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Unit 6. Polynomials FOIL and Factoring. Terminology. Poly nomial – many terms Standard form – terms are arranged from largest exponent to smallest exponent Degree of a Polynomial – largest exponent Leading Coefficient – the first coefficient when written in standard form.

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Unit 6

Unit 6

Polynomials

FOIL and Factoring


Terminology
Terminology

  • Polynomial – many terms

  • Standard form – terms are arranged from largest exponent to smallest exponent

  • Degree of a Polynomial – largest exponent

  • Leading Coefficient – the first coefficient when written in standard form.

  • Classification

BY NUMBER OF TERMS

Monomial : one term

Binomial : two terms

Trinomial : three terms

n-nomial: n terms

(more than three terms)

BY DEGREE

Zero: constant

One: linear

Two: quadratic

Three: cubic

Four: quartic

n>4: nth degree


Polynomial addition and subtraction
Polynomial Addition and Subtraction

  • Addition: ignore parentheses and combine like terms.

    (2x3-5x2-7x+4) + (-6x3-2x2+x+6) = -4x3-7x2-6x+10

  • Subtraction: distribute the minus to all terms in parentheses behind then combine like terms.

  • (2x3-5x2-7x+4) - (-6x3-2x2+x+6)

  • = (2x3-5x2-7x+4) + (--6x3--2x2-+x-+6)

  • = (2x3-5x2-7x+4) + (6x3+2x2-x-6)

  • = 8x3-3x2-8x-2


Polynomial multiplication
Polynomial Multiplication

  • General Rule: Multiply every term of one polynomial by every term of the other

  • Special Polynomial Multiplications:

    • Distributive Property: (Monomial)·(any polynomial)

    • -3x2(5x2-6x+2) = (-3x2)(5x2)+ (-3x2)(-6x)+ (-3x2)(2)

    • = -15x4 + 18x3 - 6x2

  • FOIL: (Binomial)·(binomial)

  • FirstOuterInner Last

  • (4x – 5)(2x + 7) = (4x)(2x) + (4x)(7) + (-5)(2x) + (-5)(7)

  • = 8x2+ 28x – 10x -35

  • = 8x2 + 18x -35


Punnett squares method
Punnett Squares method

(3x + 2) (4x – 5)

3x + 2

4x

-5


2 special patterns of foil
2 Special Patterns of FOIL

Sum and Difference Pattern

(a+b)(a–b) = a2–ab+ab–b2

= a2–b2

(3x+5)(3x-5) = (3x)2 – (5)2

= 9x2 - 25

  • Square of a Binomial Pattern

    (a +b)2 = (a+b)(a+b)

    = a2+ab+ab+b2

    = a2+2ab+b2

(3x+5)2 = (3x)2+2(3x)(5)+(5)2

= 9x2 + 30x + 25

(3x-5)2 =(3x+-5)2= (3x)2+2(3x)(-5)+(-5)2

= 9x2 - 30x + 25


Factoring splitting polynomials into factors
Factoring: Splitting polynomials into factors

  • You may recall factoring numbers in the following way:

    60

    6 10

    2 3 2 5

    So 60 written in factored form is 2·2·3·5

Polynomials can be factored in a similar fashion.

Polynomials can be written in factored form as the product of linear factors.


Common monomial factoring always check for first
Common Monomial Factoringalways check for first

(reverse of Distributive Property;

factor out the common stuff)

6x – 9 = 2·3·x - 3·3 = 3(2x – 3)

5x2 + 8x = 5·x·x + 2·2·2·x = x(5x+8)

10x3–15x2=2·5·x·x·x-3·5·x·x=5x2(2x-3)

x2 + 3x – 4 = x·x + 3·x - 2·2 = x2 + 3x – 4

(nothing common)


Factor by grouping 4 terms
Factor by Grouping (4 terms)

  • Group first two terms; make sure third term is addition; group last two terms

  • Common Monomial Factor both parentheses

    (inside stuff must be same in both parentheses)

