Answer :
Sure! Let's find the product of the expression step-by-step.
Given expression:
[tex]\[
\left(7 x^2\right)\left(2 x^3 + 5\right)\left(x^2 - 4 x - 9\right)
\][/tex]
We need to multiply these three expressions together. To do this systematically, we can follow these steps:
1. Multiply the first two expressions:
[tex]\[
\left(7 x^2\right)\left(2 x^3 + 5\right)
\][/tex]
Distribute [tex]\(7x^2\)[/tex] to each term inside the parentheses:
[tex]\[
(7 x^2) \cdot (2 x^3) + (7 x^2) \cdot (5)
\][/tex]
Calculate each term:
[tex]\[
7 x^2 \cdot 2 x^3 = 14 x^5
\][/tex]
[tex]\[
7 x^2 \cdot 5 = 35 x^2
\][/tex]
So, the result of this multiplication is:
[tex]\[
14 x^5 + 35 x^2
\][/tex]
2. Multiply the result by the third expression:
[tex]\[
\left(14 x^5 + 35 x^2\right) \left(x^2 - 4 x - 9\right)
\][/tex]
Distribute each term in the first polynomial to every term in the second polynomial:
[tex]\[
(14 x^5) \cdot (x^2) + (14 x^5) \cdot (-4 x) + (14 x^5) \cdot (-9)
+
(35 x^2) \cdot (x^2) + (35 x^2) \cdot (-4 x) + (35 x^2) \cdot (-9)
\][/tex]
Calculate each term:
[tex]\[
14 x^5 \cdot x^2 = 14 x^7
\][/tex]
[tex]\[
14 x^5 \cdot (-4 x) = -56 x^6
\][/tex]
[tex]\[
14 x^5 \cdot (-9) = -126 x^5
\][/tex]
[tex]\[
35 x^2 \cdot x^2 = 35 x^4
\][/tex]
[tex]\[
35 x^2 \cdot (-4 x) = -140 x^3
\][/tex]
[tex]\[
35 x^2 \cdot (-9) = -315 x^2
\][/tex]
3. Combine all terms to get the final expression:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
So, the product of the given expressions is:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
Given expression:
[tex]\[
\left(7 x^2\right)\left(2 x^3 + 5\right)\left(x^2 - 4 x - 9\right)
\][/tex]
We need to multiply these three expressions together. To do this systematically, we can follow these steps:
1. Multiply the first two expressions:
[tex]\[
\left(7 x^2\right)\left(2 x^3 + 5\right)
\][/tex]
Distribute [tex]\(7x^2\)[/tex] to each term inside the parentheses:
[tex]\[
(7 x^2) \cdot (2 x^3) + (7 x^2) \cdot (5)
\][/tex]
Calculate each term:
[tex]\[
7 x^2 \cdot 2 x^3 = 14 x^5
\][/tex]
[tex]\[
7 x^2 \cdot 5 = 35 x^2
\][/tex]
So, the result of this multiplication is:
[tex]\[
14 x^5 + 35 x^2
\][/tex]
2. Multiply the result by the third expression:
[tex]\[
\left(14 x^5 + 35 x^2\right) \left(x^2 - 4 x - 9\right)
\][/tex]
Distribute each term in the first polynomial to every term in the second polynomial:
[tex]\[
(14 x^5) \cdot (x^2) + (14 x^5) \cdot (-4 x) + (14 x^5) \cdot (-9)
+
(35 x^2) \cdot (x^2) + (35 x^2) \cdot (-4 x) + (35 x^2) \cdot (-9)
\][/tex]
Calculate each term:
[tex]\[
14 x^5 \cdot x^2 = 14 x^7
\][/tex]
[tex]\[
14 x^5 \cdot (-4 x) = -56 x^6
\][/tex]
[tex]\[
14 x^5 \cdot (-9) = -126 x^5
\][/tex]
[tex]\[
35 x^2 \cdot x^2 = 35 x^4
\][/tex]
[tex]\[
35 x^2 \cdot (-4 x) = -140 x^3
\][/tex]
[tex]\[
35 x^2 \cdot (-9) = -315 x^2
\][/tex]
3. Combine all terms to get the final expression:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
So, the product of the given expressions is:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
