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Answer :
To solve the problem of multiplying the polynomials [tex]\( (4x^2 + 3x + 7)(8x - 5) \)[/tex], we need to use the distributive property, also known as the FOIL method for binomials.
Here's how you can step through the multiplication:
1. Distribute each term of the first polynomial to each term of the second polynomial.
- Multiply [tex]\( 4x^2 \)[/tex] by each term in [tex]\( (8x - 5) \)[/tex]:
- [tex]\( 4x^2 \times 8x = 32x^3 \)[/tex]
- [tex]\( 4x^2 \times (-5) = -20x^2 \)[/tex]
- Multiply [tex]\( 3x \)[/tex] by each term in [tex]\( (8x - 5) \)[/tex]:
- [tex]\( 3x \times 8x = 24x^2 \)[/tex]
- [tex]\( 3x \times (-5) = -15x \)[/tex]
- Multiply [tex]\( 7 \)[/tex] by each term in [tex]\( (8x - 5) \)[/tex]:
- [tex]\( 7 \times 8x = 56x \)[/tex]
- [tex]\( 7 \times (-5) = -35 \)[/tex]
2. Combine all the terms together:
- Combine the results from the distribution:
[tex]\[
32x^3 + (-20x^2) + 24x^2 + (-15x) + 56x + (-35)
\][/tex]
3. Simplify by combining like terms:
- For the [tex]\( x^2 \)[/tex] terms: [tex]\(-20x^2 + 24x^2 = 4x^2\)[/tex]
- For the [tex]\( x \)[/tex] terms: [tex]\(-15x + 56x = 41x\)[/tex]
- The constant term: [tex]\(-35\)[/tex]
4. Write out the final polynomial:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
So, the correct answer is [tex]\( 32x^3 + 4x^2 + 41x - 35 \)[/tex], which matches with option D.
Here's how you can step through the multiplication:
1. Distribute each term of the first polynomial to each term of the second polynomial.
- Multiply [tex]\( 4x^2 \)[/tex] by each term in [tex]\( (8x - 5) \)[/tex]:
- [tex]\( 4x^2 \times 8x = 32x^3 \)[/tex]
- [tex]\( 4x^2 \times (-5) = -20x^2 \)[/tex]
- Multiply [tex]\( 3x \)[/tex] by each term in [tex]\( (8x - 5) \)[/tex]:
- [tex]\( 3x \times 8x = 24x^2 \)[/tex]
- [tex]\( 3x \times (-5) = -15x \)[/tex]
- Multiply [tex]\( 7 \)[/tex] by each term in [tex]\( (8x - 5) \)[/tex]:
- [tex]\( 7 \times 8x = 56x \)[/tex]
- [tex]\( 7 \times (-5) = -35 \)[/tex]
2. Combine all the terms together:
- Combine the results from the distribution:
[tex]\[
32x^3 + (-20x^2) + 24x^2 + (-15x) + 56x + (-35)
\][/tex]
3. Simplify by combining like terms:
- For the [tex]\( x^2 \)[/tex] terms: [tex]\(-20x^2 + 24x^2 = 4x^2\)[/tex]
- For the [tex]\( x \)[/tex] terms: [tex]\(-15x + 56x = 41x\)[/tex]
- The constant term: [tex]\(-35\)[/tex]
4. Write out the final polynomial:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
So, the correct answer is [tex]\( 32x^3 + 4x^2 + 41x - 35 \)[/tex], which matches with option D.
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