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Answer :
To find the product of the expressions [tex]\((5x - 3)\)[/tex] and [tex]\((x^3 - 5x + 2)\)[/tex], we will use the distributive property, often known as the FOIL method for polynomials:
1. Multiply each term in the first expression by each term in the second expression.
2. Sum all the resulting products.
Let's break it down step by step:
Step 1: Expand [tex]\((5x - 3)(x^3 - 5x + 2)\)[/tex]:
- First, distribute [tex]\(5x\)[/tex] across each term in [tex]\((x^3 - 5x + 2)\)[/tex]:
- [tex]\(5x \cdot x^3 = 5x^4\)[/tex]
- [tex]\(5x \cdot (-5x) = -25x^2\)[/tex]
- [tex]\(5x \cdot 2 = 10x\)[/tex]
- Next, distribute [tex]\(-3\)[/tex] across each term in [tex]\((x^3 - 5x + 2)\)[/tex]:
- [tex]\(-3 \cdot x^3 = -3x^3\)[/tex]
- [tex]\(-3 \cdot (-5x) = 15x\)[/tex]
- [tex]\(-3 \cdot 2 = -6\)[/tex]
Step 2: Combine all the products:
- Start with the highest degree term: [tex]\(5x^4\)[/tex]
- Next, the [tex]\(x^3\)[/tex] term: [tex]\(-3x^3\)[/tex]
- Then, the [tex]\(x^2\)[/tex] term: [tex]\(-25x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(10x + 15x = 25x\)[/tex]
- Finally, the constant term: [tex]\(-6\)[/tex]
Now, sum up the terms:
[tex]\[
5x^4 - 3x^3 - 25x^2 + 25x - 6
\][/tex]
Hence, the product of the expressions is given by the polynomial:
[tex]\[ 5x^4 - 3x^3 - 25x^2 + 25x - 6 \][/tex]
The correct choice from the options provided is:
C) [tex]\(5x^4 - 3x^3 - 25x^2 + 25x - 6\)[/tex]
1. Multiply each term in the first expression by each term in the second expression.
2. Sum all the resulting products.
Let's break it down step by step:
Step 1: Expand [tex]\((5x - 3)(x^3 - 5x + 2)\)[/tex]:
- First, distribute [tex]\(5x\)[/tex] across each term in [tex]\((x^3 - 5x + 2)\)[/tex]:
- [tex]\(5x \cdot x^3 = 5x^4\)[/tex]
- [tex]\(5x \cdot (-5x) = -25x^2\)[/tex]
- [tex]\(5x \cdot 2 = 10x\)[/tex]
- Next, distribute [tex]\(-3\)[/tex] across each term in [tex]\((x^3 - 5x + 2)\)[/tex]:
- [tex]\(-3 \cdot x^3 = -3x^3\)[/tex]
- [tex]\(-3 \cdot (-5x) = 15x\)[/tex]
- [tex]\(-3 \cdot 2 = -6\)[/tex]
Step 2: Combine all the products:
- Start with the highest degree term: [tex]\(5x^4\)[/tex]
- Next, the [tex]\(x^3\)[/tex] term: [tex]\(-3x^3\)[/tex]
- Then, the [tex]\(x^2\)[/tex] term: [tex]\(-25x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(10x + 15x = 25x\)[/tex]
- Finally, the constant term: [tex]\(-6\)[/tex]
Now, sum up the terms:
[tex]\[
5x^4 - 3x^3 - 25x^2 + 25x - 6
\][/tex]
Hence, the product of the expressions is given by the polynomial:
[tex]\[ 5x^4 - 3x^3 - 25x^2 + 25x - 6 \][/tex]
The correct choice from the options provided is:
C) [tex]\(5x^4 - 3x^3 - 25x^2 + 25x - 6\)[/tex]
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