Download Chapter 13 Summary

Chapter 13 Summary
Key Terms
• polynomial (13.1)
• term (13.1)
• coefficient (13.1)
• monomial (13.1)
• binomial (13.1)
• trinomial (13.1)
• degree of a term (13.1)
• degree of a polynomial
(13.1)
• Zero Product Property (13.4)
13.1
• Converse of Multiplication
• square root (13.6)
• positive square root (13.6)
• principal square root (13.6)
• negative square root (13.6)
• extract the square root
Property of Zero (13.4)
• roots (13.4)
• difference of two squares
(13.5)
• perfect square trinomial
(13.5)
• difference of two cubes
(13.5)
• sum of two cubes (13.5)
(13.6)
• radical expression (13.6)
• radicand (13.6)
• completing the square (13.7)
Identifying Characteristics of Polynomial Expressions
A polynomial is an expression involving the sum of powers in one or more variables
multiplied by coefficients. A polynomial in one variable is the sum of terms of the form axk
where a, called the coefficient, is a real number and k is a non-negative integer. In general,
a polynomial is of the form a1​xk​​1 a2​xk​ 2 1​1 . . . an​x0​ ​. Each of the products in a polynomial is
called a term. Polynomials are named according to the number of terms: monomials have
exactly 1 term, binomials have exactly 2 terms, and trinomials have exactly 3 terms. The
exponent of a term is the degree of the term, and the greatest exponent in a polynomial is
the degree of the polynomial. When a polynomial is written in standard form, the terms are
written in descending order, with the term of the greatest degree first and ending with the
term of the least degree.
Example
© Carnegie Learning
The characteristics of the polynomial 13x3 1 5x 1 9 are shown.
1st term
2nd term
3rd term
13x3
5x
9
Coefficient
13
5
9
Power
x3
x​ 1​ ​
x0
Exponent
3
1
0
Term
13
There are 3 terms in this polynomial. Therefore, this polynomial is a trinomial. This trinomial
has a degree of 3 because 3 is the greatest degree of the terms in the trinomial.
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13.1
Adding and Subtracting Polynomial Expressions
Polynomials can be added or subtracted by identifying the like terms of the polynomial
functions, using the Associative Property to group the like terms together, and combining the
like terms to simplify the expression.
Example 1
Expression: (7x2 2 2x 1 12) 1 (8x3 1 2x2 2 3x)
The like terms are 7x2 and 2x2 and 22x and 23x. The terms 8x3 and 12 are not like terms.
(7x2 2 2x 1 12) 1 (8x3 1 2x2 2 3x)
8x3 1 (7x2 1 2x2) 1 (22x 2 3x) 1 12
8x3 1 9x2 2 5x 1 12
Example 2
Expression: (4x4 1 7x2 2 3) 2 (2x2 2 5)
The like terms are 7x2 and 2x2 and 23 and 25. The term 4x4 does not have a like term.
(4x4 1 7x2 2 3) 2 (2x2 2 5)
4x4 1 (7x2 2 2x2) 1 (23 1 5)
4x4 1 5x2 1 2
13.2
Modeling the Product of Polynomials
The product of 2 binomials can be determined by using an area model with algebra tiles.
Another way to model the product of 2 binomials is a multiplication table which organizes
the two terms of the binomials as factors of multiplication expressions.
Example 1
13
x
1 1 1
x
x2
x x x
x
x2
x x x
1
x
1 1 1
© Carnegie Learning
(2x 1 1)(x 1 3)
(2x 1 1)(x 1 3) 5 2x2 1 7x 1 3
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Example 2
(9x 2 1)(5x 1 7)
•
9x
21
5x
45x2
25x
7
63x
27
(9x 2 1)(5x 1 7) 5 45x2 2 5x 1 63x 2 7
5 45x2 1 58x 2 7
13.2
Using the Distributive Property to Multiply Polynomials
The Distributive Property can be used to multiply polynomials. Depending on the number of
terms in the polynomials, the Distributive Property may need to be used multiple times.
