Answer :
To multiply the polynomials [tex]\((7x^2 + 9x + 7)(9x - 4)\)[/tex], we'll use the distributive property, also known as the FOIL method for binomials, to ensure we multiply each term in the first polynomial by each term in the second polynomial.
Here is a step-by-step breakdown of the multiplication:
1. Distribute [tex]\(9x\)[/tex] to each term in [tex]\(7x^2 + 9x + 7\)[/tex]:
- Multiply [tex]\(9x\)[/tex] by [tex]\(7x^2\)[/tex] to get [tex]\(63x^3\)[/tex].
- Multiply [tex]\(9x\)[/tex] by [tex]\(9x\)[/tex] to get [tex]\(81x^2\)[/tex].
- Multiply [tex]\(9x\)[/tex] by [tex]\(7\)[/tex] to get [tex]\(63x\)[/tex].
2. Distribute [tex]\(-4\)[/tex] to each term in [tex]\(7x^2 + 9x + 7\)[/tex]:
- Multiply [tex]\(-4\)[/tex] by [tex]\(7x^2\)[/tex] to get [tex]\(-28x^2\)[/tex].
- Multiply [tex]\(-4\)[/tex] by [tex]\(9x\)[/tex] to get [tex]\(-36x\)[/tex].
- Multiply [tex]\(-4\)[/tex] by [tex]\(7\)[/tex] to get [tex]\(-28\)[/tex].
3. Combine all the terms obtained from the distribution:
- Combine [tex]\(63x^3\)[/tex], [tex]\(81x^2\)[/tex], and [tex]\(-28x^2\)[/tex] which results in [tex]\(63x^3 + 53x^2\)[/tex].
- Combine [tex]\(63x\)[/tex] and [tex]\(-36x\)[/tex] which results in [tex]\(27x\)[/tex].
- [tex]\( -28\)[/tex] remains as it is constant.
4. Write down the final expression:
So, the fully multiplied polynomial is:
[tex]\[ 63x^3 + 53x^2 + 27x - 28 \][/tex]
The correct multiple-choice answer is:
[tex]\[ \text{D. } 63x^3 + 53x^2 + 27x - 28 \][/tex]
Here is a step-by-step breakdown of the multiplication:
1. Distribute [tex]\(9x\)[/tex] to each term in [tex]\(7x^2 + 9x + 7\)[/tex]:
- Multiply [tex]\(9x\)[/tex] by [tex]\(7x^2\)[/tex] to get [tex]\(63x^3\)[/tex].
- Multiply [tex]\(9x\)[/tex] by [tex]\(9x\)[/tex] to get [tex]\(81x^2\)[/tex].
- Multiply [tex]\(9x\)[/tex] by [tex]\(7\)[/tex] to get [tex]\(63x\)[/tex].
2. Distribute [tex]\(-4\)[/tex] to each term in [tex]\(7x^2 + 9x + 7\)[/tex]:
- Multiply [tex]\(-4\)[/tex] by [tex]\(7x^2\)[/tex] to get [tex]\(-28x^2\)[/tex].
- Multiply [tex]\(-4\)[/tex] by [tex]\(9x\)[/tex] to get [tex]\(-36x\)[/tex].
- Multiply [tex]\(-4\)[/tex] by [tex]\(7\)[/tex] to get [tex]\(-28\)[/tex].
3. Combine all the terms obtained from the distribution:
- Combine [tex]\(63x^3\)[/tex], [tex]\(81x^2\)[/tex], and [tex]\(-28x^2\)[/tex] which results in [tex]\(63x^3 + 53x^2\)[/tex].
- Combine [tex]\(63x\)[/tex] and [tex]\(-36x\)[/tex] which results in [tex]\(27x\)[/tex].
- [tex]\( -28\)[/tex] remains as it is constant.
4. Write down the final expression:
So, the fully multiplied polynomial is:
[tex]\[ 63x^3 + 53x^2 + 27x - 28 \][/tex]
The correct multiple-choice answer is:
[tex]\[ \text{D. } 63x^3 + 53x^2 + 27x - 28 \][/tex]
