Answer :
To multiply the polynomials [tex]\((7x^2 + 5x + 7)\)[/tex] and [tex]\((4x - 6)\)[/tex], let's go through the multiplication step-by-step:
1. Distribute each term in the first polynomial by each term in the second polynomial:
- Start with the first term of the first polynomial [tex]\(7x^2\)[/tex]:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex] to get [tex]\(28x^3\)[/tex].
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex] to get [tex]\(-42x^2\)[/tex].
- Next, move to the second term of the first polynomial [tex]\(5x\)[/tex]:
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex] to get [tex]\(20x^2\)[/tex].
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex] to get [tex]\(-30x\)[/tex].
- Finally, use the last term of the first polynomial [tex]\(7\)[/tex]:
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex] to get [tex]\(28x\)[/tex].
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex] to get [tex]\(-42\)[/tex].
2. Combine all the terms:
Now let's collect all the terms:
- [tex]\(28x^3\)[/tex] (the only cubic term)
- Combine [tex]\(20x^2 - 42x^2\)[/tex] to get [tex]\(-22x^2\)[/tex].
- Combine [tex]\(-30x + 28x\)[/tex] to get [tex]\(-2x\)[/tex].
- The constant term is [tex]\(-42\)[/tex].
Now, putting it all together, the resulting polynomial is:
[tex]\[28x^3 - 22x^2 - 2x - 42\][/tex]
Therefore, the correct option is B. [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
1. Distribute each term in the first polynomial by each term in the second polynomial:
- Start with the first term of the first polynomial [tex]\(7x^2\)[/tex]:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex] to get [tex]\(28x^3\)[/tex].
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex] to get [tex]\(-42x^2\)[/tex].
- Next, move to the second term of the first polynomial [tex]\(5x\)[/tex]:
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex] to get [tex]\(20x^2\)[/tex].
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex] to get [tex]\(-30x\)[/tex].
- Finally, use the last term of the first polynomial [tex]\(7\)[/tex]:
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex] to get [tex]\(28x\)[/tex].
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex] to get [tex]\(-42\)[/tex].
2. Combine all the terms:
Now let's collect all the terms:
- [tex]\(28x^3\)[/tex] (the only cubic term)
- Combine [tex]\(20x^2 - 42x^2\)[/tex] to get [tex]\(-22x^2\)[/tex].
- Combine [tex]\(-30x + 28x\)[/tex] to get [tex]\(-2x\)[/tex].
- The constant term is [tex]\(-42\)[/tex].
Now, putting it all together, the resulting polynomial is:
[tex]\[28x^3 - 22x^2 - 2x - 42\][/tex]
Therefore, the correct option is B. [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
