Download Mod 2 Ch 5

Mth 95
Module 2
Spring 2014
Section 5.3 – Polynomials and Polynomial Functions
Vocabulary of Polynomials
A term is a number, a variable, or a product of numbers and variables raised to powers.
Terms in an expression are separated by addition and subtraction signs.
Examples:
A monomial is a term, which has only nonnegative integer exponents. (No addition or
subtraction signs …no division by variables)
Monomials
Not Monomials
The degree of a monomial is the sum of the exponents of its variables. A constant has
degree 0, unless the term is 0, which has undefined degree.
Give the degree of each monomial.
2x 3
17
2x3 y z2
0
A coefficient is the numerical constant (including its sign) in a monomial.
Give the coefficient in each monomial
cd 4
4x 3
A polynomial is a monomial or a sum of monomials.
Polynomials
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Not Polynomials
A binomial is a polynomial which has _____________ terms
Examples:
A trinomial is a polynomial that has ______________ terms.
Examples:
The degree of a polynomial equals the degree of the monomial with the highest degree.
4xy5 + 5 x3y4
2xy – 6x3y + y5
Degree of each term
Polynomial’s degree
Chapter 5
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Mth 95
Module 2
Spring 2014
The leading coefficient of a polynomial of one variable is the coefficient of the monomial
with the highest degree
Degree
Type
Example
Leading Coefficient
0
Constant
14
1
Linear
3x – 1
2
Quadratic
-3y + 5y2 – 4
3
Cubic
-2x3 – 4x2 – x + 1
Standard form of a single variable polynomial is written in descending order of the
powers of the variable.
3x  6  7 x 2
4n  5n3  6  9n 2
Standard form
Degree
Leading Term
Leading Coefficient
Number of Terms
To add two polynomials combine like terms. Like terms contain the same variables raised to
the same power. (So, x and x2 are NOT like terms.)
Horizontal method
(3x2 – 4x + 3) + (5x2 + 4x + 12)
Try
Vertical method
(3n + n3 + 4) + (3n3 – 8)
To subtract two polynomials, add the first polynomial and the opposite of the second
polynomial. The opposite of a polynomial is found by changing the sign of every term.
(Remember, subtraction is NOT commutative.)
(3x + 9) – (-2x + 4)
(-x2 + 6) – (2x2 + 4x – 1)
Polynomial functions create smooth, continuous graphs.
Remember: Since these are functions, they pass the vertical line test, i.e. for each input there is
a unique output.
Scale on the axes below is one. The horizontal axis is x; the vertical axis is f(x).
x
f(x)
f(x) = 3
f(x) = 2x – 1
f(x) = x2 – 2
Type:
Chapter 5
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Mth 95
Module 2
Spring 2014
f(x) = x3
other
Type:
Symbolical Evaluating Polynomial Functions
Evaluate f(x) = 4 – x2
at x = -1 and x = 3
Evaluate f(x) = -3x3 + 2x – 4
at x = 2 and x = -3
Which of the following represent polynomial functions?
f(x) = x2 + 1
f(x) = x -3
f(x) = ½ + 3x – x2
f(x) =
5
x 3
4
5.4 - Multiplication of Polynomials
Remember distribution
3(x – 4) =
-2(x – 5y) =
Properties of Exponents: For any nonzero numbers a and b and integers m and n,
am · an = a m + n
(am)n = amn
(ab)n = anbn
Simplifying Polynomials
-6x4· 4x3 =
(-2x2y3)(-3xy2) =
4m2(2m – m3) =
-xy(x3 – y) =
(-2n3m)3 =
(-xy2)4 =
4y(2y2 + 3y – 5) =
-y3(y2 – 2y + 6) =
Multiplying Binomials
Geometric area model
(x + 2)(x + 1)
Chapter 5
Distribution
(x + 2)(x + 1) = x (x + 1) + 2(x + 1)
= x2 + x + 2x + 2
=
3
Mth 95
Module 2
Table
FOIL – a shortcut to multiply every term in
the first binomial by every term in the
second binomial.
