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Answer :
To find the product of the polynomials [tex]\((7x^2)\)[/tex], [tex]\((2x^3 + 5)\)[/tex], and [tex]\((x^2 - 4x - 9)\)[/tex], we will multiply them together step-by-step.
### Step 1: Multiply the first two polynomials
Start with [tex]\(7x^2\)[/tex] and [tex]\(2x^3 + 5\)[/tex]:
[tex]\[
(7x^2) \times (2x^3 + 5) = 7x^2 \times 2x^3 + 7x^2 \times 5
\][/tex]
Calculating each term gives:
- [tex]\(7x^2 \times 2x^3 = 14x^{2+3} = 14x^5\)[/tex]
- [tex]\(7x^2 \times 5 = 35x^2\)[/tex]
Thus, the result of the first multiplication is:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result by the third polynomial
Now, take the result [tex]\(14x^5 + 35x^2\)[/tex] and multiply it by the third polynomial [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
(14x^5 + 35x^2) \times (x^2 - 4x - 9)
\][/tex]
Distribute each term in [tex]\(14x^5 + 35x^2\)[/tex] across [tex]\(x^2 - 4x - 9\)[/tex] as follows:
1. Multiply [tex]\(14x^5\)[/tex] by each term in [tex]\(x^2 - 4x - 9\)[/tex]:
- [tex]\(14x^5 \times x^2 = 14x^{5+2} = 14x^7\)[/tex]
- [tex]\(14x^5 \times (-4x) = -56x^{5+1} = -56x^6\)[/tex]
- [tex]\(14x^5 \times (-9) = -126x^5\)[/tex]
2. Multiply [tex]\(35x^2\)[/tex] by each term in [tex]\(x^2 - 4x - 9\)[/tex]:
- [tex]\(35x^2 \times x^2 = 35x^{2+2} = 35x^4\)[/tex]
- [tex]\(35x^2 \times (-4x) = -140x^{2+1} = -140x^3\)[/tex]
- [tex]\(35x^2 \times (-9) = -315x^2\)[/tex]
Now, combine all these results to form the complete expanded polynomial:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This polynomial represents the final product of multiplying the original three polynomials together.
### Step 1: Multiply the first two polynomials
Start with [tex]\(7x^2\)[/tex] and [tex]\(2x^3 + 5\)[/tex]:
[tex]\[
(7x^2) \times (2x^3 + 5) = 7x^2 \times 2x^3 + 7x^2 \times 5
\][/tex]
Calculating each term gives:
- [tex]\(7x^2 \times 2x^3 = 14x^{2+3} = 14x^5\)[/tex]
- [tex]\(7x^2 \times 5 = 35x^2\)[/tex]
Thus, the result of the first multiplication is:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result by the third polynomial
Now, take the result [tex]\(14x^5 + 35x^2\)[/tex] and multiply it by the third polynomial [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
(14x^5 + 35x^2) \times (x^2 - 4x - 9)
\][/tex]
Distribute each term in [tex]\(14x^5 + 35x^2\)[/tex] across [tex]\(x^2 - 4x - 9\)[/tex] as follows:
1. Multiply [tex]\(14x^5\)[/tex] by each term in [tex]\(x^2 - 4x - 9\)[/tex]:
- [tex]\(14x^5 \times x^2 = 14x^{5+2} = 14x^7\)[/tex]
- [tex]\(14x^5 \times (-4x) = -56x^{5+1} = -56x^6\)[/tex]
- [tex]\(14x^5 \times (-9) = -126x^5\)[/tex]
2. Multiply [tex]\(35x^2\)[/tex] by each term in [tex]\(x^2 - 4x - 9\)[/tex]:
- [tex]\(35x^2 \times x^2 = 35x^{2+2} = 35x^4\)[/tex]
- [tex]\(35x^2 \times (-4x) = -140x^{2+1} = -140x^3\)[/tex]
- [tex]\(35x^2 \times (-9) = -315x^2\)[/tex]
Now, combine all these results to form the complete expanded polynomial:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This polynomial represents the final product of multiplying the original three polynomials together.
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