  • Answer: (Outside stuff)·(Inside stuff)

  • 5x2 – 3x – 10x + 6 = (5x2 – 3x) + (–10x + 6)

  • = x(5x-3) – 2(5x – 3)

  • = (x – 2)(5x – 3)


R s method without shortcut 3 terms a x 2 b x c
r & s method without shortcut(3 terms: ax2 + bx + c)

  • Find two numbers, r & s, so that r + s = b and r · s =a · c

  • Rewrite ax2 + bx + c as ax2 + rx+ sx+ c

  • Use factor by grouping rules to complete

a=2 b=7 c=-15

r+s = 7

r·s = 2·-15 = -30

1·-30=-30 1+-30=-29

2 ·-15=-30 2+-15=-13

3 ·-10=-30 3+-10=-7

5 ·-6=-30 5+-6=-1

6 ·-5=-30 6+-5=1

10 ·-3=-30 10+-3=7

15 ·-2=-30 15+-2=13

30 ·-1=-30 30+-1=29

2x2 + 7x – 15

=2x2+ 10x – 3x -15

=(2x2 +10x) + (-3x – 15)

=2x(x+5)-3(x+5)

=(2x-3)(x+5)


R s method with shortcut 3 terms x 2 b x c a 1
r & s method with shortcut(3 terms: x2 + bx + c;a=1)

  • Find two numbers, r & s, so that r + s = b and r · s=c

  • Answer: (x + r)(x + s)

x2 + 5x – 24

= (x+8)(x-3)

a=1 b=5 c=-24

r+s = 5

r·s = -24

1·-24=-24 1+-24=-23

2 ·-12=-24 2+-12=-10

3 ·-8=-24 3+-8=-5

4 ·-6=-24 4+-6=-2

6 ·-4=-24 6+-4=2

8 ·-3=-24 8+-3=5

12 ·-2=-24 12+-2=10

24·-1=-24 24+-1=23


Difference of two squares two terms a 2 b 2
Difference of two squaresTwo terms: a2-b2

  • Find square roots of both terms

  • Answer: (a + b)(a – b)

25x2 - 49

=(5x)2 – (7)2

=(5x + 7)(5x – 7)


Perfect square trinomials three terms a 2 2 ab b 2
Perfect Square TrinomialsThree terms: a2±2ab+b2

  • Find square roots of first and last terms

  • If 2ab matches the middle term, then answer is (a ± b)2; use sign of middle term.

  • If 2ab does not match then it is not a perfect square and you must use another method.

9x2-30x+25

=(3x)2–30x+(5)2

=(3x–5)2

b/c 2(3x)(5)=30x

4x2+36x+81

=(2x)2+36x+(9)2

=(2x+9)2

b/c 2(2x)(9)=36x


Perfect square trinomials three terms a 2 2 ab b 2 additional example
Perfect Square TrinomialsThree terms: a2±2ab+b2 additional example

x2+34x+64

=(x)2+34x+(8)2

≠(x+8)2

b/c 2(x)(8)=16x ≠ 34x

x2+34x+64 is not a perfect square;

since a=1 use r&s with shortcut

32·2=64 and 32+2=34 so

(x+32)(x+2)


Review 6 factor types studied
REVIEW: 6 factor types studied

  • Common Monomial: always look for first

  • Factor by Grouping: four terms

  • r&s without shortcut: three terms, a ≠1

  • r&s with shortcut: three terms, a =1

  • Difference of Two Squares: minus sign between two terms, know square roots of both

  • Perfect Square Trinomial: three terms, know square roots of first and last terms.


Flowchart
Flowchart

Common Monomial

Difference of

Two Squares

Number of terms

Factor by Grouping

2 3 4

r & s method

with shortcut

Does a = 1?

yes no

Do you know square roots

of first and last terms?

r & s method

without shortcut

Perfect Square Trinomial

yes no

Does 2ab part work?

yes no


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