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MATH 110 Lecture Notes - Lecture 15: Negative Number, Factor 5, Greatest Common Divisor

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silvercicada472

10 Jul 2018

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University of Alabama

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Document Summary

To factor means to find two or more quantities whose product equals the original quantity. To factor out a common factor,(1) find the largest common monomial factor of each term and (2) divide the original polynomial by this factor to obtain the second factor. Factor: 5 x2 + 4 x = x(5 x + 4, 2 y3 (cid:1679) 6 y = 2 y( y2 (cid:1679) 3, x5 (cid:1679) 4 x 3 + x2 = x2( x3 (cid:1679) 4 x + 1) When the common monomial factor is the last term, 1 is used as a place holder in the second factor. Factor: x2 (cid:1679) 144 = ( x + 12)( x (cid:1679) 12) Note: x2 + 144 is not factorable: a2 (cid:1679) b2 = ( a + b)( a (cid:1679) b, 9 y2 (cid:1679) 1 = (3 y + 1)(3 y (cid:1679) 1) Factoring polynomials having three terms of the form ax2 + bx + c.

Related textbook solutions

Calculus

ISBN: 9781319050733

Single Variable Calculus: Early Transcendentals

4th Edition, 2018

ISBN: 9781337687805

CALCULUS:EARLY TRANSCENDENTALS

ISBN: 9781319050740

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Related Questions

Hello I need help with example 5

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1. Factor out the GCF from thepolynomial.
20m^9 + 6m^7 + 20m^5

2(10m^9 + 3m^7 +10m^5)
No common factor
m^5 (20m^4 + 6m^2 +20)
2m^5 (10m^4 + 3m^2 +10)

2. Find the solution to the system by addition(elimination) method.
4x + 15y = 15
8x - 3y = -3

(1, 0)
(0, 0)
(0, 1)
(1, 1)

3. (11p + 4)(11p - 4)

p^2 - 16
121p^2 + 88p - 16
121p^2 - 16
121p^2 - 88p -16

4. Find the solution to the system by addition(elimination) method.
2x + 15y = -91
9x + 5y = 28

(-5, 7)
(7, -7)
(9, -9)
(-7, 7)

5. Factor by grouping.
18x^3 + 15x^2 + 12x + 10

(3x^2 + 2)(6x + 5)
(18x^2 - 2)(x - 5)
(3x^2 - 2)(6x - 5)
x(18x + 2)(x +5)

6. (x + 3y)(x + 5y)

x^2 + 5xy + 15y^2
x^2 + 8xy + 8y^2
x^2 + 8xy + 15y^2
x + 8xy + 15y

7. 6x^7 (9x^7 + 12x^2 - 7)

54x^7 + 72x^2 - 42
54x^14 + 72x^9
54x^14 + 72x^9 -42x^7
54x^14 + 12x^2 -7

8. 15x^2 - 12x + 10x - 8

(15x + 2)(x - 4)
(15x - 2)(x + 4)
(3x - 2)(5x + 4)
(3x + 2)(5x -4)

9. For the polynomial, state the degree andwhether or not the polynomial is a monomial, a binomial, or atrinomial.
5x^3y^4 - 8x^5y^6 + 8

Binomial, degree 7
Trinomial, degree 3
Trinomial, degree18
Trinomial, degree11

10. (3x^8 + 2x^7 - 4x^6 + 8) + (7x^8 - 4x^7 +8x^6 + 1)

10x^8 - 2x^7 + 4x^6 +9
12x^42 + 9
10x^26 - 2x^14 + 4x^12 +9
4x^8 + 4x^7 + 4x^6 +9

11. (2n^5 - 3n^3 - 6) - (5n^5 - 14n^3 -3)

