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Answer :
To multiply the polynomials [tex]\((8x^2 + 6x + 8)(6x - 5)\)[/tex], we will use the distributive property to expand the expression step by step.
Here’s how you do it:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- First, multiply [tex]\(8x^2\)[/tex] by both terms in [tex]\(6x - 5\)[/tex]:
- [tex]\(8x^2 \times 6x = 48x^3\)[/tex]
- [tex]\(8x^2 \times (-5) = -40x^2\)[/tex]
- Next, multiply [tex]\(6x\)[/tex] by both terms in [tex]\(6x - 5\)[/tex]:
- [tex]\(6x \times 6x = 36x^2\)[/tex]
- [tex]\(6x \times (-5) = -30x\)[/tex]
- Lastly, multiply [tex]\(8\)[/tex] by both terms in [tex]\(6x - 5\)[/tex]:
- [tex]\(8 \times 6x = 48x\)[/tex]
- [tex]\(8 \times (-5) = -40\)[/tex]
2. Combine all the terms you’ve calculated:
- You’ll have these terms: [tex]\(48x^3\)[/tex], [tex]\(-40x^2\)[/tex], [tex]\(36x^2\)[/tex], [tex]\(-30x\)[/tex], [tex]\(48x\)[/tex], and [tex]\(-40\)[/tex].
3. Combine like terms:
- The terms with [tex]\(x^2\)[/tex] are [tex]\(-40x^2\)[/tex] and [tex]\(36x^2\)[/tex]:
- When you add these, you get [tex]\((-40 + 36)x^2 = -4x^2\)[/tex].
- The terms with [tex]\(x\)[/tex] are [tex]\(-30x\)[/tex] and [tex]\(48x\)[/tex]:
- When you add these, you get [tex]\((-30 + 48)x = 18x\)[/tex].
4. Write down the final expanded expression:
- Combine all your results:
- [tex]\(48x^3\)[/tex] (from the first multiplication)
- [tex]\(-4x^2\)[/tex] (from combining the [tex]\(x^2\)[/tex] terms)
- [tex]\(18x\)[/tex] (from combining the [tex]\(x\)[/tex] terms)
- [tex]\(-40\)[/tex] (the constant term)
The expanded polynomial is [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
Therefore, the correct option is:
D. [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
Here’s how you do it:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- First, multiply [tex]\(8x^2\)[/tex] by both terms in [tex]\(6x - 5\)[/tex]:
- [tex]\(8x^2 \times 6x = 48x^3\)[/tex]
- [tex]\(8x^2 \times (-5) = -40x^2\)[/tex]
- Next, multiply [tex]\(6x\)[/tex] by both terms in [tex]\(6x - 5\)[/tex]:
- [tex]\(6x \times 6x = 36x^2\)[/tex]
- [tex]\(6x \times (-5) = -30x\)[/tex]
- Lastly, multiply [tex]\(8\)[/tex] by both terms in [tex]\(6x - 5\)[/tex]:
- [tex]\(8 \times 6x = 48x\)[/tex]
- [tex]\(8 \times (-5) = -40\)[/tex]
2. Combine all the terms you’ve calculated:
- You’ll have these terms: [tex]\(48x^3\)[/tex], [tex]\(-40x^2\)[/tex], [tex]\(36x^2\)[/tex], [tex]\(-30x\)[/tex], [tex]\(48x\)[/tex], and [tex]\(-40\)[/tex].
3. Combine like terms:
- The terms with [tex]\(x^2\)[/tex] are [tex]\(-40x^2\)[/tex] and [tex]\(36x^2\)[/tex]:
- When you add these, you get [tex]\((-40 + 36)x^2 = -4x^2\)[/tex].
- The terms with [tex]\(x\)[/tex] are [tex]\(-30x\)[/tex] and [tex]\(48x\)[/tex]:
- When you add these, you get [tex]\((-30 + 48)x = 18x\)[/tex].
4. Write down the final expanded expression:
- Combine all your results:
- [tex]\(48x^3\)[/tex] (from the first multiplication)
- [tex]\(-4x^2\)[/tex] (from combining the [tex]\(x^2\)[/tex] terms)
- [tex]\(18x\)[/tex] (from combining the [tex]\(x\)[/tex] terms)
- [tex]\(-40\)[/tex] (the constant term)
The expanded polynomial is [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
Therefore, the correct option is:
D. [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
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