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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

- 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

- 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

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

- 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 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)

- 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: 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: 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 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 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 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

- 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

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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