Answer :
Sure, let's find the product of the given polynomials step-by-step:
We have three terms to multiply: [tex]\((7x^2)\)[/tex], [tex]\((2x^3 + 5)\)[/tex], and [tex]\((x^2 - 4x - 9)\)[/tex].
Step 1: Multiply the first two polynomials:
[tex]\[ (7x^2)(2x^3 + 5) \][/tex]
Distribute [tex]\(7x^2\)[/tex] into each term inside the parentheses:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(2x^3\)[/tex]:
[tex]\[ 7x^2 \cdot 2x^3 = 14x^{2+3} = 14x^5 \][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[ 7x^2 \cdot 5 = 35x^2 \][/tex]
After distributing, the result is:
[tex]\[ 14x^5 + 35x^2 \][/tex]
Step 2: Multiply the result from Step 1 with the third polynomial:
Now we need to multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex].
Distribute each term in [tex]\((14x^5 + 35x^2)\)[/tex] across each term in [tex]\((x^2 - 4x - 9)\)[/tex].
- Multiply [tex]\(14x^5\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[ 14x^5 \cdot x^2 = 14x^{5+2} = 14x^7 \][/tex]
[tex]\[ 14x^5 \cdot (-4x) = -56x^{5+1} = -56x^6 \][/tex]
[tex]\[ 14x^5 \cdot (-9) = -126x^5 \][/tex]
- Multiply [tex]\(35x^2\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[ 35x^2 \cdot x^2 = 35x^{2+2} = 35x^4 \][/tex]
[tex]\[ 35x^2 \cdot (-4x) = -140x^{2+1} = -140x^3 \][/tex]
[tex]\[ 35x^2 \cdot (-9) = -315x^2 \][/tex]
Step 3: Combine all terms:
Combine all the terms we found:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
And there we have the product of the three given polynomials.
We have three terms to multiply: [tex]\((7x^2)\)[/tex], [tex]\((2x^3 + 5)\)[/tex], and [tex]\((x^2 - 4x - 9)\)[/tex].
Step 1: Multiply the first two polynomials:
[tex]\[ (7x^2)(2x^3 + 5) \][/tex]
Distribute [tex]\(7x^2\)[/tex] into each term inside the parentheses:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(2x^3\)[/tex]:
[tex]\[ 7x^2 \cdot 2x^3 = 14x^{2+3} = 14x^5 \][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[ 7x^2 \cdot 5 = 35x^2 \][/tex]
After distributing, the result is:
[tex]\[ 14x^5 + 35x^2 \][/tex]
Step 2: Multiply the result from Step 1 with the third polynomial:
Now we need to multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex].
Distribute each term in [tex]\((14x^5 + 35x^2)\)[/tex] across each term in [tex]\((x^2 - 4x - 9)\)[/tex].
- Multiply [tex]\(14x^5\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[ 14x^5 \cdot x^2 = 14x^{5+2} = 14x^7 \][/tex]
[tex]\[ 14x^5 \cdot (-4x) = -56x^{5+1} = -56x^6 \][/tex]
[tex]\[ 14x^5 \cdot (-9) = -126x^5 \][/tex]
- Multiply [tex]\(35x^2\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[ 35x^2 \cdot x^2 = 35x^{2+2} = 35x^4 \][/tex]
[tex]\[ 35x^2 \cdot (-4x) = -140x^{2+1} = -140x^3 \][/tex]
[tex]\[ 35x^2 \cdot (-9) = -315x^2 \][/tex]
Step 3: Combine all terms:
Combine all the terms we found:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
And there we have the product of the three given polynomials.
