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Answer :
To find the product of the polynomials [tex]\((7x^2 - 2)\)[/tex] and [tex]\((3x^2 - 5x)\)[/tex], we need to use the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last) when dealing with binomials. Let's break it down step-by-step:
1. First terms: Multiply the first terms of each polynomial.
[tex]\[ 7x^2 \cdot 3x^2 = 21x^4 \][/tex]
2. Outer terms: Multiply the outer terms of each polynomial.
[tex]\[ 7x^2 \cdot (-5x) = -35x^3 \][/tex]
3. Inner terms: Multiply the inner terms of each polynomial.
[tex]\[ -2 \cdot 3x^2 = -6x^2 \][/tex]
4. Last terms: Multiply the last terms of each polynomial.
[tex]\[ -2 \cdot (-5x) = 10x \][/tex]
Next, combine all these results to get the product of the polynomials:
[tex]\[ 21x^4 - 35x^3 - 6x^2 + 10x \][/tex]
Therefore, the expanded product of the polynomials [tex]\((7x^2 - 2)\)[/tex] and [tex]\((3x^2 - 5x)\)[/tex] is:
[tex]\[ 21x^4 - 35x^3 - 6x^2 + 10x \][/tex]
This matches one of the provided options: [tex]\(21x^4 - 35x^3 - 6x^2 + 10x\)[/tex].
1. First terms: Multiply the first terms of each polynomial.
[tex]\[ 7x^2 \cdot 3x^2 = 21x^4 \][/tex]
2. Outer terms: Multiply the outer terms of each polynomial.
[tex]\[ 7x^2 \cdot (-5x) = -35x^3 \][/tex]
3. Inner terms: Multiply the inner terms of each polynomial.
[tex]\[ -2 \cdot 3x^2 = -6x^2 \][/tex]
4. Last terms: Multiply the last terms of each polynomial.
[tex]\[ -2 \cdot (-5x) = 10x \][/tex]
Next, combine all these results to get the product of the polynomials:
[tex]\[ 21x^4 - 35x^3 - 6x^2 + 10x \][/tex]
Therefore, the expanded product of the polynomials [tex]\((7x^2 - 2)\)[/tex] and [tex]\((3x^2 - 5x)\)[/tex] is:
[tex]\[ 21x^4 - 35x^3 - 6x^2 + 10x \][/tex]
This matches one of the provided options: [tex]\(21x^4 - 35x^3 - 6x^2 + 10x\)[/tex].
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