For the function $$ f(x)=x^{4}-3 x^{3}+x^{2}+2 x+83 $$ use long division to determine whether $$ x+4 $$ is a factor of $$ f(x) $$.

Question

UpStudy AI · Accepted Answer

1. Write the polynomial in descending order:

$$
f(x)=x^4-3x^3+x^2+2x+83
$$

and we want to divide by

$$
x+4.
$$

2. Divide the first term of $$ f(x) $$, $$ x^4 $$, by $$ x $$ from $$ x+4 $$. This gives:

$$
x^4 \div x = x^3.
$$

So, $$ x^3 $$ is the first term of the quotient.

3. Multiply the divisor by $$ x^3 $$:

$$
x^3(x+4)=x^4+4x^3.
$$

Subtract this from $$ f(x) $$:

$$
\begin{aligned}
(x^4-3x^3)&-(x^4+4x^3)\
&= x^4 -3x^3- x^4-4x^3\
&=-7x^3.
\end{aligned}
$$

Bring down the next term, $$ +x^2 $$, to obtain:

$$
-7x^3+x^2.
$$

4. Divide the new leading term $$ -7x^3 $$ by $$ x $$:

$$
-7x^3 \div x = -7x^2.
$$

So, $$ -7x^2 $$ is the next term of the quotient.

5. Multiply the divisor by $$ -7x^2 $$:

$$
-7x^2(x+4)=-7x^3-28x^2.
$$

Subtract this from the current polynomial:

$$
\begin{aligned}
(-7x^3+x^2)-(-7x^3-28x^2)&= -7x^3+x^2+7x^3+28x^2\
&=29x^2.
\end{aligned}
$$

Bring down the next term, $$ +2x $$, to obtain:

$$
29x^2+2x.
$$

6. Divide $$ 29x^2 $$ by $$ x $$:

$$
29x^2 \div x = 29x.
$$

So, $$ 29x $$ is the next term of the quotient.

7. Multiply the divisor by $$ 29x $$:

$$
29x(x+4)=29x^2+116x.
$$

Subtract this from the current polynomial:

$$
\begin{aligned}
(29x^2+2x)- (29x^2+116x) &=29x^2+2x-29x^2-116x\
&=-114x.
\end{aligned}
$$

Bring down the next term, $$ +83 $$, to obtain:

$$
-114x+83.
$$

8. Divide $$ -114x $$ by $$ x $$:

$$
-114x \div x = -114.
$$

So, $$ -114 $$ is the next term of the quotient.

9. Multiply the divisor by $$ -114 $$:

$$
-114(x+4)=-114x-456.
$$

Subtract this from the current remainder:

$$
\begin{aligned}
(-114x+83)-(-114x-456)&=-114x+83+114x+456\
&=539.
\end{aligned}
$$

10. The remainder is $$ 539 $$. Since the remainder is not zero, $$ x+4 $$ is not a factor of $$ f(x) $$.

Thus, $$ x+4 $$ is not a factor of the polynomial $$ f(x) $$.
$$ x+4 $$ is not a factor of $$ f(x) $$.

For the function \( f(x)=x^{4}-3 x^{3}+x^{2}+2 x+83 \) use long division to determine whether \( x+4 \) is a factor of \( f(x) \).