Answer :
To find the product [tex]\(\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)\)[/tex], we need to multiply the three expressions together. Let's break it down step-by-step:
### Step 1: Multiply the First Two Expressions
Start by multiplying [tex]\((7x^2)\)[/tex] with [tex]\((2x^3 + 5)\)[/tex].
1. Distribute [tex]\(7x^2\)[/tex] across each term inside the parenthesis:
- [tex]\(7x^2 \times 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \times 5 = 35x^2\)[/tex]
This gives us the intermediate result:
[tex]\[14x^5 + 35x^2\][/tex]
### Step 2: Multiply the Result with the Third Expression
Next, multiply the expression [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex].
1. Distribute each term in [tex]\((14x^5 + 35x^2)\)[/tex] across [tex]\((x^2 - 4x - 9)\)[/tex].
For [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]
For [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 the Terms
Now, add all the terms together:
- [tex]\(14x^7\)[/tex]
- [tex]\(-56x^6\)[/tex]
- [tex]\(-126x^5\)[/tex]
- [tex]\(35x^4\)[/tex]
- [tex]\(-140x^3\)[/tex]
- [tex]\(-315x^2\)[/tex]
After combining these terms, we get the final expression:
[tex]\[14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\][/tex]
Thus, the product of the given expression is:
[tex]\[14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\][/tex]
### Step 1: Multiply the First Two Expressions
Start by multiplying [tex]\((7x^2)\)[/tex] with [tex]\((2x^3 + 5)\)[/tex].
1. Distribute [tex]\(7x^2\)[/tex] across each term inside the parenthesis:
- [tex]\(7x^2 \times 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \times 5 = 35x^2\)[/tex]
This gives us the intermediate result:
[tex]\[14x^5 + 35x^2\][/tex]
### Step 2: Multiply the Result with the Third Expression
Next, multiply the expression [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex].
1. Distribute each term in [tex]\((14x^5 + 35x^2)\)[/tex] across [tex]\((x^2 - 4x - 9)\)[/tex].
For [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]
For [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 the Terms
Now, add all the terms together:
- [tex]\(14x^7\)[/tex]
- [tex]\(-56x^6\)[/tex]
- [tex]\(-126x^5\)[/tex]
- [tex]\(35x^4\)[/tex]
- [tex]\(-140x^3\)[/tex]
- [tex]\(-315x^2\)[/tex]
After combining these terms, we get the final expression:
[tex]\[14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\][/tex]
Thus, the product of the given expression is:
[tex]\[14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\][/tex]
