Let's begin with the task of finding [tex]\( P(-4) \)[/tex] for the polynomial [tex]\( P(x) = 4x^4 - 9x^2 + 8x + 2 \)[/tex] by direct substitution.
### Step-by-Step Solution
Step 1: Understand the polynomial and the value you need to substitute.
- The polynomial given is [tex]\( P(x) = 4x^4 - 9x^2 + 8x + 2 \)[/tex].
- We are asked to find [tex]\( P(-4) \)[/tex], so we will substitute [tex]\( x = -4 \)[/tex] into the polynomial.
Step 2: Substitute [tex]\( x = -4 \)[/tex] into the polynomial.
- Substitute [tex]\( -4 \)[/tex] for [tex]\( x \)[/tex] in each term of the polynomial:
[tex]\[
P(-4) = 4(-4)^4 - 9(-4)^2 + 8(-4) + 2
\][/tex]
Step 3: Compute each term individually.
- Calculate [tex]\((-4)^4 \)[/tex]:
[tex]\[
(-4)^4 = 256
\][/tex]
So, [tex]\( 4 \cdot 256 = 1024 \)[/tex].
- Calculate [tex]\((-4)^2 \)[/tex]:
[tex]\[
(-4)^2 = 16
\][/tex]
So, [tex]\( 9 \cdot 16 = 144 \)[/tex].
- Calculate [tex]\( 8(-4) \)[/tex]:
[tex]\[
8 \cdot (-4) = -32
\][/tex]
- The constant term in the polynomial is [tex]\( +2 \)[/tex].
Step 4: Combine the computed values.
- Substitute the calculated values back into the polynomial expression:
[tex]\[
P(-4) = 1024 - 144 - 32 + 2
\][/tex]
Step 5: Simplify the expression.
- Perform the arithmetic step-by-step:
[tex]\[
1024 - 144 = 880
\][/tex]
[tex]\[
880 - 32 = 848
\][/tex]
[tex]\[
848 + 2 = 850
\][/tex]
So, [tex]\( P(-4) = 850 \)[/tex].
### Conclusion
Therefore, by direct substitution, [tex]\( P(-4) = 850 \)[/tex].
