Answer :
To find the product [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we will multiply these expressions step by step.
1. Identifying Terms:
- First, recognize each component of the expression:
- [tex]\(7x^2\)[/tex] is a single term.
- [tex]\(2x^3 + 5\)[/tex] is a binomial.
- [tex]\(x^2 - 4x - 9\)[/tex] is a trinomial.
2. Multiply the First Two Expressions:
- Start by multiplying [tex]\(7x^2\)[/tex] with each term in [tex]\((2x^3 + 5)\)[/tex]:
- [tex]\((7x^2) \times (2x^3) = 14x^5\)[/tex]
- [tex]\((7x^2) \times 5 = 35x^2\)[/tex]
- The result of the first multiplication is:
[tex]\[14x^5 + 35x^2\][/tex]
3. Multiply the Result by the Third Expression:
- Now, multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex]. Distribute each term in the binomial across each term in the trinomial:
- Distributing [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]
- Distributing [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]
4. Combine the Like Terms:
- Now, we combine all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
5. Final Result:
- The expanded form of multiplying the given expressions is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the polynomial expression representing the product.
1. Identifying Terms:
- First, recognize each component of the expression:
- [tex]\(7x^2\)[/tex] is a single term.
- [tex]\(2x^3 + 5\)[/tex] is a binomial.
- [tex]\(x^2 - 4x - 9\)[/tex] is a trinomial.
2. Multiply the First Two Expressions:
- Start by multiplying [tex]\(7x^2\)[/tex] with each term in [tex]\((2x^3 + 5)\)[/tex]:
- [tex]\((7x^2) \times (2x^3) = 14x^5\)[/tex]
- [tex]\((7x^2) \times 5 = 35x^2\)[/tex]
- The result of the first multiplication is:
[tex]\[14x^5 + 35x^2\][/tex]
3. Multiply the Result by the Third Expression:
- Now, multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex]. Distribute each term in the binomial across each term in the trinomial:
- Distributing [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]
- Distributing [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]
4. Combine the Like Terms:
- Now, we combine all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
5. Final Result:
- The expanded form of multiplying the given expressions is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the polynomial expression representing the product.
