Answer:
Evaluating Polynomials:
a. f(1) = -10
b. f(-3) = -239
c. f(2)² = 3125
Factoring Polynomials:
a. (x + 1)(x² - 5x + 6) = (x + 1)(x - 3)(x - 2)
b. (x² - x - 6)(x² + 6x + 9) = (x - 3)(x + 2)(x + 3)(x + 3)
c. x³ + 3x² - 4x - 12 = (x + 3)(x - 2)(x + 2)(x - 2)(x + 2)
A polynomial can be evaluated and factored by long division, factoring and synthetic division methods.
1. Evaluate polynomials
[tex]\mathbf{f(x) = x^4 - 7x^2 + 2x - 6}[/tex]
Substitute 1 for x
[tex]\mathbf{f(1) = 1^4 - 7(1)^2 + 2(1) - 6}[/tex]
[tex]\mathbf{f(1) = 1 - 7 + 2 - 6}[/tex]
[tex]\mathbf{f(1) = -10}[/tex]
[tex]\mathbf{f(x) = 10x^3 + 4x^2 - 5}[/tex]
Substitute -3 for x
[tex]\mathbf{f(-3) = 10(-3)^3 + 4(-3)^2 - 5}[/tex]
[tex]\mathbf{f(-3) = -270 + 36 - 5}[/tex]
[tex]\mathbf{f(-3) = -239}[/tex]
[tex]\mathbf{f(x) =x^4 + 5x^3 - 2x + 3}[/tex]
Substitute 2 for x
[tex]\mathbf{f(2) =2^4 + 5(2)^3 - 2(2) + 3}[/tex]
[tex]\mathbf{f(2) =16 + 40 - 4 + 3}[/tex]
[tex]\mathbf{f(2) =55}[/tex]
Square both sides
[tex]\mathbf{f(2)^2 =3025}[/tex]
Hence, the results of evaluating the polynomials are [tex]\mathbf{f(1) = -10}[/tex], [tex]\mathbf{f(-3) = -239}[/tex] and [tex]\mathbf{f(2)^2 =3025}[/tex]
2. Factor the polynomials
[tex]\mathbf{(x + 1)(x^2 - 5x + 6)}[/tex]
Expand
[tex]\mathbf{(x + 1)(x^2 - 5x + 6) =(x + 1)(x^2 - 2x - 3x + 6) }[/tex]
Factorize
[tex]\mathbf{(x + 1)(x^2 - 5x + 6) =(x + 1)(x(x - 2) - 3(x - 2) )}[/tex]
Factor out x - 2
[tex]\mathbf{(x + 1)(x^2 - 5x + 6) =(x + 1)(x - 3)(x - 2) }[/tex]
[tex]\mathbf{(x^2 - x - 6)(x^2 + 6x + 9)}[/tex]
Expand
[tex]\mathbf{(x^2 - x - 6)(x^2 + 6x + 9) = (x^2 +2x - 3x - 6)(x^2 + 3x +3x + 9)}[/tex]
Factorize
[tex]\mathbf{(x^2 - x - 6)(x^2 + 6x + 9) = [x(x +2) - 3(x + 2)][x(x + 3) +3(x + 3)]}[/tex]
Factor out x + 2 and x + 3
[tex]\mathbf{(x^2 - x - 6)(x^2 + 6x + 9) = [(x - 3) (x + 2)][(x + 3)(x + 3)]}[/tex]
Remove square brackets
[tex]\mathbf{(x^2 - x - 6)(x^2 + 6x + 9) = (x - 3) (x + 2)(x + 3)(x + 3)}[/tex]
[tex]\mathbf{x^3 + 3x^2 - 4x - 12}[/tex]
Factorize
[tex]\mathbf{x^3 + 3x^2 - 4x - 12 = x^2(x + 3) - 4(x + 3)}[/tex]
Factor out x + 3
[tex]\mathbf{x^3 + 3x^2 - 4x - 12 = (x^2 -4) (x + 3)}[/tex]
Apply the difference of two squares
[tex]\mathbf{x^3 + 3x^2 - 4x - 12 = (x -2)(x+2) (x + 3)}[/tex]
Hence, the results of factoring the polynomials are [tex]\mathbf{(x + 1)(x^2 - 5x + 6) =(x + 1)(x - 3)(x - 2) }[/tex], [tex]\mathbf{(x^2 - x - 6)(x^2 + 6x + 9) = (x - 3) (x + 2)(x + 3)(x + 3)}[/tex] and [tex]\mathbf{x^3 + 3x^2 - 4x - 12 = (x -2)(x+2) (x + 3)}[/tex]
Read more about polynomials at:
https://brainly.com/question/11536910