(x + 2)(x + 1)
Multiply the FIRST terms x ·x = x2
Multiply the OUTER terms x ·1 = x
Multiply the INNER terms 2 ·x = 2x
Multiply the LAST terms
2·1=2
Write as a sum
x2 + 3x + 2
Vertical
x+2
x+1
(3x – 2)(x + 4)
(5 – 2x)(3 – 5x)
 x  7 x 11
(n – 5)2
Multiplying other polynomials
(x + 5)(x2 – 4x + 5)
Horizontal method
Vertical method
x(x – 4x + 5) + 5(x – 4x + 5)
2
2
Spring 2014
(y2 – 1)(2y – 3)
(x – 5)(x + 5)
(3x – 2)(2x2 + 5x – 4)
x – 4x + 5
x+5
2
Special Products
Multiplying binomials that are the sum and difference of the same two terms.
a  ba  b   a2  ab  ab  b2 
( difference of squares)
 x  3 x  3
 x  4 x  4
 y 11 y 11
3x  23x  2
Chapter 5
1 
1

 x  2  x  2 



(2w + 5v2)(2w – 5v2)
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Mth 95
Module 2
Spring 2014
Squaring a binomial:
Squaring a binomial sum:
 x  4
a  b 
2
2
  a  b  a  b  
(3x + 2)2
What if it was squaring a difference instead of a sum?
 2x  3 
2
 x  3
a  b 
2
  a  b  a  b  
2
Review: Simplify. Begin by multiplying to change each product to a sum.
2
3x  x 2  3x  2
5 y  2
3x  23x  2
 2x  3 y 5x  y 
Section 5.5 The Greatest Common Factor and Factoring by Grouping
Factoring polynomials is important because we will use factoring to solve
polynomial equations.
Factoring is “undistributing” or “unfoiling.” It is the process of writing a polynomial
as the product of polynomials.
Distribution 2x (3x + 5) = 6x2 + 10x
Foiling
(x + 3) (x + 2) = x2 + 5x + 6
2
Factoring
6x + 10x = 2x (3x + 5)
Factoring
x2 + 5x + 6 = (x + 3) (x + 2)
Factoring out a monomial greatest common factor
Always look for the GCF before using other factoring patterns.
10x3  6 x2
What factors are common to both terms?
25xxx  23xx
Factor each monomial and find the GCF, the highest degree
(2 x  x)(5x)  (2xx)(3) of each variable in common and the largest common
coefficient.
2
2x (5x 3)
Write as a product of the greatest common monomial and a
polynomial
Check your factoring by distribution
Find the greatest common factor (GCF)
2y7 , 4y
-2, -4, 8x4 y
Chapter 5
11x3 y2 z4 , 33x y5 z6
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Mth 95
Module 2
Spring 2014
Factor the following to change each polynomial from a sum to a product.
6x3y2 + 9xy
12x4  8x2
8m3 – 2m2 + 6m
40x2 y + 15 x y2
(When the lead coefficient is negative, factor out a negative GCF.)
x2y3  7xy2
-2x3 – 6x2 + 4x
Factoring out a binomial greatest common factor
This looks a little messier: 2(x + 3) + x(x + 3) What is the common factor?
( 2 + x) (x + 3)
Write as a product.
Factoring out a binomial greatest common factor
3  x  7   5y  x  7 
6 x  2x  3y   5  2 x  3 y 
x 2  x  5  3  x  5
2x  x  3    x  3 
 x  4  3x  x  4
3  x  2  x  x  2
6 x  2x  3y    2x  3y 
4w w  3   w  3 
Factoring by Grouping
(Use this if you have four terms after you’ve checked for a GCF of all four terms)
1. Group terms in pairs that have a common factor
2. Factor out the common monomial factor from each pair
3. Factor out, if it exists, the remaining common binomial factor
4. Check factors by multiplying
xy  5x  9y + 45
3x2 + 6x + 5x + 10
2x3  10 + 4x2  5x
Chapter 5
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Mth 95
Module 2
ab  2a  5b  10
xy  4 y  3 x  12
Spring 2014
2x 2  3 xy  4 x  6y
15 xy  20 x  6 y  8
x 3  2x 2  3 x  6
3 x 2  12x  4 xy  16y
Review: Multiply to change each product to a sum. Simplify.
6 x  2x 2  3 x  5 
3x  2 2x  5
 x  3 y  x  3 y 
3 y  4
2
Factor each polynomial using GCF or factoring by grouping.
5x 2 y5 +10x 3 y
6 xy  9 x  8 y  12
Section 5.6 Factoring Trinomials
Remember factoring is undoing distribution. It is writing the sum as a product.
Leading Coefficient is One:
To factor x2 + bx + c, find integers m and n that satisfy
m  n = c and m + n = b,
2
then x + bx + c = ( x + m)(x + n).
For x2 + 6x + 8 ask, “What pair of integer factors for 8 add to 6?
Possible factor pairs are: 1 and 8, 2 and 4, -1 and –8, and – 2 and -4
Sums: 1 + 8 = 9
2+4=6
-1 + -8 = -9 -2 + -4 = -6
Choose 2 and 4 for m and n. Therefore, x2 + 6x + 8 = (x + 2)(x + 4).