Example
(2x2 1 5x 2 10)(x 1 7)
(2x2 1 5x 2 10)(x) 1 (2x2 1 5x 2 10)(7)
(2x2)(x) 1 (5x)(x) 210(x) 1 (2x2)(7) 1 (5x)(7) 2 10(7)
2x3 1 5x2 2 10x 1 14x2 1 35x 2 70
2x3 1 19x2 1 25x 2 70
13.3
Factoring Polynomials by Determining the Greatest
Common Factor
Factoring a polynomial means to rewrite the expression as a product of factors. The first
step in factoring any polynomial expression is to determine whether or not the expression
has a greatest common factor.
Example
© Carnegie Learning
Expression: 12x3 1 4x2 1 16x
The greatest common factor is 4x.
13
12x3 1 4x2 1 16x 5 4x(3x2) 1 4x(x) 1 4x(4)
5 4x(3x2 1 x 1 4)
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13.3
Factoring Trinomials
A quadratic expression can be written in factored form, ax2 1 bx 1 c 5 a(x 2 r1)(x 2 r2), by
using an area model with algebra tiles, multiplication tables, or trial and error. Factoring a
quadratic expression means to rewrite it as a product of two linear expressions.
Example 1
Trinomial: x2 1 3x 1 2
Represent each part of the trinomial as a piece of the area model. Then use the parts to form
a rectangle.
x2
x
x
x
1
1
x12
x11
x2
x x
x
1 1
The factors of this trinomial are the length and width of the rectangle.
Therefore, x2 1 3x 1 2 5 (x 1 1)(x 1 2).
Example 2
Trinomial: x2 1 15x 1 54
•
x
9
x
x2
9x
6
6x
54
13.4
Solving Quadratic Equations Using Factoring
The Zero Product Property states that if the product of two or more factors is equal to 0,
at least one factor must be equal to 0. The property is also known as the Converse of the
Multiplication Property of Zero. This property can be used to solve a quadratic equation.
The solutions to a quadratic equation are called roots. To calculate the roots of a quadratic
equation using factoring:
13
© Carnegie Learning
So, x2 1 15x 1 54 5 (x 1 6)(x 1 9).
• Perform transformations so that one side of the equation is equal to 0.
• Factor the quadratic expression on the other side of the equation.
• Set each factor equal to 0.
• Solve the resulting equation for the roots. Check each solution in the original equation.
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Example
Equation: 2x2 1 x 5 6
2x2 1 x 5 6
2x2 1 x 2 6 5 6 2 6
2x2 1 x 2 6 5 0
(2x 2 3)(x 1 2) 5 0
2x 2 3 5 0 or x 1 2 5 0
3  ​
___
​ 2x ​ 5 ​ __
2
2
x 5 1.5 13.4
x 5 22
Connecting the Zeros of a Function to the x-intercepts of a Graph
The x-intercepts of the graph of the quadratic function f(x) 5 ax2 1 bx 1 c and the zeros of
the function are the same as the roots of the equation ax2 1 bx 1 c 5 0.
Example
Function: f(x) 5 x2 1 6x 2 55
x2 1 6x 2 55 5 0
(x 1 11)(x 2 5) 5 0
x 1 11 5 0 or x 2 5 5 0
x 5 211 x55
The zeros of the function f(x) 5 x2 1 6x 2 55 are x 5 211 and x 5 5.
13.5
Identifying Special Products of Degree 2
© Carnegie Learning
There are special products of degree 2 that have certain characteristics. A perfect square
trinomial is a trinomial formed by multiplying a binomial by itself. A perfect square trinomial is
in the form a2 1 2ab 1 b2 or a2 2 2ab 1 b2. A binomial is a difference of two squares if it is in
the form a2 2 b2 and can be factored as (a 1 b)(a 2 b).
Example 1
Expression: 4x2 1 12x 1 9
13
4x2 1 12x 1 9
(2x 1 3)(2x 1 3)
(2x 1 3)2
This expression is a perfect square trinomial.
