Answer :
To multiply the polynomials [tex]\((5x^2 + 2x + 8)\)[/tex] and [tex]\((7x - 6)\)[/tex], follow these steps:
1. Distribute each term of the first polynomial with each term of the second polynomial:
- Multiply the first term of the first polynomial [tex]\(5x^2\)[/tex] by each term in the second polynomial:
[tex]\[
5x^2 \cdot 7x = 35x^3
\][/tex]
[tex]\[
5x^2 \cdot (-6) = -30x^2
\][/tex]
- Multiply the second term of the first polynomial [tex]\(2x\)[/tex] by each term in the second polynomial:
[tex]\[
2x \cdot 7x = 14x^2
\][/tex]
[tex]\[
2x \cdot (-6) = -12x
\][/tex]
- Multiply the third term of the first polynomial [tex]\(8\)[/tex] by each term in the second polynomial:
[tex]\[
8 \cdot 7x = 56x
\][/tex]
[tex]\[
8 \cdot (-6) = -48
\][/tex]
2. Combine all the products:
[tex]\[
35x^3 - 30x^2 + 14x^2 - 12x + 56x - 48
\][/tex]
3. Simplify by combining like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-12x + 56x = 44x\)[/tex]
4. Write the simplified expression:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
So, the product of the polynomials [tex]\((5x^2 + 2x + 8)\)[/tex] and [tex]\((7x - 6)\)[/tex] is [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]. Therefore, the correct answer is C.
1. Distribute each term of the first polynomial with each term of the second polynomial:
- Multiply the first term of the first polynomial [tex]\(5x^2\)[/tex] by each term in the second polynomial:
[tex]\[
5x^2 \cdot 7x = 35x^3
\][/tex]
[tex]\[
5x^2 \cdot (-6) = -30x^2
\][/tex]
- Multiply the second term of the first polynomial [tex]\(2x\)[/tex] by each term in the second polynomial:
[tex]\[
2x \cdot 7x = 14x^2
\][/tex]
[tex]\[
2x \cdot (-6) = -12x
\][/tex]
- Multiply the third term of the first polynomial [tex]\(8\)[/tex] by each term in the second polynomial:
[tex]\[
8 \cdot 7x = 56x
\][/tex]
[tex]\[
8 \cdot (-6) = -48
\][/tex]
2. Combine all the products:
[tex]\[
35x^3 - 30x^2 + 14x^2 - 12x + 56x - 48
\][/tex]
3. Simplify by combining like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-12x + 56x = 44x\)[/tex]
4. Write the simplified expression:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
So, the product of the polynomials [tex]\((5x^2 + 2x + 8)\)[/tex] and [tex]\((7x - 6)\)[/tex] is [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]. Therefore, the correct answer is C.
