Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.
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Transcript of Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.
![Page 1: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/1.jpg)
Chapter 4
Products and Factors of Polynomials
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Section 4-1
Polynomials
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Constant -
A number
-2, 3/5, 0
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Monomial-
A constant, a variable, or a product of a constant and one or more variables
-7 5u (1/3)m2 -s2t3 x
![Page 5: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/5.jpg)
Coefficient-
The constant (or numerical) factor in a monomial
3m2 coefficient = 3 u coefficient = 1 - s2t3 coefficient = -1
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Degree of a Variable-
The number of times the variable occurs as a factor in the monomial
For Example – 6xy3
What is the degree of x? y?
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Degree of a monomial-
The sum of the degrees of the variables in the monomial. A nonzero constant has degree 0.
The constant 0 has no degree.
![Page 8: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/8.jpg)
Examples-
6xy3 degree = 4 -s2t3 degree =
5 u degree = 1 -7 degree = 0
![Page 9: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/9.jpg)
Similar Monomials-
Monomials that are identical or that differ only in their coefficients
Also called like terms Are - s2t3 and 2s2t3 similar?
![Page 10: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/10.jpg)
Polynomial-
A monomial or a sum of monomials.
The monomials in a polynomial are called the terms of the polynomial.
![Page 11: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/11.jpg)
Examples-
x2 + (-4)x + 5x2 – 4x + 5What are the terms?x2, -4x, and 5
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Simplified Polynomial-
A polynomial in which no two terms are similar.
The terms are usually arranged in order of decreasing degree of one of the variables
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Are they Simplified?
2x3 – 5 + 4x + x3
3x3 + 4x – 54x2 – x + 3x4 – 5 + x2
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Degree of a Polynomial-
The greatest of the degrees of its terms after it has been simplified
What is the degree?x4 + 3x 2x3 + 3x – 7x – 5x2 + 1 7x + 1x4 – 2x2y3 + 6y -11
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Adding Polynomials
To add two or more polynomials, write their sum and then simplify by combining like terms
Add the following- (x2 + 4x – 3) + (x3 – 2x2 + 6x – 7)
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Subtracting Polynomials
To subtract one polynomial from another, add the opposite of each term of the polynomial you’re subtracting
(x3 – 5x2 + 2x – 5) – (2x2 – 3x + 5)
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Section 4-2
Using Laws of Exponents
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Laws of Exponents
Let a and b be real numbers and m and n be positive integers in all the following laws
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Law 1
am · an = am+n
x2 · x4 = x6
y3 · y5 = ?m · m4 = ?
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Law 2
(ab)m = ambm
(xy)3 = x3y3
(3st)2 = ?(xy)5 = ?
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Law 3
(am)n = amn
(x3)2 = x6
(x2y3)4 = ?(2mn2)3 = ?
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Using Distributive Law
Distribute the variable using exponent laws
3t2(t3 – 2t2 + t – 4) = ?– 2x2(x3 – 3x + 4) = ?
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Section 4-3
Multiplying Polynomials
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Binomial
A polynomial that has two terms
2x + 3 4x – 3y3xy – 14 613 + 39z
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Trinomial
A polynomial that has three terms
2x2 – 3x + 1 14 + 32z – 3xmn – m2 + n2
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Multiplying binomialsWhen multiplying two binomials both terms of each binomial must be multiplied by the other two terms
Using the F.O.I.L method helps you remember the steps when multiplying
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F.O.I.L. MethodF – multiply First termsO – multiply Outer terms I – multiply Inner termsL – multiply Last terms
Add all terms to get product
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Binomial
A polynomial that has two terms
2x + 3 4x – 3y3xy – 14 613 + 39z
![Page 29: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/29.jpg)
Trinomial
A polynomial that has three terms
2x2 – 3x + 1 14 + 32z – 3xmn – m2 + n2
![Page 30: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/30.jpg)
Multiplying binomialsWhen multiplying two binomials both terms of each binomial must be multiplied by the other two terms
Using the F.O.I.L method helps you remember the steps when multiplying
![Page 31: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/31.jpg)
F.O.I.L. MethodF – multiply First termsO – multiply Outer terms I – multiply Inner termsL – multiply Last terms
Add all terms to get product
![Page 32: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/32.jpg)
Example - (2a – b)(3a + 5b)
F – 2a · 3aO – 2a · 5b I – (-b) ▪ 3aL - (-b) ▪ 5b6a2 + 10ab – 3ab – 5b2 6a2 + 7ab – 5b2
![Page 33: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/33.jpg)
Example – (x + 6)(x +4)
F – x ▪ xO – x ▪ 4 I – 6 ▪ xL – 6 ▪ 4
x2 + 4x + 6x + 24x2 + 10x + 24
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Special Products
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
(a + b)(a – b) = a2 - b2
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Section 4-4
Using Prime Factorization
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Factor
A number over a set of numbers, you write it as a product of numbers chosen from that set
The set is called a factor set
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Example
The number 15 can be factored in the following ways
(1)(15) (-1)(-15)(5)(3) (-3)(-5)
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Prime Number
An integer greater than 1 whose only positive integral factors are itself and 1
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Prime Factorization
If the factor set is restricted to the set of primes
To find it you write the integer as a product of primes
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Example
350 = 2 x 175 = 2 x 5 x 35 = 2 x 5 x 5 x 7 So the prime factorization of
350 is 2 x 52 x 7
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Greatest Common Factor
The greatest integer that is a factor of each number.
To find the GCF, take the least power of each common prime factor.
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Example
What is the GCF of 100, 120, and 90?
10
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Least Common MultipleThe least positive integer having each as a factor
To find the LCM, take the greatest power of each common prime factor.