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Example 2
Expression: x2 2 49y2
x2 2 49y2
x2 2 (7y)2
(x 1 7y)(x 2 7y)
This binomial is the difference of two squares.
13.5
Identifying Special Products of Degree 3
There are also special products of degree 3 that have certain characteristics. The difference of
two cubes is an expression in the form a3 2 b3 that can be factored as (a 2 b)(a2 1 ab 1 b2).
The sum of two cubes is an expression in the form of a3 1 b3 that can be factored as
(a 1 b)(a2 2 ab 1 b2).
Example 1
Expression: 27x3 2 125
27x3 2 125
(3x)3 2 53
(3x 2 5)​( (3x)2 1 (3x)(5) 1 52 )​
(3x 2 5)(9x2 1 15x 1 25)
Example 2
Expression: 64x3 1 1
64x3 1 1
(4x)3 1 13
(4x 1 1)​( (4x)2 2 (4x)(1) 1 12 )​
© Carnegie Learning
(4x 1 1)(16x2 2 4x 1 1)
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13.6
Determining Approximate Square Roots of Given Values
A number b is a square root of a if b2 5 a. There are 2 square roots for every whole number:
a positive square root, which is also called the principal square root, and a negative square
root. To determine approximate square roots for given values, determine the perfect square
that is closest to, but less than, the given value. Also, determine the perfect square that is
closest to, but greater than, the given value. You can use these square roots to approximate
the square root of the given number.
Example
___
Determine the approximate value of √
​ 40 ​. 
36 # 40 # 49
___
___
___
​√ 36 ​ 5 6​√40 ​ 5 ?​√49 ​ 5 7
6.32 5 39.69
6.42 5 40.96
___
The approximate value of √
​ 40 ​ is 6.3.
13.6
Simplifying Square Roots
To simplify a square root, extract any perfect squares within the expression.
Example
___
Simplify √
​ 40 ​. 
___
______
​√ 40 ​ 5 √
​ 4 • 10 ​ 
__
___
5 ​√ 4 ​ • √
​ 10 ​ 
___
5 2​√10 ​ 
13.6
Extracting Square Roots to Solve Equations
© Carnegie Learning
The solution to an equation where one term contains a variable and a constant term that are
squared can be determined by extracting the square root. To do so, take the square root of
both sides of the equation, and then isolate the variable to determine the value of the
variable.
Example
(x 2 7)2 5 75
13
_______
___
5√
​ 75 ​ 
√​ (x 2 7)2 ​ 
___
x 2 7 5 6​√75 ​ 
___
x 5 7 6 ​√75 ​ 
x ¯ 7 6 8.7
x ¯ 15.7 or 21.7
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Determining the Roots of a Quadratic Equation by Completing
the Square
13.7
For a quadratic function that has zeros but cannot be factored, there exists another method
for calculating the zeros of the function or solving the quadratic equation. Completing the
square is a process for writing a quadratic expression in vertex form which then allows you
to solve for the zeros.
b
When a function is written in standard form, ax2 1 bx 1 c, the axis of symmetry is x 5 2​___
    ​. 
2a
Example
Function: f(x) 5 x2 1 4x 1 1
x2 1 4x 1 1 5 0
x2 1 4x 5 21
x2 1 4x 1 4 5 21 1 4
x2 1 4x 1 4 5 3
(x 1 2)2 5 3
_______
__
5 6​√ 3 ​ 
√​ (x 1 2)2 ​ 
Check:
__
x 1 2 5 6​√ 3 ​ 
__
x 5 22 6 ​√ 3 ​ 
x ¯ 20.268, 23.732
__
x 5 20.268
?
(20.268)2 1 4(20.268) 1 1 5 0
?
0.0718 2 1.072 1 1 5 0
x 5 23.732
?
(23.732)2 1 4(23.732) 1 1 5 0
?
13.9278 2 14.928 1 1 5 0
20.0002 ¯ 0
20.0002 ¯ 0
The roots are 22 6​√ 3 ​. 
4
__
© Carnegie Learning
The axis of symmetry is x 5 2​   ​ , or x 5 22.
2
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