Answer :
To find the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we will multiply these polynomials together step by step.
Step 1: Multiply the first two polynomials
First, we'll multiply [tex]\((7x^2)\)[/tex] by [tex]\((2x^3 + 5)\)[/tex].
[tex]\[
7x^2 \times (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
[tex]\[
= 14x^5 + 35x^2
\][/tex]
This gives us the intermediate polynomial: [tex]\(14x^5 + 35x^2\)[/tex].
Step 2: Multiply the result by the third polynomial
Next, we multiply the polynomial from Step 1, [tex]\( (14x^5 + 35x^2) \)[/tex], by [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 the polynomial [tex]\((x^2 - 4x - 9)\)[/tex]:
1. Distributing [tex]\(14x^5\)[/tex]:
[tex]\[
14x^5 \times x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \times (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \times (-9) = -126x^5
\][/tex]
2. Distributing [tex]\(35x^2\)[/tex]:
[tex]\[
35x^2 \times x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \times (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \times (-9) = -315x^2
\][/tex]
Step 3: Combine all terms
Now, we add all these terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the final expanded form of the given expression, which represents the product of the three polynomials.
Step 1: Multiply the first two polynomials
First, we'll multiply [tex]\((7x^2)\)[/tex] by [tex]\((2x^3 + 5)\)[/tex].
[tex]\[
7x^2 \times (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
[tex]\[
= 14x^5 + 35x^2
\][/tex]
This gives us the intermediate polynomial: [tex]\(14x^5 + 35x^2\)[/tex].
Step 2: Multiply the result by the third polynomial
Next, we multiply the polynomial from Step 1, [tex]\( (14x^5 + 35x^2) \)[/tex], by [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 the polynomial [tex]\((x^2 - 4x - 9)\)[/tex]:
1. Distributing [tex]\(14x^5\)[/tex]:
[tex]\[
14x^5 \times x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \times (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \times (-9) = -126x^5
\][/tex]
2. Distributing [tex]\(35x^2\)[/tex]:
[tex]\[
35x^2 \times x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \times (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \times (-9) = -315x^2
\][/tex]
Step 3: Combine all terms
Now, we add all these terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the final expanded form of the given expression, which represents the product of the three polynomials.