![Page 44: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/44.jpg)
Example
What is the LCM of 100, 120, and 90?
1800
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Summary
GCF – take the least power of each common prime factor.
LCM – take the greatest power of each prime factor
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Section 4-5
Factoring Polynomials
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Factor
To factor a polynomial you express it as a product of other polynomials
We will factor using polynomials with integral coefficients
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Greatest Monomial Factor
The GCF of the terms
What is the GCF of 2x4 – 4x3 + 8x2?2x2
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Now factor:
2x4 – 4x3 + 8x2
Factor out 2x2
2x2(x2 – 2x + 4)
![Page 50: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/50.jpg)
Perfect Square Trinomials
The polynomials in the form of a2 + 2ab + b2 and a2 – 2ab + b2 are the result of squaring a + b and a – b respectively
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Difference of Squares
The polynomial a2 – b2 is the product of a + b and a - b
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Factor Each Polynomial
z2 + 6z + 94s2 – 4 st + t2
25x2 – 16a2
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Factored Form:
(z + 3)2
(2s – t)2
(5x + 4a)(5x – 4a)
![Page 54: Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.](https://reader034.fdocuments.in/reader034/viewer/2022052219/56649e0b5503460f94af2f5e/html5/thumbnails/54.jpg)
Sum and Difference of Cubes
a3 + b3 = (a + b)(a2 - ab + b2)
a3 – b3 = (a – b)(a2 + ab + b2)
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Factor Each Polynomial
y3 - 1
8u3 + v3
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Factor by Grouping
Factor each polynomial by grouping terms that have a common factor
Then factor out the common factor and write the polynomial as a product of two factors
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Factor each Polynomial
3xy - 4 - 6x + 2y
xy + 3y + 2x + 6
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Section 4-6
Factoring Quadratic Polynomials
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Quadratic PolynomialsPolynomials of the form ax2 + bx + c
Also called second- degree polynomials
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Terms
ax2 - quadratic termbx - linear termc - constant term
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Quadratic Trinomial
A quadratic polynomial for which a, b, and c are all nonzero integers
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Factoring Quadratic Trinomials
ax2 + bx + c can be factored into the form
(px + q)(rx + s) where p, q, r, and s are integers
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Factors
a = prb = ps + qrc = qs
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Factor the Polynomial
x2 + 2x - 15a = 1, so pr = 1c = -15, so qs = -15b = 2, so ps + qr = 2
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Factor the Polynomials15t2 - 16t + 4
3 - 2z - z2
x2 + 4x - 3
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Irreducible If a polynomial has more than one term and cannot be expressed as a product of polynomials of lower degree taken from a given factor set, it is irreducible
x2 + 4x - 3 is irreducible
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Factored CompletelyA polynomial is factored completely when it is written as a product of factors and each factor is either a monomial, a prime polynomial, or a power of a prime polynomial
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Greatest Common FactorThe GCF of two or more polynomials is the common factor having the greatest degree and the greatest constant factor
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Least Common Multiple
The LCM of two or more polynomials is the common multiple having the least degree and least positive constant factor
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Section 4-7
Solving Polynomial Equations
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Polynomial Equation
An equation that is equivalent to one with a polynomial as one side and 0 as the other
x2 = 5x + 24
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Root
The value of a variable that satisfies the equation
Also called the solution
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Solving a polynomial Equation
You can factor the polynomial to solve the equation
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Steps to Solving a polynomial Equation
Write the equation with 0 as one side
Factor the other side of the equation
Solve the equation obtained by setting each factor equal to 0
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Example 1
Solve (x – 5)(x + 2) = 0Step 1: already = 0Step 2: already factoredStep 3: set each factor = 0x - 5 = 0 x + 2 = 0 x = 5 x = -2
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Example 2
Solve x2 = x + 301: x2 - x – 30 = 02: (x – 6)(x + 5) = 03: x – 6 = 0 x + 5 = 0 x = 6 x = -5The solution set is {6, -5}
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Zeros
A number r is a zero of a function f if f(r) = 0
You can find zeros using the same method that is used to solve polynomial equations
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Example
Find the zeros of f(x) = (x – 4)3 – 4(3x – 16)
1: simplify2: factor3: set each factor = 0
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Double Zero
A number that occurs as a zero of a function twice
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Double Root
A number that occurs twice as a root of a polynomial equation
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Solve
x2 + 25 = 10x
12 + 4m = m2
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Section 4-8
Problem Solving Using Polynomial Equations
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Example 1
A graphic artist is designing a poster that consists of a rectangular print with a uniform border. The print is to be twice as tall as it is wide, and the border is to be 3 in. wide. If the area of the poster is to be 680 in2, find the dimensions of the print.
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Solution
1.Draw a diagram2.Let w = width and 2w = height3.The dimensions are 6 in.
greater than the print, so they are w + 6 and 2w + 6
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Solution
4. The area is represented by
(w + 6)(2w + 6) = 6805. Solve the equation.
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Example 2
The sum of two numbers is 9. The sum of their squares is 101. Find the numbers.
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Solution
1.Let x = one number2.Then 9 – x = the other
number3.x2 + (9 – x)2 = 101
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Section 4-9
Solving Polynomial Inequalities
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Polynomial InequalityAn inequality that is equivalent to an inequality with a polynomial as one side and 0 as the other side.
x2 > x + 6
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Solve by factoring
The product is positive if both factors are positive, or both factors are negative
The product is negative if the factors have opposite signs
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Example 1
Solve and graph x2 – 1 > x + 5x2 – x – 6 > 0Both factors must be positive or negative
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Example 2
Solve and graph3t < 4 – t2
t2 + 3t – 4 < 0 The factors must have opposite signs