Answer :
To multiply the polynomials [tex]\( (7x^2 + 9x + 7)(9x - 4) \)[/tex], we will use the distributive property (also known as the FOIL method when dealing with binomials, but here we need to extend it to a trinomial and binomial).
Here's a step-by-step guide:
1. Distribute each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[
7x^2 \times 9x = 63x^3
\][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[
7x^2 \times -4 = -28x^2
\][/tex]
- Multiply [tex]\(9x\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[
9x \times 9x = 81x^2
\][/tex]
- Multiply [tex]\(9x\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[
9x \times -4 = -36x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[
7 \times 9x = 63x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[
7 \times -4 = -28
\][/tex]
2. Combine all the resulting terms:
[tex]\[
63x^3 - 28x^2 + 81x^2 - 36x + 63x - 28
\][/tex]
3. Combine like terms:
- For the [tex]\(x^2\)[/tex] terms: [tex]\(-28x^2 + 81x^2 = 53x^2\)[/tex]
- For the [tex]\(x\)[/tex] terms: [tex]\(-36x + 63x = 27x\)[/tex]
4. Write the final combined expression:
[tex]\[
63x^3 + 53x^2 + 27x - 28
\][/tex]
So, the product of the polynomials is [tex]\( 63x^3 + 53x^2 + 27x - 28 \)[/tex]. This matches the third option.
Here's a step-by-step guide:
1. Distribute each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[
7x^2 \times 9x = 63x^3
\][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[
7x^2 \times -4 = -28x^2
\][/tex]
- Multiply [tex]\(9x\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[
9x \times 9x = 81x^2
\][/tex]
- Multiply [tex]\(9x\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[
9x \times -4 = -36x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[
7 \times 9x = 63x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[
7 \times -4 = -28
\][/tex]
2. Combine all the resulting terms:
[tex]\[
63x^3 - 28x^2 + 81x^2 - 36x + 63x - 28
\][/tex]
3. Combine like terms:
- For the [tex]\(x^2\)[/tex] terms: [tex]\(-28x^2 + 81x^2 = 53x^2\)[/tex]
- For the [tex]\(x\)[/tex] terms: [tex]\(-36x + 63x = 27x\)[/tex]
4. Write the final combined expression:
[tex]\[
63x^3 + 53x^2 + 27x - 28
\][/tex]
So, the product of the polynomials is [tex]\( 63x^3 + 53x^2 + 27x - 28 \)[/tex]. This matches the third option.
