College

Multiply the polynomials:

[tex]\left(4x^2 + 3x + 7\right)(8x - 5)[/tex]

A. [tex]32x^3 - 4x^2 - 41x + 35[/tex]

B. [tex]32x^3 - 44x^2 - 71x - 35[/tex]

C. [tex]32x^3 + 4x^2 + 41x - 35[/tex]

D. [tex]32x^3 + 4x^2 + 41x + 35[/tex]

Answer :

Sure! Let's multiply the polynomials [tex]\((4x^2 + 3x + 7)\)[/tex] and [tex]\((8x - 5)\)[/tex] step by step.

To multiply these polynomials, we'll use the distributive property to expand the expression:

1. Multiply each term in the first polynomial by each term in the second polynomial:
- First term: [tex]\(4x^2 \times 8x = 32x^3\)[/tex]
- Second term: [tex]\(4x^2 \times (-5) = -20x^2\)[/tex]
- Third term: [tex]\(3x \times 8x = 24x^2\)[/tex]
- Fourth term: [tex]\(3x \times (-5) = -15x\)[/tex]
- Fifth term: [tex]\(7 \times 8x = 56x\)[/tex]
- Sixth term: [tex]\(7 \times (-5) = -35\)[/tex]

2. Combine like terms:
- For [tex]\(x^3\)[/tex], we have [tex]\(32x^3\)[/tex].
- For [tex]\(x^2\)[/tex], combine [tex]\(-20x^2\)[/tex] and [tex]\(24x^2\)[/tex] to get [tex]\(4x^2\)[/tex].
- For [tex]\(x\)[/tex], combine [tex]\(-15x\)[/tex] and [tex]\(56x\)[/tex] to get [tex]\(41x\)[/tex].
- The constant term is [tex]\(-35\)[/tex].

Putting it all together, the resulting polynomial is:

[tex]\[ 32x^3 + 4x^2 + 41x - 35 \][/tex]

So, the correct answer is:

C. [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex]