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Answer :
To find the product of the polynomials [tex]\((7x^2 + x + 3)(x - 1)\)[/tex], we can use the distributive property, also known as the FOIL method in the context of binomials. Here, we'll distribute each term in the first polynomial across each term in the second polynomial.
Here's a step-by-step breakdown:
1. Distribute [tex]\(x\)[/tex] from the second polynomial to each term in the first polynomial:
- Multiply [tex]\(x\)[/tex] by [tex]\(7x^2\)[/tex]:
[tex]\[
x \cdot 7x^2 = 7x^3
\][/tex]
- Multiply [tex]\(x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
x \cdot x = x^2
\][/tex]
- Multiply [tex]\(x\)[/tex] by [tex]\(3\)[/tex]:
[tex]\[
x \cdot 3 = 3x
\][/tex]
2. Distribute [tex]\(-1\)[/tex] from the second polynomial to each term in the first polynomial:
- Multiply [tex]\(-1\)[/tex] by [tex]\(7x^2\)[/tex]:
[tex]\[
-1 \cdot 7x^2 = -7x^2
\][/tex]
- Multiply [tex]\(-1\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
-1 \cdot x = -x
\][/tex]
- Multiply [tex]\(-1\)[/tex] by [tex]\(3\)[/tex]:
[tex]\[
-1 \cdot 3 = -3
\][/tex]
3. Combine all the terms from the distributions:
[tex]\[
7x^3 + x^2 + 3x - 7x^2 - x - 3
\][/tex]
4. Simplify by combining like terms:
- The [tex]\(x^2\)[/tex] terms: [tex]\(x^2 - 7x^2 = -6x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(3x - x = 2x\)[/tex]
5. Write the simplified expression:
[tex]\[
7x^3 - 6x^2 + 2x - 3
\][/tex]
Therefore, the product of [tex]\((7x^2 + x + 3)(x - 1)\)[/tex] is [tex]\(7x^3 - 6x^2 + 2x - 3\)[/tex], which matches the second option in the multiple choice list.
Here's a step-by-step breakdown:
1. Distribute [tex]\(x\)[/tex] from the second polynomial to each term in the first polynomial:
- Multiply [tex]\(x\)[/tex] by [tex]\(7x^2\)[/tex]:
[tex]\[
x \cdot 7x^2 = 7x^3
\][/tex]
- Multiply [tex]\(x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
x \cdot x = x^2
\][/tex]
- Multiply [tex]\(x\)[/tex] by [tex]\(3\)[/tex]:
[tex]\[
x \cdot 3 = 3x
\][/tex]
2. Distribute [tex]\(-1\)[/tex] from the second polynomial to each term in the first polynomial:
- Multiply [tex]\(-1\)[/tex] by [tex]\(7x^2\)[/tex]:
[tex]\[
-1 \cdot 7x^2 = -7x^2
\][/tex]
- Multiply [tex]\(-1\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
-1 \cdot x = -x
\][/tex]
- Multiply [tex]\(-1\)[/tex] by [tex]\(3\)[/tex]:
[tex]\[
-1 \cdot 3 = -3
\][/tex]
3. Combine all the terms from the distributions:
[tex]\[
7x^3 + x^2 + 3x - 7x^2 - x - 3
\][/tex]
4. Simplify by combining like terms:
- The [tex]\(x^2\)[/tex] terms: [tex]\(x^2 - 7x^2 = -6x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(3x - x = 2x\)[/tex]
5. Write the simplified expression:
[tex]\[
7x^3 - 6x^2 + 2x - 3
\][/tex]
Therefore, the product of [tex]\((7x^2 + x + 3)(x - 1)\)[/tex] is [tex]\(7x^3 - 6x^2 + 2x - 3\)[/tex], which matches the second option in the multiple choice list.
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