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Answer :
To identify the degree and leading term of the polynomial for the volume function [tex]\( V(x) = (14 - 2x)(8.5 - 2x)(x) \)[/tex], we'll expand the expression step by step.
1. Expand the first two terms:
We start by expanding [tex]\((14 - 2x)(8.5 - 2x)\)[/tex].
[tex]\[
(14 - 2x)(8.5 - 2x) = 14 \times 8.5 - 14 \times 2x - 2x \times 8.5 + 4x^2
\][/tex]
Calculate each term:
- [tex]\(14 \times 8.5 = 119\)[/tex]
- [tex]\(14 \times 2x = 28x\)[/tex]
- [tex]\(2x \times 8.5 = 17x\)[/tex]
- [tex]\(4x^2\)[/tex] is already calculated
So, the expansion becomes:
[tex]\[
119 - 28x - 17x + 4x^2 = 119 - 45x + 4x^2
\][/tex]
2. Expand with the third term:
Now multiply the result by [tex]\(x\)[/tex]:
[tex]\[
(119 - 45x + 4x^2)(x) = 119x - 45x^2 + 4x^3
\][/tex]
3. Identify the degree and leading term:
- The degree of the polynomial is the highest power of [tex]\(x\)[/tex], which is 3 because of the term [tex]\(4x^3\)[/tex].
- The leading term is the term with the highest power, which is [tex]\(4x^3\)[/tex].
Therefore, the degree of the polynomial is 3, and the leading term is [tex]\(4x^3\)[/tex].
1. Expand the first two terms:
We start by expanding [tex]\((14 - 2x)(8.5 - 2x)\)[/tex].
[tex]\[
(14 - 2x)(8.5 - 2x) = 14 \times 8.5 - 14 \times 2x - 2x \times 8.5 + 4x^2
\][/tex]
Calculate each term:
- [tex]\(14 \times 8.5 = 119\)[/tex]
- [tex]\(14 \times 2x = 28x\)[/tex]
- [tex]\(2x \times 8.5 = 17x\)[/tex]
- [tex]\(4x^2\)[/tex] is already calculated
So, the expansion becomes:
[tex]\[
119 - 28x - 17x + 4x^2 = 119 - 45x + 4x^2
\][/tex]
2. Expand with the third term:
Now multiply the result by [tex]\(x\)[/tex]:
[tex]\[
(119 - 45x + 4x^2)(x) = 119x - 45x^2 + 4x^3
\][/tex]
3. Identify the degree and leading term:
- The degree of the polynomial is the highest power of [tex]\(x\)[/tex], which is 3 because of the term [tex]\(4x^3\)[/tex].
- The leading term is the term with the highest power, which is [tex]\(4x^3\)[/tex].
Therefore, the degree of the polynomial is 3, and the leading term is [tex]\(4x^3\)[/tex].
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