Answer :
To find the equivalent expressions for [tex]\(\left(-7 x^3 + 9 x^2 - 3\right) \cdot \left(-2 x^2 - 5 x + 6\right)\)[/tex], we need to multiply each term in the first polynomial by each term in the second polynomial and then combine like terms. Let's go through this step-by-step.
### Step 1: Distribute and Multiply Each Pair
1. Distribute [tex]\(-7x^3\)[/tex] to every term in the second polynomial:
[tex]\[
-7x^3 \cdot (-2x^2) = 14x^5
\][/tex]
[tex]\[
-7x^3 \cdot (-5x) = 35x^4
\][/tex]
[tex]\[
-7x^3 \cdot 6 = -42x^3
\][/tex]
2. Distribute [tex]\(9x^2\)[/tex] to every term in the second polynomial:
[tex]\[
9x^2 \cdot (-2x^2) = -18x^4
\][/tex]
[tex]\[
9x^2 \cdot (-5x) = -45x^3
\][/tex]
[tex]\[
9x^2 \cdot 6 = 54x^2
\][/tex]
3. Distribute [tex]\(-3\)[/tex] to every term in the second polynomial:
[tex]\[
-3 \cdot (-2x^2) = 6x^2
\][/tex]
[tex]\[
-3 \cdot (-5x) = 15x
\][/tex]
[tex]\[
-3 \cdot 6 = -18
\][/tex]
### Step 2: Combine Like Terms
Now, let's combine the terms by matching the same powers of [tex]\(x\)[/tex]:
[tex]\[
14x^5
\][/tex]
[tex]\[
35x^4 - 18x^4 = 17x^4
\][/tex]
[tex]\[
-42x^3 - 45x^3 = -87x^3
\][/tex]
[tex]\[
54x^2 + 6x^2 = 60x^2
\][/tex]
[tex]\[
15x
\][/tex]
[tex]\[
-18
\][/tex]
Putting it all together, we get:
[tex]\[
14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18
\][/tex]
### Checking the Given Options
Based on our calculations, let's verify the options:
1. [tex]\( 14x^5 - 17x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex]
- This is incorrect because the coefficient of [tex]\(x^4\)[/tex] should be [tex]\(+17\)[/tex], not [tex]\(-17\)[/tex].
2. [tex]\( 14x^5 + 35x^4 - 18x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex]
- This can be simplified to [tex]\(14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex], so it's correct.
3. [tex]\( 14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex]
- This matches our combined terms, so it's correct.
4. [tex]\( 14x^5 + 17x^4 - 87x^3 + 40x^2 + 20x^2 + 15x - 18 \)[/tex]
- This can be simplified to [tex]\( 14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex], so it's correct.
5. [tex]\( 14x^5 + 53x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex]
- This is incorrect; the coefficient of [tex]\(x^4\)[/tex] should be [tex]\(+17\)[/tex], not [tex]\(+53\)[/tex].
### Correct Options
The equivalent expressions are:
- [tex]\( 14x^5 + 35x^4 - 18x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex]
- [tex]\( 14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex]
- [tex]\( 14x^5 + 17x^4 - 87x^3 + 40x^2 + 20x^2 + 15x - 18 \)[/tex]
### Step 1: Distribute and Multiply Each Pair
1. Distribute [tex]\(-7x^3\)[/tex] to every term in the second polynomial:
[tex]\[
-7x^3 \cdot (-2x^2) = 14x^5
\][/tex]
[tex]\[
-7x^3 \cdot (-5x) = 35x^4
\][/tex]
[tex]\[
-7x^3 \cdot 6 = -42x^3
\][/tex]
2. Distribute [tex]\(9x^2\)[/tex] to every term in the second polynomial:
[tex]\[
9x^2 \cdot (-2x^2) = -18x^4
\][/tex]
[tex]\[
9x^2 \cdot (-5x) = -45x^3
\][/tex]
[tex]\[
9x^2 \cdot 6 = 54x^2
\][/tex]
3. Distribute [tex]\(-3\)[/tex] to every term in the second polynomial:
[tex]\[
-3 \cdot (-2x^2) = 6x^2
\][/tex]
[tex]\[
-3 \cdot (-5x) = 15x
\][/tex]
[tex]\[
-3 \cdot 6 = -18
\][/tex]
### Step 2: Combine Like Terms
Now, let's combine the terms by matching the same powers of [tex]\(x\)[/tex]:
[tex]\[
14x^5
\][/tex]
[tex]\[
35x^4 - 18x^4 = 17x^4
\][/tex]
[tex]\[
-42x^3 - 45x^3 = -87x^3
\][/tex]
[tex]\[
54x^2 + 6x^2 = 60x^2
\][/tex]
[tex]\[
15x
\][/tex]
[tex]\[
-18
\][/tex]
Putting it all together, we get:
[tex]\[
14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18
\][/tex]
### Checking the Given Options
Based on our calculations, let's verify the options:
1. [tex]\( 14x^5 - 17x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex]
- This is incorrect because the coefficient of [tex]\(x^4\)[/tex] should be [tex]\(+17\)[/tex], not [tex]\(-17\)[/tex].
2. [tex]\( 14x^5 + 35x^4 - 18x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex]
- This can be simplified to [tex]\(14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex], so it's correct.
3. [tex]\( 14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex]
- This matches our combined terms, so it's correct.
4. [tex]\( 14x^5 + 17x^4 - 87x^3 + 40x^2 + 20x^2 + 15x - 18 \)[/tex]
- This can be simplified to [tex]\( 14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex], so it's correct.
5. [tex]\( 14x^5 + 53x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex]
- This is incorrect; the coefficient of [tex]\(x^4\)[/tex] should be [tex]\(+17\)[/tex], not [tex]\(+53\)[/tex].
### Correct Options
The equivalent expressions are:
- [tex]\( 14x^5 + 35x^4 - 18x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex]
- [tex]\( 14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18 \)[/tex]
- [tex]\( 14x^5 + 17x^4 - 87x^3 + 40x^2 + 20x^2 + 15x - 18 \)[/tex]
