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Answer :
To multiply the polynomials [tex]\((5x^2 + 2x + 8)(7x - 6)\)[/tex], we'll apply the distributive property (also known as the FOIL method for binomials) to ensure that every term in the first polynomial is multiplied by each term in the second polynomial. Let's break it down step-by-step:
1. Multiply each term in [tex]\((5x^2 + 2x + 8)\)[/tex] by [tex]\(7x\)[/tex]:
- [tex]\(5x^2 \cdot 7x = 35x^3\)[/tex]
- [tex]\(2x \cdot 7x = 14x^2\)[/tex]
- [tex]\(8 \cdot 7x = 56x\)[/tex]
So, we get:
[tex]\(35x^3 + 14x^2 + 56x\)[/tex]
2. Multiply each term in [tex]\((5x^2 + 2x + 8)\)[/tex] by [tex]\(-6\)[/tex]:
- [tex]\(5x^2 \cdot (-6) = -30x^2\)[/tex]
- [tex]\(2x \cdot (-6) = -12x\)[/tex]
- [tex]\(8 \cdot (-6) = -48\)[/tex]
So, we get:
[tex]\(-30x^2 - 12x - 48\)[/tex]
3. Combine all the obtained terms:
[tex]\(35x^3 + 14x^2 + 56x - 30x^2 - 12x - 48\)[/tex]
4. Combine like terms:
- Combine [tex]\(14x^2\)[/tex] and [tex]\(-30x^2\)[/tex]: [tex]\(14x^2 - 30x^2 = -16x^2\)[/tex]
- Combine [tex]\(56x\)[/tex] and [tex]\(-12x\)[/tex]: [tex]\(56x - 12x = 44x\)[/tex]
So, the expression simplifies to:
[tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]
Therefore, the answer is:
A. [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]
1. Multiply each term in [tex]\((5x^2 + 2x + 8)\)[/tex] by [tex]\(7x\)[/tex]:
- [tex]\(5x^2 \cdot 7x = 35x^3\)[/tex]
- [tex]\(2x \cdot 7x = 14x^2\)[/tex]
- [tex]\(8 \cdot 7x = 56x\)[/tex]
So, we get:
[tex]\(35x^3 + 14x^2 + 56x\)[/tex]
2. Multiply each term in [tex]\((5x^2 + 2x + 8)\)[/tex] by [tex]\(-6\)[/tex]:
- [tex]\(5x^2 \cdot (-6) = -30x^2\)[/tex]
- [tex]\(2x \cdot (-6) = -12x\)[/tex]
- [tex]\(8 \cdot (-6) = -48\)[/tex]
So, we get:
[tex]\(-30x^2 - 12x - 48\)[/tex]
3. Combine all the obtained terms:
[tex]\(35x^3 + 14x^2 + 56x - 30x^2 - 12x - 48\)[/tex]
4. Combine like terms:
- Combine [tex]\(14x^2\)[/tex] and [tex]\(-30x^2\)[/tex]: [tex]\(14x^2 - 30x^2 = -16x^2\)[/tex]
- Combine [tex]\(56x\)[/tex] and [tex]\(-12x\)[/tex]: [tex]\(56x - 12x = 44x\)[/tex]
So, the expression simplifies to:
[tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]
Therefore, the answer is:
A. [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]
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