Question

A polynomial f (x) has the given zeros of 7, –1, and –3.

Part A: Using the Factor Theorem, determine the polynomial f (x) in expanded form. Show all necessary calculations. (3 points)

Part B: Divide the polynomial f (x) by (x2 – x – 2) to create a rational function g(x) in simplest factored form. Determine g(x) and find its slant asymptote. (4 points)

Part C: List all locations and types of discontinuities of the function g(x).
Ty<3

Answer 1

Answer:

$\textsf{A)} \quad f(x) =x^3-3x^2-25x-21$

$\textsf{B)} \quad g(x)=\dfrac{(x-7)(x+3)}{x-2}$

    Slant asymptote:  y = x - 2

C)  Removable discontinuity at x = -1  ⇒  (-1, ¹⁶/₃)

     Infinite discontinuity at x = 2

Step-by-step explanation:

Factor Theorem

If f(x) is a polynomial, and f(a) = 0, then (x – a)  is a factor of f(x).

Therefore, if the polynomial f(x) has the given zeros of 7, -1 and -3 then:

$f(x) = (x - 7)(x + 1)(x + 3)$

Expanded:

$\implies f(x) = (x - 7)(x + 1)(x + 3)$

$\implies f(x) = (x - 7)(x^2+4x+3)$

$\implies f(x) = x(x^2+4x+3) - 7(x^2+4x+3)$

$\implies f(x) =x^3+4x^2+3x - 7x^2-28x-21$

$\implies f(x) =x^3-3x^2-25x-21$

Part B

Factor $x^2-x-2$ :

$\implies x^2-2x+x-2$

$\implies x(x-2)+1(x-2)$

$\implies (x+1)(x-2)$

Therefore:

$\begin{aligned}\implies g(x) & = \dfrac{f(x)}{x^2-x-2}\\\\ & = \dfrac{x^3-3x^2-25x-21}{x^2-x-2}\\\\& = \dfrac{(x-7)(x+1)(x+3)}{(x+1)(x-2)}\\\\& = \dfrac{(x-7)(x+3)}{x-2}\end{aligned}$

A slant asymptote occurs when the degree of the numerator polynomial is greater than the degree of the denominator polynomial.

To find the slant asymptote, divide the numerator by the denominator.

$\large \begin{array}{r}x-2\phantom{)}\\x-2{\overline{\smash{\big)}\,x^2-4x-21\phantom{)}}}\\\underline{-~\phantom{(}(x^2-2x)\phantom{-b)))}}\\0-2x-21\phantom{)}\\\underline{-~\phantom{()}(-2x+4)\phantom{)}}\\-25\phantom{)}\\\end{array}$

Therefore, the slant asymptote is y = x - 2.

Part C

Discontinuous function:  A function that is not continuous.

Removable Discontinuity (holes):  When a rational function has a factor with an x that is in both the numerator and the denominator.

Jump Discontinuity:  The function jumps from one point to another along the curve of the function, often splitting the curve into two separate sections.

Infinite Discontinuity:  When a function has a vertical asymptote (a line that the curve gets infinitely close to, but never touches).

Therefore, the discontinuities of function g(x) are:

• Removable discontinuity at x = -1  ⇒  (-1, ¹⁶/₃).

• Infinite discontinuity at x = 2.