Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

The polynomial function
f is defined as
f(c)=(c-k)(c^(2)-4c+4) where
k is a constant. The value 2 is a zero of
f. What is the remainder of
f(c) when divided by
(c-2) ?

◻

The polynomial function $f$ is defined as $f(c)=(c-k)(c^{2}-4c+4)$ where $k$ is a constant. The value $2$ is a zero of $f$ . What is the remainder of $f(c)$ when divided by $(c-2)?$ $\newline$ ◻

View step-by-step help

View step-by-step help

Full solution

Q. The polynomial function

f

is defined as

f(c)=(c-k)(c^{2}-4c+4)

where

k

is a constant. The value

2

is a zero of

f

. What is the remainder of

f(c)

when divided by

(c-2)?

\newline

◻

Identify Given Information: Identify the given information and the question prompt. $\newline$ We are given that $f(c) = (c-k)(c^2 - 4c + 4)$ and that $2$ is a zero of $f$ . We need to find the remainder when $f(c)$ is divided by $(c-2)$ .
Determine Value of k: Use the fact that $2$ is a zero of $f$ to determine the value of $k$ . If $2$ is a zero of $f$ , then $f(2) = 0$ . We can substitute $c$ with $2$ in the equation $f(c) = (c-k)(c^2 - 4c + 4)$ to find $k$ . $f$ $0$
Apply Remainder Theorem: Simplify the equation to find $k$ . $f(2) = (2-k)(4 - 8 + 4) = (2-k)(0) = 0$ Since $(2-k)(0) = 0$ for any value of $k$ , we cannot determine $k$ from this equation. However, we do not need the value of $k$ to find the remainder when $f(c)$ is divided by $(c-2)$ .
Substitute to Find Remainder: Apply the Remainder Theorem to find the remainder. $\newline$ The Remainder Theorem states that the remainder of a polynomial $f(c)$ when divided by $(c-a)$ is $f(a)$ . Since we want to divide $f(c)$ by $(c-2)$ , we need to find $f(2)$ .
Calculate Remainder: Substitute $c$ with $2$ in the polynomial $f(c)$ to find the remainder. $\newline$ $f(2) = (2-k)(2^2 - 4\cdot2 + 4)$ $\newline$ Simplify the expression. $\newline$ $f(2) = (2-k)(4 - 8 + 4) = (2-k)(0)$
Calculate Remainder: Substitute $c$ with $2$ in the polynomial $f(c)$ to find the remainder. $\newline$ $f(2) = (2-k)(2^2 - 4\cdot2 + 4)$ $\newline$ Simplify the expression. $\newline$ $f(2) = (2-k)(4 - 8 + 4) = (2-k)(0)$ Calculate the remainder. $\newline$ $f(2) = (2-k)(0) = 0$ $\newline$ The remainder of $f(c)$ when divided by $(c-2)$ is $0$ .

More problems from Nth roots

Question

x^{2}-y^{2}=28

\newline

What is the value of

\frac{d^{2} y}{d x^{2}}

(8,-6)

\newline

Give an exact number.

Posted 3 months ago

Question

f(x)=2 \cos \left(\frac{x}{2}\right)

\newline

f^{\prime \prime}(x)

\newline

1

\newline

-\cos \left(\frac{x}{2}\right)

\newline

-\sin \left(\frac{x}{2}\right)

\newline

-8 \cos \left(\frac{x}{2}\right)

\newline

-\frac{1}{2} \cos \left(\frac{x}{2}\right)

Posted 3 months ago

Question

Last week, the thickness of the ice on a lake increased by

\frac{25}{t+3}

centimeters per day (where

t

is the number of days since the ice first formed).

\newline

By how many centimeters did the thickness of the ice increase between

t=0

t=4

\newline

1

\newline

\frac{1,000}{441}

\newline

\frac{100}{21}

\newline

25 \ln \left(\frac{7}{3}\right)

\newline

25 \ln (4)

Posted 3 months ago

Question

What is the area of the region bound by the graphs of

f(x)=-x^{2}, g(x)=3 x-10

x=0

in quadrant IV?

\newline

1

\newline

2

\newline

\frac{8}{3}

\newline

\frac{16}{3}

\newline

\frac{34}{3}

Posted 3 months ago

Question

f

be a twice differentiable function, and let

f(1)=-7

f^{\prime}(1)=0

f^{\prime \prime}(1)=-2

\newline

What occurs in the graph of

f

(1,-7)

\newline

1

\newline

(1,-7)

is a minimum point.

\newline

(1,-7)

is a maximum point.

\newline

(C) There's not enough information to tell.

Posted 3 months ago

Question

f

be a twice differentiable function, and let

f(-3)=-4

f^{\prime}(-3)=0

f^{\prime \prime}(-3)=1

\newline

What occurs in the graph of

f

(-3,-4)

\newline

1

\newline

(-3,-4)

is a minimum point.

\newline

(-3,-4)

is a maximum point.

\newline

(C) There's not enough information to tell.

Posted 3 months ago

Question

What is the area of the region between the graphs of

f(x)=x^{2}-4 x+1

g(x)=3 x-5

x=1

x=4

\newline

1

\newline

\frac{125}{6}

\newline

\frac{27}{2}

\newline

4

\newline

\frac{81}{2}

Posted 3 months ago

Question

What is the area of the region between the graphs of

f(x)=x^{2}-3 x

g(x)=2 x

x=0

x=5

\newline

1

\newline

\frac{175}{6}

\newline

25

\newline

\frac{325}{6}

\newline

\frac{125}{6}

Posted 3 months ago

Question

What is the area of the region bound by the graphs of

f(x)=\sqrt{x-2}, g(x)=14-x

x=2

\newline

1

\newline

\frac{99}{2}

\newline

\frac{19}{6}

\newline

\frac{151}{2}

\newline

\frac{45}{2}

Posted 3 months ago

Question

\frac{d}{d x}\left[\frac{x^{3}}{\sin (x)}\right]=

Posted 3 months ago