Answer :
To find the simplest form of the expression [tex]\((4x^3 + 6x - 7) + (3x^3 - 5x^2 - 5x + 9)\)[/tex], we need to combine like terms. Here's how you can do it step-by-step:
1. Group the like terms:
- Cubic terms ([tex]\(x^3\)[/tex]): [tex]\(4x^3\)[/tex] and [tex]\(3x^3\)[/tex].
- Quadratic terms ([tex]\(x^2\)[/tex]): There's no [tex]\(x^2\)[/tex] term in the first polynomial, and [tex]\(-5x^2\)[/tex] in the second polynomial.
- Linear terms ([tex]\(x\)[/tex]): [tex]\(6x\)[/tex] and [tex]\(-5x\)[/tex].
- Constant terms: [tex]\(-7\)[/tex] and [tex]\(+9\)[/tex].
2. Combine the like terms:
- Cubic terms: Add the coefficients of the [tex]\(x^3\)[/tex] terms: [tex]\(4 + 3 = 7\)[/tex]. So, the combined [tex]\(x^3\)[/tex] term is [tex]\(7x^3\)[/tex].
- Quadratic terms: The only quadratic term is [tex]\(-5x^2\)[/tex], so it remains [tex]\(-5x^2\)[/tex].
- Linear terms: Add the coefficients of the [tex]\(x\)[/tex] terms: [tex]\(6 - 5 = 1\)[/tex]. So, the combined [tex]\(x\)[/tex] term is [tex]\(1x\)[/tex], or simply [tex]\(x\)[/tex].
- Constant terms: Add the constant terms: [tex]\(-7 + 9 = 2\)[/tex].
3. Write the final simplified expression:
Combine all the terms you found:
[tex]\[
7x^3 - 5x^2 + x + 2
\][/tex]
Thus, the simplest form of the expression is [tex]\(7x^3 - 5x^2 + x + 2\)[/tex]. The correct answer is option A: [tex]\(7x^3 - 5x^2 - x + 2\)[/tex].
1. Group the like terms:
- Cubic terms ([tex]\(x^3\)[/tex]): [tex]\(4x^3\)[/tex] and [tex]\(3x^3\)[/tex].
- Quadratic terms ([tex]\(x^2\)[/tex]): There's no [tex]\(x^2\)[/tex] term in the first polynomial, and [tex]\(-5x^2\)[/tex] in the second polynomial.
- Linear terms ([tex]\(x\)[/tex]): [tex]\(6x\)[/tex] and [tex]\(-5x\)[/tex].
- Constant terms: [tex]\(-7\)[/tex] and [tex]\(+9\)[/tex].
2. Combine the like terms:
- Cubic terms: Add the coefficients of the [tex]\(x^3\)[/tex] terms: [tex]\(4 + 3 = 7\)[/tex]. So, the combined [tex]\(x^3\)[/tex] term is [tex]\(7x^3\)[/tex].
- Quadratic terms: The only quadratic term is [tex]\(-5x^2\)[/tex], so it remains [tex]\(-5x^2\)[/tex].
- Linear terms: Add the coefficients of the [tex]\(x\)[/tex] terms: [tex]\(6 - 5 = 1\)[/tex]. So, the combined [tex]\(x\)[/tex] term is [tex]\(1x\)[/tex], or simply [tex]\(x\)[/tex].
- Constant terms: Add the constant terms: [tex]\(-7 + 9 = 2\)[/tex].
3. Write the final simplified expression:
Combine all the terms you found:
[tex]\[
7x^3 - 5x^2 + x + 2
\][/tex]
Thus, the simplest form of the expression is [tex]\(7x^3 - 5x^2 + x + 2\)[/tex]. The correct answer is option A: [tex]\(7x^3 - 5x^2 - x + 2\)[/tex].
