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Answer :
To multiply the two expressions [tex]\((x^2 - 5x)\)[/tex] and [tex]\((2x^2 + x - 3)\)[/tex], we'll use the distributive property, also known as the FOIL method for binomials. However, these are not binomials, so we'll distribute each term in the first polynomial to every term in the second polynomial. Here is a step-by-step breakdown:
1. Distribute [tex]\(x^2\)[/tex]:
- Multiply [tex]\(x^2\)[/tex] by [tex]\(2x^2\)[/tex]: [tex]\(x^2 \cdot 2x^2 = 2x^4\)[/tex]
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex]: [tex]\(x^2 \cdot x = x^3\)[/tex]
- Multiply [tex]\(x^2\)[/tex] by [tex]\(-3\)[/tex]: [tex]\(x^2 \cdot (-3) = -3x^2\)[/tex]
2. Distribute [tex]\(-5x\)[/tex]:
- Multiply [tex]\(-5x\)[/tex] by [tex]\(2x^2\)[/tex]: [tex]\(-5x \cdot 2x^2 = -10x^3\)[/tex]
- Multiply [tex]\(-5x\)[/tex] by [tex]\(x\)[/tex]: [tex]\(-5x \cdot x = -5x^2\)[/tex]
- Multiply [tex]\(-5x\)[/tex] by [tex]\(-3\)[/tex]: [tex]\(-5x \cdot (-3) = 15x\)[/tex]
3. Combine all these terms:
- Write out all the terms: [tex]\(2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x\)[/tex]
4. Combine like terms:
- Combine [tex]\(x^3\)[/tex] terms: [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
5. Write the final expression:
- The combined expression is: [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
Now, compare this with the options provided:
A. [tex]\(2x^4 + 9x^3 - 8x^2 + 15x\)[/tex]
B. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
C. [tex]\(4x^4 + 9x^3 - 8x^2 + 15x\)[/tex]
D. [tex]\(2x^4 - 9x^3 - 9x^2 - 15x\)[/tex]
The correct answer is B: [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
1. Distribute [tex]\(x^2\)[/tex]:
- Multiply [tex]\(x^2\)[/tex] by [tex]\(2x^2\)[/tex]: [tex]\(x^2 \cdot 2x^2 = 2x^4\)[/tex]
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex]: [tex]\(x^2 \cdot x = x^3\)[/tex]
- Multiply [tex]\(x^2\)[/tex] by [tex]\(-3\)[/tex]: [tex]\(x^2 \cdot (-3) = -3x^2\)[/tex]
2. Distribute [tex]\(-5x\)[/tex]:
- Multiply [tex]\(-5x\)[/tex] by [tex]\(2x^2\)[/tex]: [tex]\(-5x \cdot 2x^2 = -10x^3\)[/tex]
- Multiply [tex]\(-5x\)[/tex] by [tex]\(x\)[/tex]: [tex]\(-5x \cdot x = -5x^2\)[/tex]
- Multiply [tex]\(-5x\)[/tex] by [tex]\(-3\)[/tex]: [tex]\(-5x \cdot (-3) = 15x\)[/tex]
3. Combine all these terms:
- Write out all the terms: [tex]\(2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x\)[/tex]
4. Combine like terms:
- Combine [tex]\(x^3\)[/tex] terms: [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
5. Write the final expression:
- The combined expression is: [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
Now, compare this with the options provided:
A. [tex]\(2x^4 + 9x^3 - 8x^2 + 15x\)[/tex]
B. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
C. [tex]\(4x^4 + 9x^3 - 8x^2 + 15x\)[/tex]
D. [tex]\(2x^4 - 9x^3 - 9x^2 - 15x\)[/tex]
The correct answer is B: [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
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