Use the remainder theorem to find the remainder when is divided by . Then use the factor theorem to determine whether is a factor of .

The remainder is .

Ask by Williams Reyes. in the United States

Dec 04,2024

Question

Use the remainder theorem to find the remainder when is divided by . Then use the factor theorem to determine whether is a factor of .

The remainder is .

Ask by Williams Reyes. in the United States

Dec 04,2024

UpStudy AI · Accepted Answer

To find the remainder when $$ f(x) = 3x^3 - 2x^2 + 7x - 2 $$ is divided by $$ x - 1 $$, we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of the division of a polynomial $$ f(x) $$ by $$ x - c $$ is given by $$ f(c) $$.

In this case, we need to evaluate $$ f(1) $$:

$$
f(1) = 3(1)^3 - 2(1)^2 + 7(1) - 2
$$

Calculating each term:

$$
f(1) = 3(1) - 2(1) + 7(1) - 2
$$
$$
= 3 - 2 + 7 - 2
$$
$$
= 3 - 2 = 1
$$
$$
1 + 7 = 8
$$
$$
8 - 2 = 6
$$

Thus, the remainder when $$ f(x) $$ is divided by $$ x - 1 $$ is $$ 6 $$.

Now, to determine whether $$ x - 1 $$ is a factor of $$ f(x) $$, we can use the Factor Theorem. The Factor Theorem states that $$ x - c $$ is a factor of $$ f(x) $$ if and only if $$ f(c) = 0 $$.

Since we found that $$ f(1) = 6 $$, which is not equal to zero, we conclude that $$ x - 1 $$ is not a factor of $$ f(x) $$.

In summary:
- The remainder is $$ \boxed{6} $$.
- $$ x - 1 $$ is not a factor of $$ f(x) $$.
The remainder is 6.

Use the remainder theorem to find the remainder when is divided by . Then use the factor theorem to determine whether is a factor of .

The remainder is .

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Use the remainder theorem to find the remainder when is divided by . Then use the factor theorem to determine whether is a factor of . The remainder is .

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Use the remainder theorem to find the remainder when is divided by . Then use the factor theorem to determine whether is a factor of .

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