Download FACTORING PERFECT SQUARE TRINOMIALS

APRIL 19, 2017
UNIT 2: FACTORS AND PRODUCTS
SECTION 3.8:
FACTORING SPECIAL
POLYNOMIALS
M. MALTBY INGERSOLL
NUMBERS, RELATIONS AND FUNCTIONS 10
1
WHAT'S THE POINT OF TODAY'S LESSON?
We will continue 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 ­ Factor completely:
12x2 ­ 27x + 6
= 3(4x ­ 1)(x ­ 2)
4
HOMEWORK QUESTIONS???
(page 177, #8 and #12;
page 178, #16 TO #19;
page 195, #12bef)
5
MULTIPLYING BINOMIALS:
Previously, we EXPANDED binomials using 'FOIL' to remember the 4 multiplications that were required.
For example:
(x + 4)2
= (x + 4)(x + 4)
= x2 + 4x + 4x + 16
= x2 + 8x + 16
(Can you remember our shortcut here?)
(PERFECT SQUARE TRINOMIAL; what
can you tell me about x 2 and 16? 6
FACTORING PERFECT SQUARE TRINOMIALS :
For example:
x2 + 8x + 16 (PERFECT SQUARE TRINOMIAL)
STEP 1: Does the trinomial have a GCF other than 1?
NO!
STEP 2: Factor using the "chart" for polynomials of form x2 + bx + c. OR
STEP 3: Use perfect squares and what we know about
the centre term of perfect square trinomials
to factor this trinomial. What is the square root of: x2? 16?
= x = 4
Does (4)(x) doubled = 8x?
YES!
= (x + 4)(x + 4)
= (x + 4)2
Check:
(x + 4)(x + 4)
= x2 + 4x + 4x + 16
= x2 + 8x + 16
7
MULTIPLYING BINOMIALS:
Previously, we EXPANDED binomials using 'FOIL' to remember the 4 multiplications that were required.
For example:
(3x + 4) 2
= (3x + 4)(3x + 4)
= 9x2 + 12x + 12x + 16
= 9x2 + 24x + 16
(Can you remember our shortcut here?)
(PERFECT SQUARE TRINOMIAL; what
can you tell me about 9 and 16? What about
9x2?)
8
FACTORING PERFECT SQUARE TRINOMIALS :
For example:
9x2 + 24x + 16 (PERFECT SQUARE TRINOMIAL)
STEP 1: Does the trinomial have a GCF other than 1?
NO!
STEP 2: Factor using the "Mutch Method". OR
STEP 3: Use perfect squares and what we know about
the centre term of perfect square trinomials
to factor this trinomial. What is the square root of: 9x2? 16? = 3x = 4
Does (4)(3x) doubled = 24x?
YES!
= (3x + 4) 2
Check:
(3x + 4)(3x + 4)
= 9x2 + 12x + 12x + 16
= 9x2 + 24x + 16
9
MULTIPLYING BINOMIALS:
For example:
(x ­ 4)2
= (x ­ 4)(x ­ 4)
= x2 ­ 4x ­ 4x + 16
= x2 ­ 8x + 16
(Can you remember our shortcut here?)
(PERFECT SQUARE TRINOMIAL; what
can you still tell me about x 2 and 16? 10
FACTORING PERFECT SQUARE TRINOMIALS :
For example:
x2 ­ 8x + 16 (PERFECT SQUARE TRINOMIAL)
STEP 1: Does the trinomial have a GCF other than 1?
NO!
STEP 2: Factor using the "chart" for polynomials
form x2 + bx + c. OR
STEP 3: Use perfect squares and what we know about
the centre term of perfect square trinomials
to factor this trinomial. Keep in mind the middle term is negative!! What is the square root of: x2? 16?
= x = ­4 OR +4
Does (­4)(x) doubled = ­8x?
YES!
= (x ­ 4)2
Check:
(x ­ 4)(x ­ 4)
= x2 ­ 4x ­ 4x + 16
= x2 ­ 8x + 16
11
MULTIPLYING BINOMIALS:
For example:
(3x ­ 4) 2
= (3x ­ 4)(3x ­ 4)
= 9x2 ­ 12x ­ 12x + 16
= 9x2 ­ 24x + 16
(Can you remember our shortcut here?)
(PERFECT SQUARE TRINOMIAL; what
can you still tell me about 9 and 16? What about 9x 2?)
12
FACTORING PERFECT SQUARE TRINOMIALS :
For example:
9x2 ­ 24x + 16 (PERFECT SQUARE TRINOMIAL)
STEP 1: Does the trinomial have a GCF other than 1?
NO!
STEP 2: Factor using the "Mutch Method". OR
STEP 3: Use perfect squares and what we know about
the centre term of perfect square trinomials
to factor this trinomial. Keep in mind the middle term is negative!! What is the square root of: 9x2? 16?
= 3x = ­4 OR +4
Does (­4)(3x) doubled = ­24x?
YES!
= (3x ­ 4) 2
Check:
(3x ­ 4)(3x ­ 4)
= 9x2 ­ 12x ­ 12x + 16
= 9x2 ­ 24x + 16
13
FACTORING PERFECT SQUARE TRINOMIALS :
For example:
9x2 ­ 24xy + 16y2 (PERFECT SQUARE TRINOMIAL)
STEP 1: Does the trinomial have a GCF other than 1?
NO!
STEP 2: Factor using the "Mutch Method". OR
STEP 3: Use perfect squares and what we know about
the centre term of perfect square trinomials
to factor this trinomial. Keep in mind the middle term is negative!! What is the square root of: 9x2? 16y2?
= 3x = ­4y OR +4y
Does (­4y)(3x) doubled = ­24xy?
YES!
= (3x ­ 4y) 2
Check:
(3x ­ 4y)(3x ­ 4y)
= 9x2 ­ 12xy ­ 12xy + 16y 2
= 9x2 ­ 24xy + 16y 2
14
FACTORING "COMPLETELY":
For example:
4x2 ­ 40x + 100 STEP 1: Does the trinomial have a GCF other than 1?
YES, 4!!!
2
= 4(x ­ 10x + 25)
STEP 2: Can we continue factoring?
YES ­ we have a PERFECT SQUARE TRINOMIAL! = 4(x ­ 5) 2
Check:
4(x ­ 5) 2
= 4(x2 ­ 10x + 25)
= 4x2 ­ 40x + 100
15
FACTORING PERFECT SQUARE TRINOMIALS :
For example:
x2 + 13x + 36 (PERFECT SQUARE TRINOMIAL???)
STEP 1: Does the trinomial have a GCF other than 1?
NO!
STEP 2: Can we use perfect squares and what we know about the centre term of perfect square trinomials to factor this trinomial?
NO! STEP 3: Factor this polynomial of form ax2 + bx + c
using the "chart". = (x + 4)(x + 9)
Check:
(x + 4)(x + 9)
= x2 + 9x + 4x + 36
= x2 + 13x + 36
16
CONCEPT REINFORCEMENT:
FPCM 10:
Page 194:
Page 195:
#7a & #8
#11
17