Answer :
To multiply the polynomials [tex]\((7x^2 + 5x + 7)(4x - 6)\)[/tex], we use the distributive property, also known as the FOIL method for binomials, to ensure each term in the first polynomial is multiplied by each term in the second polynomial.
1. Multiply each term of the first polynomial by each term of the second polynomial:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7x^2 \cdot 4x = 28x^3
\][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7x^2 \cdot (-6) = -42x^2
\][/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
5x \cdot 4x = 20x^2
\][/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
5x \cdot (-6) = -30x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7 \cdot 4x = 28x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7 \cdot (-6) = -42
\][/tex]
2. Combine all these terms:
- Start with the cubic term:
[tex]\[
28x^3
\][/tex]
- Combine the quadratic terms [tex]\( -42x^2 \)[/tex] and [tex]\( 20x^2 \)[/tex]:
[tex]\[
-42x^2 + 20x^2 = -22x^2
\][/tex]
- Combine the linear terms [tex]\( -30x \)[/tex] and [tex]\( 28x \)[/tex]:
[tex]\[
-30x + 28x = -2x
\][/tex]
- Finally, add the constant term:
[tex]\[
-42
\][/tex]
3. Write the final expression:
The product of the polynomials is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
The correct choice is D: [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
1. Multiply each term of the first polynomial by each term of the second polynomial:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7x^2 \cdot 4x = 28x^3
\][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7x^2 \cdot (-6) = -42x^2
\][/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
5x \cdot 4x = 20x^2
\][/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
5x \cdot (-6) = -30x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7 \cdot 4x = 28x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7 \cdot (-6) = -42
\][/tex]
2. Combine all these terms:
- Start with the cubic term:
[tex]\[
28x^3
\][/tex]
- Combine the quadratic terms [tex]\( -42x^2 \)[/tex] and [tex]\( 20x^2 \)[/tex]:
[tex]\[
-42x^2 + 20x^2 = -22x^2
\][/tex]
- Combine the linear terms [tex]\( -30x \)[/tex] and [tex]\( 28x \)[/tex]:
[tex]\[
-30x + 28x = -2x
\][/tex]
- Finally, add the constant term:
[tex]\[
-42
\][/tex]
3. Write the final expression:
The product of the polynomials is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
The correct choice is D: [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
