Certainly! Let's solve each part of the problem step-by-step:
Part 1: Creating the Polynomial
Given the polynomial structure:
[tex]\[ 2x^2 + (specific \ integer \ from \ the \ set) * x + 5x + 1 \][/tex]
1. First, combine like terms for the [tex]\(x\)[/tex] coefficients:
- We have [tex]\(3x + 5x = 8x\)[/tex]
2. Now, the polynomial form is:
[tex]\[ 2x^2 + 8x + 1 \][/tex]
3. We need to check if it's a positive integer by substituting [tex]\( x = 1 \)[/tex]:
[tex]\[
2(1)^2 + 8(1) + 1 = 2 + 8 + 1 = 11
\][/tex]
Therefore, the polynomial [tex]\(2x^2 + 8x + 1\)[/tex] evaluated at [tex]\(x=1\)[/tex] equals 11.
Part 2: Choosing [tex]\( a \)[/tex] from the Given Set
Given the second polynomial structure:
[tex]\[ 2x + (a - 1) \][/tex]
1. Choose [tex]\( a = 7 \)[/tex]. Since we need an integer from the set {4, 5, 6, 7, 8, 9} at most once, using 7 works well here.
2. Substitute [tex]\( a \)[/tex] into the expression:
[tex]\[ 2x + (7 - 1) = 2x + 6 \][/tex]
3. Evaluate the polynomial at [tex]\( x = 2 \)[/tex]:
[tex]\[
2(2) + 6 = 4 + 6 = 10
\][/tex]
Summary:
- For [tex]\(2x^2 + 8x + 1\)[/tex], evaluating at [tex]\( x = 1 \)[/tex] gives a result of 11.
- For [tex]\(2x + 6\)[/tex], evaluating at [tex]\( x = 2 \)[/tex] gives a result of 10.
Thus, the final answers are:
[tex]\[
(11, 10)
\][/tex]
