Answer :
To multiply the polynomials [tex]\((7x^2 + 5x + 7)\)[/tex] and [tex]\((4x - 6)\)[/tex], we follow the distributive property, which involves multiplying each term in the first polynomial by each term in the second polynomial and then adding the results.
Let's break it down step-by-step:
1. Multiply each term in [tex]\((7x^2 + 5x + 7)\)[/tex] by each term in [tex]\((4x - 6)\)[/tex]:
- First term in the first polynomial: [tex]\(7x^2\)[/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\(7x^2 \times 4x = 28x^3\)[/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\(7x^2 \times -6 = -42x^2\)[/tex]
- Second term in the first polynomial: [tex]\(5x\)[/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\(5x \times 4x = 20x^2\)[/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\(5x \times -6 = -30x\)[/tex]
- Third term in the first polynomial: [tex]\(7\)[/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\(7 \times 4x = 28x\)[/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\(7 \times -6 = -42\)[/tex]
2. Combine all the resulting terms:
After distributing and multiplying each term, we combine all like terms:
- The [tex]\(x^3\)[/tex] term:
[tex]\(28x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms:
[tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex]
- The [tex]\(x\)[/tex] terms:
[tex]\(-30x + 28x = -2x\)[/tex]
- The constant term:
[tex]\(-42\)[/tex]
3. Write the final expression:
By combining and simplifying, the resulting polynomial is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
So, the correct answer is option B: [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
Let's break it down step-by-step:
1. Multiply each term in [tex]\((7x^2 + 5x + 7)\)[/tex] by each term in [tex]\((4x - 6)\)[/tex]:
- First term in the first polynomial: [tex]\(7x^2\)[/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\(7x^2 \times 4x = 28x^3\)[/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\(7x^2 \times -6 = -42x^2\)[/tex]
- Second term in the first polynomial: [tex]\(5x\)[/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\(5x \times 4x = 20x^2\)[/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\(5x \times -6 = -30x\)[/tex]
- Third term in the first polynomial: [tex]\(7\)[/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\(7 \times 4x = 28x\)[/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\(7 \times -6 = -42\)[/tex]
2. Combine all the resulting terms:
After distributing and multiplying each term, we combine all like terms:
- The [tex]\(x^3\)[/tex] term:
[tex]\(28x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms:
[tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex]
- The [tex]\(x\)[/tex] terms:
[tex]\(-30x + 28x = -2x\)[/tex]
- The constant term:
[tex]\(-42\)[/tex]
3. Write the final expression:
By combining and simplifying, the resulting polynomial is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
So, the correct answer is option B: [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
