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The function $f$ is defined by $f(x) = 2x^3 + 3x^2 + cx + 8$ where $c$ is a constant. $\newline$ In the xy-plane, the graph of $f$ intersects the x-axis at the three points $\newline$ $(-4, 0)$ , $(-1, 0)$ , and $(p, 0)$ . What is the value of $c$ ? $\newline$ A) $-18$ $\newline$ B) $-2$ $\newline$ C) $f(x) = 2x^3 + 3x^2 + cx + 8$ $0$ $\newline$ D) $f(x) = 2x^3 + 3x^2 + cx + 8$ $1$

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Transformations of quadratic functions

Full solution

Q. The function

f

is defined by

f(x) = 2x^3 + 3x^2 + cx + 8

where

c

is a constant.

\newline

In the xy-plane, the graph of

f

intersects the x-axis at the three points

\newline

(-4, 0)

,

(-1, 0)

, and

(p, 0)

. What is the value of

c

?

\newline

A)

-18

\newline

B)

-2

\newline

C)

f(x) = 2x^3 + 3x^2 + cx + 8

0

\newline

D)

f(x) = 2x^3 + 3x^2 + cx + 8

1

Identify Roots and Equation: Identify the roots of the polynomial and set up the equation based on the given x-intercepts. $\newline$ Since the x-intercepts are given as $(-4, 0)$ , $(-1, 0)$ , and $(p, 0)$ , the polynomial can be expressed in factored form as: $\newline$ $f(x) = 2(x + 4)(x + 1)(x - p)$
Expand Factored Form: Expand the factored form to find an expression for $f(x)$ in standard polynomial form. $\newline$ Expanding, we get: $\newline$ $f(x) = 2(x + 4)(x + 1)(x - p)$ $\newline$ $= 2[(x^2 + 5x + 4)(x - p)]$ $\newline$ $= 2(x^3 - px^2 + 5x^2 - 5px + 4x - 4p)$ $\newline$ $= 2x^3 + (10 - 2p)x^2 + (8 - 10p)x - 8p$
Compare Coefficients: Compare the expanded form with the given polynomial to find $c$ . The given polynomial is $2x^3 + 3x^2 + cx + 8$ . By comparing coefficients from the expanded form: $3 = 10 - 2p$ (for $x^2$ coefficient) $c = 8 - 10p$ (for $x$ coefficient)
Solve for $p$ : Solve for $p$ using the equation for the $x^2$ coefficient. $\newline$ $3 = 10 - 2p$ $\newline$ $2p = 10 - 3$ $\newline$ $2p = 7$ $\newline$ $p = 3.5$
Substitute $p$ into $c$ : Substitute $p = 3.5$ into the equation for $c$ .
$c = 8 - 10(3.5)$
$c = 8 - 35$
$c = -27$

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