Sure, let's solve the problem step by step.
1. Understand the polynomial and simplify if necessary:
Given: [tex]\( P(x) = x^4 + 3x^2 - 5x^2 + 6x \)[/tex]
First, simplify the polynomial:
[tex]\[
P(x) = x^4 + (3x^2 - 5x^2) + 6x
\][/tex]
[tex]\[
P(x) = x^4 - 2x^2 + 6x
\][/tex]
2. Substitute [tex]\( x = 2 \)[/tex] into the polynomial:
[tex]\[
P(2) = (2)^4 - 2(2)^2 + 6(2)
\][/tex]
[tex]\[
P(2) = 16 - 2 \cdot 4 + 12
\][/tex]
[tex]\[
P(2) = 16 - 8 + 12
\][/tex]
[tex]\[
P(2) = 20
\][/tex]
3. Given the equation [tex]\( P(x) = 3 \)[/tex] and need to find the remainder when substituting [tex]\( x = 2 \)[/tex]:
The polynomial equation at [tex]\( x = 2 \)[/tex] must satisfy:
[tex]\[
P(2) = 3
\][/tex]
4. Calculate the remainder by subtracting 3 from [tex]\( P(2) \)[/tex]:
We have already found [tex]\( P(2) = 20 \)[/tex], so the remainder is:
[tex]\[
\text{Remainder} = P(2) - 3
\][/tex]
[tex]\[
\text{Remainder} = 20 - 3
\][/tex]
[tex]\[
\text{Remainder} = 17
\][/tex]
Therefore, the value of the polynomial at [tex]\( x = 2 \)[/tex] is 20, and the remainder when the polynomial is set to 3 and [tex]\( x = 2 \)[/tex] is substituted is 17.