-3n^5 + 2n^3 - 9
-3n^5 + 11n^3 - 3
5n^8
-3n^5 + 11n^3 -9

12. Use the substitution method to solve thesystem.
x - 4y = -13
7x - 5y = -22

(1, 4)
(-2, 4)
(-1, 3)
No solution

13. Use the substitution method to solve thesystem.
8x - 6y = 32
2x - 2y = 8

(3, 1)
(4, 1)
(4, 0)
No solution

14. Terms in a polynomial are separated byaddition or subtraction.

15. A polynomial is an algebraic expression ofone or more terms.

16. A solution of two linear equations in twovariables is an ordered pair that is a solution to eachequation.

17. The greatest common factor will always begreater than the largest factor of each term of theexpression.

18. A system of equations in two variables mayhave infinite solutions.

19. When two or more algebraic expressions aremultiplied, each expression is called a term.

20. Factoring is multiplication inreverse.

Algebra

Complete the sentence below. If r is a real zero of even multiplicity of a function f then the graph of f the x-axis at r. If r is a real zero of even multiplicity of a function f then the graph of f the x-axis at r. Complete the sentence below. The graphs of power functions of the form f(x) = x n: where n is an even integer, always contain the points and The graphs of power functions of the form f(x) = x n: where n is an even integer, always contain the points and If r is a solution to the equation f(x) = 05 name three additional statements that can be made about f and r assuming f is a polynomial function. Choose the correct answer below. If r is a solution to the equation f(x) = 0, then r is a real zero of the polynomial function r is an x-intercept of the graph of and x - r is a factor of f. If r is a solution to the equation f(x) = 0, then r is a complex zero of the polynomial function r is an x-intercept of the graph of and x + r is a factor of f. If r is a solution to the equation f(x) = 0, then r is a real zero of the polynomial function r is a y-intercept of the graph of x and x + r is a factor of f. Explain what the notation Km f(x) = -infinity means. x rightarrow infinity Choose the correct answer below. The limit of f(x) as x approaches - infinity equals infinity. The limit of x as f(x) approaches infinity equals - infinity. The limit of f(x) as x approaches infinity equals - infinity. The limit of x as f(x) approaches - infinity equals infinity. Form a polynomial whose zeros and degree are given. Zeros: -7, multiplicity 1; - 3, multiplicity 2; degree 3 Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below. F(x) = Find the vertical horizontal and oblique asymptotes, if any, for the following rational function.R(x) = 12x / x + 20 Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = (Use a comma to separate answers as needed.) There is no vertical asymptote. Find the vertical, horizontal and oblique asymptotes, if any, of the given rational function. R(x) = x3 - 64 / x2 - 6x + 8 Find the vertical asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = (Simplify your answer. Use a comma to separate answers as needed.) There is no vertical asymptote. Find the vertical, horizontal and oblique asymptotes, if any, for the given rational function. Q(x) = 2x2 - 7x - 4 / 3x2 - 7x - 20 Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) There is no vertical asymptote. Analyze the graph of the function. R(x) = x + 14 / x(x + 15) What is the domain of R(x)? {x x 0 andx - 16} {x x 0 andx - 16 andx - 14} {x x 0 and x - 14} All real numbers Analyze the graph of the function. R(x) = 5x + 5 / 4x + 12 What is the domain of R(x)? {x|x 0 and x -3 and x - 1} {x|x -3} {x|x and x - 3} All real numbers Analyze the graph of the function.R(x) = x2 / x2 + x - 56 What is the domain of R(x)? {x|x 7 and x - 8} {x|x 0,x -1, and x 8} {x|x 0, x 7, and x - 8} All real numbers

Show transcribed image text Complete the sentence below. If r is a real zero of even multiplicity of a function f then the graph of f the x-axis at r. If r is a real zero of even multiplicity of a function f then the graph of f the x-axis at r. Complete the sentence below. The graphs of power functions of the form f(x) = x n: where n is an even integer, always contain the points and The graphs of power functions of the form f(x) = x n: where n is an even integer, always contain the points and If r is a solution to the equation f(x) = 05 name three additional statements that can be made about f and r assuming f is a polynomial function. Choose the correct answer below. If r is a solution to the equation f(x) = 0, then r is a real zero of the polynomial function r is an x-intercept of the graph of and x - r is a factor of f. If r is a solution to the equation f(x) = 0, then r is a complex zero of the polynomial function r is an x-intercept of the graph of and x + r is a factor of f. If r is a solution to the equation f(x) = 0, then r is a real zero of the polynomial function r is a y-intercept of the graph of x and x + r is a factor of f. Explain what the notation Km f(x) = -infinity means. x rightarrow infinity Choose the correct answer below. The limit of f(x) as x approaches - infinity equals infinity. The limit of x as f(x) approaches infinity equals - infinity. The limit of f(x) as x approaches infinity equals - infinity. The limit of x as f(x) approaches - infinity equals infinity. Form a polynomial whose zeros and degree are given. Zeros: -7, multiplicity 1; - 3, multiplicity 2; degree 3 Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below. F(x) = Find the vertical horizontal and oblique asymptotes, if any, for the following rational function.R(x) = 12x / x + 20 Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = (Use a comma to separate answers as needed.) There is no vertical asymptote. Find the vertical, horizontal and oblique asymptotes, if any, of the given rational function. R(x) = x3 - 64 / x2 - 6x + 8 Find the vertical asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = (Simplify your answer. Use a comma to separate answers as needed.) There is no vertical asymptote. Find the vertical, horizontal and oblique asymptotes, if any, for the given rational function. Q(x) = 2x2 - 7x - 4 / 3x2 - 7x - 20 Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) There is no vertical asymptote. Analyze the graph of the function. R(x) = x + 14 / x(x + 15) What is the domain of R(x)? {x x 0 andx - 16} {x x 0 andx - 16 andx - 14} {x x 0 and x - 14} All real numbers Analyze the graph of the function. R(x) = 5x + 5 / 4x + 12 What is the domain of R(x)? {x|x 0 and x -3 and x - 1} {x|x -3} {x|x and x - 3} All real numbers Analyze the graph of the function.R(x) = x2 / x2 + x - 56 What is the domain of R(x)? {x|x 7 and x - 8} {x|x 0,x -1, and x 8} {x|x 0, x 7, and x - 8} All real numbers

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