Check by foiling.
If a polynomial cannot be factored, we say it is a prime polynomial.
If c is positive, m and n are either both positive or both negative.
If c is negative, m and n have opposite signs.
x 2  8 x  15
x 2 10 x  24
a2  8a  16
x2  3x  28
y2 + 8y + 18
x 2  2xy  8y 2
Chapter 5
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Mth 95
Module 2
Spring 2014
Leading Coefficient is NOT One
Sometimes a common factor must be factored out first before factoring the trinomial.
Factor completely.
7x3 + 35x2 + 42x
m4 + 6m3 + 5m2
3m3  21m2 + 36m
You can factor ax2 + bx + c, where a, b, and c have no common factor
by the “ac” method or guess and check.
The “ac” method:
1. Find numbers m and n such that mn = ac and m + n = b.
2. Write the trinomial as ax2 + mx + nx + c.
3. Use grouping to factor this expression as two binomials.
Example
2x2  5x  3
Since 2(-3) = -6, ask what factors of –6 have a sum of –5.
Since -6(1) = -6 and –6 + 1 = -5 rewrite –5x as –6x + x
2x2 –6x + x  3
Next factor by grouping
2x(x – 3) + 1 (x – 3)
(2x + 1)(x – 3)
10y2 + 13y  3
4n2 + 19n + 12
10x2  28x + 16
3x2 7x  6
2y 2  5 y  3
3 x 2  13 x  4
15 x 2  39 x 18
12x 2  5 x  2
14 x 2  xy  3y 2
Chapter 5
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Mth 95
Module 2
Spring 2014
Section 5.7 - Factoring by Special Products
Difference of Two Squares:
For any real numbers a and b, a2  b2 = (a + b)(a b)
Steps in identifying and factoring a difference of squares
1) Is it a binomial (two terms) difference (subtraction)?
2) Are both terms perfect squares? (eg. 1, 4, 9, 16, …,x2, x4, …, 25x2, …)
3) Take the square root of each term and substitute into the above pattern.
4) Check by foiling
x2  9
25x2  16
4x2 + 9
9 x 2  100
(x  2)2  9
100  (n  4)2
y 2  144
9 x 2  4y 2
Binomials can only be factored by GCF and Difference of Squares.
Always check for a GCF before using other factoring patterns.
6x2  6y2
3x3  12xy2
36x2  4y2
64z 4  49z 2
Sometimes more than one factoring pattern must be applied or a pattern must be applied
more than once.
x4  y4
x3 + x2  9x  9
x 4  81y 4
4x 3  8x 2  x  2
Chapter 5
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Mth 95
Module 2
Spring 2014
Perfect Square Trinomials: For and real numbers a and b,
a2 + 2ab + b2 = (a + b)2 and a2 - 2ab + b2 = (a - b)2
Steps in identifying and factoring a difference of squares
1) Is it a trinomial (three terms) with the powers of the variable in order?
2) Are the first and third terms POSITIVE perfect squares?
3) Multiply the square roots of these two terms and double the result.
If it is a perfect square trinomial, the answer should be the same as or
the opposite of the middle term of the polynomial.
4) Factor using one of the patterns above.
5) Check by foiling.
x2 + 2x + 1
x2 – 6x + 9
9y 2  6y  4
25x2 + 30xy + 9y2
4z 2  4z  1
x 2  2x 1
x 2  20 x  100
64 x 2  16 x  1
A General Factoring Strategy
1. For two or more terms, factor out any common factor.
2.If you have only two terms, check for difference of squares.
3.If you have three terms, factor by trial and error, use the “ac” method or use one of
the perfect square trinomial patterns.
4.If you have four terms, factor by grouping.
5.Continue factoring until none of the polynomial factors can be factored further.
Factoring Flow Chart
GCF
Two terms: Use
Difference of Squares
a2  b2 = (a + b)(a b)
Chapter 5
Three terms: Use
x2 + bx +c
ax2 + bx + c, where a ≠ 1
a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2 = (a - b)2
Four terms: Use
Factoring by grouping
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Mth 95
Module 2
Spring 2014
Factor completely.
3x2y ─ 75y3
x2y ─ 16y + 32 ─ 2x2
32y2 + 4y ─ 6
4z 2  17z  4
4 x 2  32x  60
18x2 + 30x
2x2 +5x – 12
4 x 2  22x  10
75a2  48
x 2  25
n3  2n 2  n  2
5x 2  x  7
3 xy  6y  5 x  10
Chapter 5
2r 3  12r 2t  18rt 2
4r 4  t 6
11