Download Add, Subtract, Multiply Polynomials

10.1 Adding and Subtracting
Polynomials
7xy2 + 2y
A polynomial of two terms is a binomial.
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.
The degree is 3.
6x3 – 2x2 + 8x + 15
The leading coefficient is 6.
The constant term is 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.
Function
linear
Equation
f (x) = mx + b
Degree
one
quadratic
f (x) = ax2 + bx + c, a  0
two
cubic
f (x) = ax3 + bx2 + cx + d, a  0
three
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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.
(5x3 + 6x2 + 3) + (3x3 – 12x2 – 10)
= (5x3 + 3x3 ) + (6x2 – 12x2) + (3 – 10)
= 8x3 – 6x2 – 7
Rearrange and group like
terms.
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
2x3
– 3x + 10
5x3 – 7x2 + x + 5
13x3 – 7x2 + 9x – 6
Arrange terms of each polynomial in
descending order with like terms in
the same column.
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)
= (4x2 – 5xy + 2y2) + (x2 – 2xy + y2)
= (4x2 + x2) + (– 5xy – 2xy) + (2y2 + y2)
= 5x2 – 7xy + 3y2
Rewrite the subtraction as the
addition of the additive inverse.
Rearrange and group like terms.
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.
Example:
2x2 + 7x + 3
x–1
– 2x2 – 7x – 3
2x3 + 7x2 + 3x
2x3 + 5x2 – 4x – 3
Multiply – 1(2x2 + 7x + 3).
Multiply x(2x2 + 7x + 3).
Add the terms in each column.
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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).
First
Outer
Inner
Last
= 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).
First
Outer
Inner
Last
= 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)
= (3x)2 – (2)2
= 9x2 – 4
(x + 1)(x – 1)
= (x)2 – (1)2
= 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.
GCF = 6x
18x3 + 60x = 6x(3x2) + 6x(10)
= 6x(3x2 + 10)
Find the GCF.
Apply the distributive law
to factor the polynomial.
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
Check the answer.
= 4(x2 – 3x + 5)
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).
5(x + 1) – y(x + 1) = (5 – y)(x + 1)
Check.
(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
= (x + 3y)(x – 3y)
Write terms as perfect
squares.
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.
4(x + 1)2 – 25y 4
= (2(x + 1))2 – (5y2)2
= [(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.
Factor.
= 2a(a – b) + 3c(b – a)
= 2a(a – b) – 3c(a – b)
= (2a – 3c)(a – b)
b – a = – (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.
Example: Factor x2 + 3x + 2.
x2 + 3x + 2 = (x + a)(x + b)
Express the trinomial as a product of
two binomials with leading term x and
unknown constant terms a and b.
F O I L
= x2 + ax + bx + ab
= x2 + (a + b)x + ab
= x2 + (1 + 2)x + 1 · 2
Apply FOIL to multiply the binomials.
Since ab = 2 and a + b = 3, it follows
that a = 1 and b = 2.
Therefore, x2 + 3x + 2 = (x + 1)(x + 2).
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10.4 Factoring
Example: Factor x2 – 8x + 15.
= (x + a)(x + b)
= x2 + (a + b)x + ab
Therefore a + b = –8
and ab = 15.
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.
x2 + 13x + 36 = (x + 4)(x + 9)
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)
= 4x(x2 – 10x + 25)
= 4x(x – 5)(x – 5)
The GCF is 4x.
Use distributive property
to factor out the GCF.
Factor the trinomial.
Check: 4x(x – 5)(x – 5) = 4x(x2 – 5x – 5x + 25)
= 4x(x2 – 10x + 25)
= 4x3 – 40x2 + 100x
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10.4 Factoring
Example: Factor 2x2 + 5x + 3.
2x2 + 5x + 3 = (2x + a)(x + b)
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.
3x 4 + 28x2 + 9 = 3u2 + 28u + 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 it’s 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.
x–6
A = L · W = Area
L = (x + 5) ft
W = (x – 6) ft
x+5
A = (x + 5)(x – 6 ) = x2 – 6x + 5x – 30
= x2 – x – 30
The area is (x2 – x – 30) ft2.
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