10.1 Adding and Subtracting Polynomials

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10.1 Adding and Subtracting Polynomials
A polynomial of two terms is a binomial.
7xy2 2y
A polynomial of three terms is a trinomial.
8x2 12xy 2y2
The leading coefficient of a polynomial is the
coefficient of the variable with the largest
exponent.
The constant term is the term without a variable.
6x3 2x2 8x 15
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10.1 Adding and Subtracting Polynomials
The degree of a polynomial is the greatest of the
degrees of any of its terms. The degree of a term
is the sum of the exponents of the variables.
Examples
3y2 5x 7
degree 2
21x5y 3x3 2y2
degree 6
Common polynomial functions are named according
to their degree.
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10.1 Adding and Subtracting Polynomials
To add polynomials, combine like terms.
Examples Add (5x3 6x2 3) (3x3 12x2
10).
Use a horizontal format.
Rearrange and group like terms.
(5x3 6x2 3) (3x3 12x2 10)
(5x3 3x3 ) (6x2 12x2) (3 10)
8x3 6x2 7
Combine like terms.
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10.1 Adding and Subtracting Polynomials
Add (6x3 11x 21) (2x3 10 3x) (5x3 x
7x2 5).
Use a vertical format.
6x3 11x 21
Arrange terms of each polynomial in descending
order with like terms in the same column.
2x3 3x 10
5x3 7x2 x 5
13x3 7x2 9x 6
Add the terms of each column.
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10.1 Adding and Subtracting Polynomials
The additive inverse of the polynomial x2 3x
2 is (x2 3x 2).
This is equivalent to the additive inverse of
each of the terms.
(x2 3x 2) x2 3x 2
To subtract two polynomials, add the additive
inverse of the second polynomial to the first.
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10.1 Adding and Subtracting Polynomials
Example Add (4x2 5xy 2y2) ( x2 2xy
y2).
(4x2 5xy 2y2) ( x2 2xy y2)
Rewrite the subtraction as theaddition of the
additive inverse.
(4x2 5xy 2y2) (x2 2xy y2)
(4x2 x2) ( 5xy 2xy) (2y2 y2)
Rearrange and group like terms.
5x2 7xy 3y2
Combine like terms.
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10.1 Adding and Subtracting Polynomials
Let P(x) 2x2 3x 1 and R(x) x3 x 5.
Examples Find P(x) R(x).
P(x) R(x) (2x2 3x 1) ( x3 x 5)
x3 2x2 ( 3x x) (1 5)
x3 2x2 2x 6
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10.2 Multiplying Polynomials
To multiply a polynomial by a monomial, use the
distributive property and the rule for
multiplying exponential expressions.
Examples. Multiply 2x(3x2 2x 1).
2x(3x2 ) 2x(2x) 2x(1)
6x3 4x2 2x
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10.2 Multiplying Polynomials
Multiply 3x2y(5x2 2xy 7y2).
3x2y(5x2 ) 3x2y( 2xy) 3x2y(7y2)
15x4y 6x3y2 21x2y3
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10.2 Multiplying Polynomials
To multiply two polynomials, apply the
distributive property.
Example Multiply (x 1)(2x2 7x 3).
(x 1)(2x2) (x 1)(7x) (x 1)(3)
2x3 2x2 7x2 7x 3x 3
2x3 5x2 4x 3
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10.2 Multiplying Polynomials
Example Multiply (x 1)(2x2 7x 3).
Two polynomials can also be multiplied using a
vertical format.
2x2 7x 3
Multiply 1(2x2 7x 3).
2x3 7x2 3x
Multiply x(2x2 7x 3).
Add the terms in each column.
2x3 5x2 4x 3
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10.2 Multiplying Polynomials
To multiply two binomials use a method called
FOIL, which is based on the distributive
property. The letters of FOIL stand for First,
Outer, Inner, and Last.
1. Multiply the first terms.
4. Multiply the last terms.
2. Multiply the outer terms.
5. Add the products.
3. Multiply the inner terms.
6. Combine like terms.
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10.2 Multiplying Polynomials
Examples Multiply (2x 1)(7x 5).
2x(7x) 2x(5) (1)(7x) (1)( 5)
14x2 10x 7x 5
14x2 3x 5
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10.2 Multiplying Polynomials
Multiply (5x 3y)(7x 6y).
5x(7x) 5x(6y) ( 3y)(7x) ( 3y)(6y)
35x2 30xy 21yx 18y2
35x2 9xy 18y2
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10.3 Special Cases
The multiply the sum and difference of two terms,
use this pattern
(a b)(a b)
a2 ab ab b2
a2 b2
square of the second term
square of the first term
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10.3 Special Cases
Examples (3x 2)(3x 2)
(x 1)(x 1)
(3x)2 (2)2
(x)2 (1)2
9x2 4
x2 1
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10.3 Special Cases
To square a binomial, use this pattern
(a b)2 (a b)(a b)
a2 ab ab b2
a2 2ab b2
square of the first term
twice the product of the two terms
square of the last term
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10.3 Special Cases
Examples Multiply (2x 2)2 .
(2x)2 2(2x)( 2) ( 2)2
4x2 8x 4
Multiply (x 3y)2 .
(x)2 2(x)(3y) (3y)2
x2 6xy 9y2
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10.4 Factoring
The simplest method of factoring a polynomial is
