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Answer :
To multiply the polynomials [tex]\( (7x^2 + 5x + 7) \)[/tex] and [tex]\( (4x - 6) \)[/tex], we can use the distributive property, which involves multiplying each term in the first polynomial by each term in the second polynomial.
1. Multiply each term in [tex]\( (7x^2 + 5x + 7) \)[/tex] by [tex]\( 4x \)[/tex]:
- [tex]\( 7x^2 \times 4x = 28x^3 \)[/tex]
- [tex]\( 5x \times 4x = 20x^2 \)[/tex]
- [tex]\( 7 \times 4x = 28x \)[/tex]
2. Multiply each term in [tex]\( (7x^2 + 5x + 7) \)[/tex] by [tex]\(-6\)[/tex]:
- [tex]\( 7x^2 \times (-6) = -42x^2 \)[/tex]
- [tex]\( 5x \times (-6) = -30x \)[/tex]
- [tex]\( 7 \times (-6) = -42 \)[/tex]
3. Add all the resulting terms together:
- Combine like terms:
- [tex]\( 28x^3 \)[/tex] (no other cubic term to combine with)
- [tex]\( 20x^2 - 42x^2 = -22x^2 \)[/tex] (combine the quadratic terms)
- [tex]\( 28x - 30x = -2x \)[/tex] (combine the linear terms)
- The constant term is [tex]\(-42\)[/tex]
So, the polynomial resulting from this multiplication is:
[tex]\[ 28x^3 - 22x^2 - 2x - 42 \][/tex]
Therefore, the correct answer is D. [tex]\( 28x^3 - 22x^2 - 2x - 42 \)[/tex].
1. Multiply each term in [tex]\( (7x^2 + 5x + 7) \)[/tex] by [tex]\( 4x \)[/tex]:
- [tex]\( 7x^2 \times 4x = 28x^3 \)[/tex]
- [tex]\( 5x \times 4x = 20x^2 \)[/tex]
- [tex]\( 7 \times 4x = 28x \)[/tex]
2. Multiply each term in [tex]\( (7x^2 + 5x + 7) \)[/tex] by [tex]\(-6\)[/tex]:
- [tex]\( 7x^2 \times (-6) = -42x^2 \)[/tex]
- [tex]\( 5x \times (-6) = -30x \)[/tex]
- [tex]\( 7 \times (-6) = -42 \)[/tex]
3. Add all the resulting terms together:
- Combine like terms:
- [tex]\( 28x^3 \)[/tex] (no other cubic term to combine with)
- [tex]\( 20x^2 - 42x^2 = -22x^2 \)[/tex] (combine the quadratic terms)
- [tex]\( 28x - 30x = -2x \)[/tex] (combine the linear terms)
- The constant term is [tex]\(-42\)[/tex]
So, the polynomial resulting from this multiplication is:
[tex]\[ 28x^3 - 22x^2 - 2x - 42 \][/tex]
Therefore, the correct answer is D. [tex]\( 28x^3 - 22x^2 - 2x - 42 \)[/tex].
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