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g(w)=(w+13)^(3)(w+19)^(2)
The polynomial function g is defined. When g(w) is divided by (w+16), the remainder is
r. What is the value of |r|?

◻

$g(w)=(w+13)^{3}(w+19)^{2}$ $\newline$ The polynomial function $g$ is defined. When $g(w)$ is divided by $(w+16)$ , the remainder is $r$ . What is the value of $|r|$ ? $\newline$ ◻

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Transformations of functions

Full solution

Q.

g(w)=(w+13)^{3}(w+19)^{2}

\newline

The polynomial function

g

is defined. When

g(w)

is divided by

(w+16)

, the remainder is

r

. What is the value of

|r|

?

\newline

◻

Divide by $(w+16)$ : Use synthetic division or long division to divide $g(w)$ by $(w+16)$ to find the remainder $r$ .
Apply Remainder Theorem: Since we're only interested in the remainder, we can use the Remainder Theorem which states that the remainder of a polynomial $g(w)$ divided by $(w - c)$ is $g(c)$ .
Calculate $g(-16)$ : In this case, we need to find $g(-16)$ because we're dividing by $(w + 16)$ , which is the same as $(w - (-16))$ .
Simplify the expression: Calculate $g(-16)$ : $g(-16) = ((-16) + 13)^3 \times ((-16) + 19)^2$ .
Calculate the powers: Simplify the expression: $g(-16) = (-3)^3 \times (3)^2$ .
Multiply the results: Calculate the powers: $g(-16) = (-27) \times (9)$ .
Find the remainder $r$ : Multiply the results: $g(-16) = -243$ .
Calculate absolute value: The remainder $r$ is $-243$ , so the value of $|r|$ is the absolute value of $-243$ .
Calculate absolute value: The remainder $r$ is $-243$ , so the value of $|r|$ is the absolute value of $-243$ .Calculate the absolute value: $|r| = |-243| = 243$ .

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