOMIALS:
We have to deal with negative terms when factoring binomials.
NOTE: We ensure that the first term inside the
brackets is POSITIVE.
For example, FACTOR:
­8a2 ­ 12a
= ­4a(2a + 3)
(Ask yourself: What is the GCF of 8 and 12? 4!
What is the GCF of a2 and a? a! Now, to ensure
a positive first term inside the brackets, use a GCF = ­4a)
2
(Mentally divide ­8a by ­4a and ­12a by ­4a.)
Check:
­4a(2a + 3)
= ­8a2 ­ 12a
14
MULTIPLYING MONOMIALS BY TRINOMIALS:
We also learned this process last week.
For example:
6(x + 3y + 2)
= (6)(x) + (6)(3)(y) + (6)(2)
= 6x + 18y + 12
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FACTORING TRINOMIALS:
For example, FACTOR:
6x + 18y + 12
= 6(x + 3y + 2)
(Ask yourself: What is the GCF of 6, 18 and 12? 6!
(Mentally divide 6x by 6, 18y by 6 and 12 by 6.)
Check:
6(x + 3y + 2)
= 6x + 18y + 12
16
FACTORING TRINOMIALS:
For example, FACTOR:
18a2 ­ 27a + 36ab (Ask yourself: What is the GCF of 18, 27 and 36? 9!
What is the GCF of a2, a and ab? a! GCF = 9a)
= 9a(2a ­ 3 + 4b)
2
(Mentally divide 18a by 9a, ­27a by 9a and
36ab by 9a.)
Check:
9a(2a ­ 3 + 4b)
= 18a2 ­ 27a + 36ab
17
FACTORING TRINOMIALS:
For example, FACTOR:
­12x3y ­ 20xy 2 ­ 16x2y2
= ­4xy(3x 2 + 5y + 4xy)
Check:
­4xy(3x2 + 5y + 4xy)
= ­12x3y ­ 20xy2 ­ 16x2y2
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FACTORING TRINOMIALS:
For example, FACTOR:
­20c4d ­ 30c3d2 ­ 25c2d
= ­5c2d(4c2 + 6cd + 5)
Check:
­5c2d(4c2 + 6cd + 5)
= ­20c4d ­ 30c3d2 ­ 25c2d
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CONCEPT REINFORCEMENT:
FPCM 10:
Page 155:
Page 156:
#8, #10 & #14
#16
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