Answer :
Certainly! Let's find the product of the polynomials step-by-step:
We have three expressions:
1. [tex]\( 7x^2 \)[/tex]
2. [tex]\( 2x^3 + 5 \)[/tex]
3. [tex]\( x^2 - 4x - 9 \)[/tex]
Our goal is to multiply these expressions together.
### Step 1: Multiply the first two expressions
First, we'll multiply the expressions [tex]\( 7x^2 \)[/tex] and [tex]\( 2x^3 + 5 \)[/tex]:
[tex]\[
7x^2 \times (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
Calculating each term:
- [tex]\( 7x^2 \cdot 2x^3 = 14x^5 \)[/tex]
- [tex]\( 7x^2 \cdot 5 = 35x^2 \)[/tex]
So, the product of the first two expressions is:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result by the third expression
Now, we take the result from the first step and multiply it by the third expression [tex]\( x^2 - 4x - 9 \)[/tex]:
[tex]\[
(14x^5 + 35x^2) \times (x^2 - 4x - 9)
\][/tex]
We'll distribute each term in [tex]\( 14x^5 + 35x^2 \)[/tex] across each term in [tex]\( x^2 - 4x - 9 \)[/tex].
Expanding:
- [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]
Now with [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 combine all these terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is our final expanded polynomial after multiplying all three expressions together. Each term in the resulting polynomial corresponds to a combined and simplified multiplication of terms from the original expressions.
We have three expressions:
1. [tex]\( 7x^2 \)[/tex]
2. [tex]\( 2x^3 + 5 \)[/tex]
3. [tex]\( x^2 - 4x - 9 \)[/tex]
Our goal is to multiply these expressions together.
### Step 1: Multiply the first two expressions
First, we'll multiply the expressions [tex]\( 7x^2 \)[/tex] and [tex]\( 2x^3 + 5 \)[/tex]:
[tex]\[
7x^2 \times (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
Calculating each term:
- [tex]\( 7x^2 \cdot 2x^3 = 14x^5 \)[/tex]
- [tex]\( 7x^2 \cdot 5 = 35x^2 \)[/tex]
So, the product of the first two expressions is:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result by the third expression
Now, we take the result from the first step and multiply it by the third expression [tex]\( x^2 - 4x - 9 \)[/tex]:
[tex]\[
(14x^5 + 35x^2) \times (x^2 - 4x - 9)
\][/tex]
We'll distribute each term in [tex]\( 14x^5 + 35x^2 \)[/tex] across each term in [tex]\( x^2 - 4x - 9 \)[/tex].
Expanding:
- [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]
Now with [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 combine all these terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is our final expanded polynomial after multiplying all three expressions together. Each term in the resulting polynomial corresponds to a combined and simplified multiplication of terms from the original expressions.
