Answer :
To find the product [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], let's break down the process step by step.
1. First Step: Distribute the first two terms
We'll start by multiplying the first two components: [tex]\( (7x^2) \times (2x^3 + 5) \)[/tex].
- Multiply [tex]\( 7x^2 \cdot 2x^3 \)[/tex]:
[tex]\[
7 \cdot 2 = 14 \quad \text{and} \quad x^2 \cdot x^3 = x^{2+3} = x^5
\][/tex]
So, [tex]\( 7x^2 \cdot 2x^3 = 14x^5 \)[/tex].
- Multiply [tex]\( 7x^2 \cdot 5 \)[/tex]:
[tex]\[
7 \cdot 5 = 35 \quad \text{so,} \quad 7x^2 \cdot 5 = 35x^2
\][/tex]
This gives us the expanded form: [tex]\( 14x^5 + 35x^2 \)[/tex].
2. Second Step: Distribute the result with the third term
Next, multiply the expanded expression [tex]\( (14x^5 + 35x^2) \)[/tex] with [tex]\( (x^2 - 4x - 9) \)[/tex].
- Start by distributing [tex]\(14x^5\)[/tex] to each term in the second parenthesis:
[tex]\[
14x^5 \cdot x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \cdot (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \cdot (-9) = -126x^5
\][/tex]
- Next, distribute [tex]\(35x^2\)[/tex] to each term in the parenthesis:
[tex]\[
35x^2 \cdot x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \cdot (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \cdot (-9) = -315x^2
\][/tex]
3. Combine all terms
Now, put everything together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the final product of multiplying the given polynomials, and the correct answer is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
1. First Step: Distribute the first two terms
We'll start by multiplying the first two components: [tex]\( (7x^2) \times (2x^3 + 5) \)[/tex].
- Multiply [tex]\( 7x^2 \cdot 2x^3 \)[/tex]:
[tex]\[
7 \cdot 2 = 14 \quad \text{and} \quad x^2 \cdot x^3 = x^{2+3} = x^5
\][/tex]
So, [tex]\( 7x^2 \cdot 2x^3 = 14x^5 \)[/tex].
- Multiply [tex]\( 7x^2 \cdot 5 \)[/tex]:
[tex]\[
7 \cdot 5 = 35 \quad \text{so,} \quad 7x^2 \cdot 5 = 35x^2
\][/tex]
This gives us the expanded form: [tex]\( 14x^5 + 35x^2 \)[/tex].
2. Second Step: Distribute the result with the third term
Next, multiply the expanded expression [tex]\( (14x^5 + 35x^2) \)[/tex] with [tex]\( (x^2 - 4x - 9) \)[/tex].
- Start by distributing [tex]\(14x^5\)[/tex] to each term in the second parenthesis:
[tex]\[
14x^5 \cdot x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \cdot (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \cdot (-9) = -126x^5
\][/tex]
- Next, distribute [tex]\(35x^2\)[/tex] to each term in the parenthesis:
[tex]\[
35x^2 \cdot x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \cdot (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \cdot (-9) = -315x^2
\][/tex]
3. Combine all terms
Now, put everything together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the final product of multiplying the given polynomials, and the correct answer is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