to factor out the greatest common factor (GCF) of
each term.
Example Factor 18x3 60x.
Find the GCF.
GCF 6x
18x3 60x 6x (3x2) 6x (10)
Apply the distributive law to factor the
polynomial.
6x (3x2 10)
Check the answer by multiplication.
6x (3x2 10) 6x (3x2) 6x (10) 18x3 60x
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10.4 Factoring
Example Factor 4x2 12x 20.
Therefore, GCF 4.
4x2 12x 20 4x2 4 3x 4 5
4(x2 3x 5)
Check the answer.
4(x2 3x 5) 4x2 12x 20
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10.4 Factoring
A common binomial factor can be factored out of
certain expressions.
Example Factor the expression 5(x 1) y(x
1).
Check.
5(x 1) y(x 1) (5 y)(x 1)
(5 y)(x 1) 5(x 1) y(x 1)
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10.4 Factoring
A difference of squares can be factored using
the formula
a2 b2 (a b)(a b).
Example Factor x2 9y2.
(x)2 (3y)2
x2 9y2
Write terms as perfect squares.
(x 3y)(x 3y)
Use the formula.
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10.4 Factoring
The same method can be used to factor any
expression which can be written as a difference
of squares.
Example Factor 4(x 1)2 25y 4.
(2(x 1))2 (5y2)2
4(x 1)2 25y 4
(2(x 1)) (5y2)(2(x 1)) (5y2)
(2x 2 5y2)(2x 2 5y2)
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10.4 Factoring
Some polynomials can be factored by grouping
terms to produce a common binomial factor.
Examples Factor 2xy 3y 4x 6.
2xy 3y 4x 6
(2xy 3y) (4x 6)
Group terms.
(2x 3)y (2x 3)2
Factor each pair of terms.
(2x 3)( y 2)
Factor out the common binomial.
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10.4 Factoring
2a2 3bc 2ab 3ac
Factor 2a2 3bc 2ab 3ac.
2a2 2ab 3bc 3ac
Rearrange terms.
(2a2 2ab) (3bc 3ac)
Group terms.
2a(a b) 3c(b a)
Factor.
2a(a b) 3c(a b)
b a (a b).
(2a 3c)(a b)
Factor.
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10.4 Factoring
To factor a trinomial of the form x2 bx c,
express the trinomial as the product of two
binomials. For example,
x2 10x 24 (x 4)(x 6).
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10.4 Factoring
One method of factoring trinomials is based on
reversing the FOIL process.
Express the trinomial as a product of two
binomials with leading term x and unknown
constant terms a and b.
Example Factor x2 3x 2.
(x a)(x b)
x2 3x 2
F
O
I
L
x2
ax
bx
ab
Apply FOIL to multiply the binomials.
Since ab 2 and a b 3, it follows that a 1
and b 2.
x2 (a b) x ab
x2 (1 2) x 1 2
Therefore, x2 3x 2 (x 1)(x 2).
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10.4 Factoring
(x a)(x b)
Example Factor x2 8x 15.
x2 (a b)x ab
and ab 15.
Therefore a b 8
It follows that both a and b are negative.
x2 8x 15 (x 3)(x 5).
Check
(x 3)(x 5) x2 5x 3x 15
x2 8x 15.
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10.4 Factoring
Example Factor x2 13x 36.
(x a)(x b)
x2 (a b) x ab
Therefore a and b are two positive factors of 36
whose sum is 13.
(x 4)(x 9)
x2 13x 36
Check (x 4)(x 9)
x2 9x 4x 36
x2 13x 36.
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10.4 Factoring
Example Factor 4x3 40x2 100x.
4x3 40x2 100x
4x(x2) 4x(10x) 4x(25)
The GCF is 4x.
4x(x2 10x 25)
Use distributive property to factor out the GCF.
4x(x 5)(x 5)
Factor the trinomial.
4x(x 5)(x 5)
4x(x2 5x 5x 25)
Check
4x(x2 10x 25)
4x3 40x2 100x
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10.4 Factoring
Example Factor 2x2 5x 3.
(2x a)(x b)
2x2 5x 3
For some a and b.
2x2 (a 2b)x ab
2x2 5x 3 (2x 3)(x 1)
Check (2x 3)(x 1) 2x2 2x 3x 3 2x2
5x 3.
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10.4 Factoring
Example Factor 4x2 12x 5.
Since ab 5, a and b have the same sign.
This polynomial factors as (x a)(4x b) or (2x
a)(2x b).
a 1, b 5 or a 1 and b 5.
The middle term 12x equals either (4a b) x or
(2a 2b) x. Since a and b cannot both be
positive, they must both be negative.
4x2 12x 5 (2x 1)(2x 5)
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Trinomials which are quadratic in form are
factored like quadratic trinomials.
Example Factor 3x 4 28x2 9.
3u2 28u 9
3x 4 28x2 9
Let u x2.
(3u 1)(u 9)
Factor.
(3x2 1)(x2 9)
Replace u by x2.
Many trinomials cannot be factored.
Example Factor x2 3x 5.
Let x2 3x 5 (x a)(x b) x2 (a b) x
ab.
Then a b 3 and ab 5. This is impossible.
The trinomial x2 3x 5 cannot be factored.
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Factor by Grouping Example 2

• FACTOR 6mx 4m 3rx 2r
• Factor the first two terms
• 6mx 4m 2m (3x - 2)
• Factor the last two terms
• 3rx 2r r (3x - 2)
• The green parentheses are the same so its the
  common factor
• Now you have a common factor
• (3x - 2) (2m r)

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Example The length of a rectangle is (x 5) ft.
The width is (x 6) ft. Find
the area of the rectangle in terms of
the variable x.
A L W Area
L (x 5) ft
W (x 6) ft
A (x 5)(x 6 )
x2 6x 5x 30
x2 x 30
The area is (x2 x 30) ft2.