Answer :
To find the sum of the two polynomials [tex]\((x^3 + 12x^2 - 5x + 4)\)[/tex] and [tex]\((2x^3 - 5x^2 - 14)\)[/tex], we need to add the corresponding coefficients. Let's break this process down step by step.
1. Identify the terms of each polynomial:
- The first polynomial is [tex]\(x^3 + 12x^2 - 5x + 4\)[/tex].
- Coefficient of [tex]\(x^3\)[/tex] is 1.
- Coefficient of [tex]\(x^2\)[/tex] is 12.
- Coefficient of [tex]\(x\)[/tex] is -5.
- Constant term is 4.
- The second polynomial is [tex]\(2x^3 - 5x^2 - 14\)[/tex].
- Coefficient of [tex]\(x^3\)[/tex] is 2.
- Coefficient of [tex]\(x^2\)[/tex] is -5.
- Coefficient of [tex]\(x\)[/tex] is 0 (since there is no [tex]\(x\)[/tex] term).
- Constant term is -14.
2. Add the coefficients of like terms:
- For [tex]\(x^3\)[/tex], we add the coefficients: [tex]\(1 + 2 = 3\)[/tex].
- For [tex]\(x^2\)[/tex], we add the coefficients: [tex]\(12 - 5 = 7\)[/tex].
- For [tex]\(x\)[/tex], we add the coefficients: [tex]\(-5 + 0 = -5\)[/tex].
- For the constant term, we add the constants: [tex]\(4 - 14 = -10\)[/tex].
3. Write the resulting polynomial:
After adding the coefficients, the sum of the polynomials is:
[tex]\[
3x^3 + 7x^2 - 5x - 10
\][/tex]
Therefore, the sum of the polynomials [tex]\((x^3 + 12x^2 - 5x + 4)\)[/tex] and [tex]\((2x^3 - 5x^2 - 14)\)[/tex] is [tex]\(3x^3 + 7x^2 - 5x - 10\)[/tex].
1. Identify the terms of each polynomial:
- The first polynomial is [tex]\(x^3 + 12x^2 - 5x + 4\)[/tex].
- Coefficient of [tex]\(x^3\)[/tex] is 1.
- Coefficient of [tex]\(x^2\)[/tex] is 12.
- Coefficient of [tex]\(x\)[/tex] is -5.
- Constant term is 4.
- The second polynomial is [tex]\(2x^3 - 5x^2 - 14\)[/tex].
- Coefficient of [tex]\(x^3\)[/tex] is 2.
- Coefficient of [tex]\(x^2\)[/tex] is -5.
- Coefficient of [tex]\(x\)[/tex] is 0 (since there is no [tex]\(x\)[/tex] term).
- Constant term is -14.
2. Add the coefficients of like terms:
- For [tex]\(x^3\)[/tex], we add the coefficients: [tex]\(1 + 2 = 3\)[/tex].
- For [tex]\(x^2\)[/tex], we add the coefficients: [tex]\(12 - 5 = 7\)[/tex].
- For [tex]\(x\)[/tex], we add the coefficients: [tex]\(-5 + 0 = -5\)[/tex].
- For the constant term, we add the constants: [tex]\(4 - 14 = -10\)[/tex].
3. Write the resulting polynomial:
After adding the coefficients, the sum of the polynomials is:
[tex]\[
3x^3 + 7x^2 - 5x - 10
\][/tex]
Therefore, the sum of the polynomials [tex]\((x^3 + 12x^2 - 5x + 4)\)[/tex] and [tex]\((2x^3 - 5x^2 - 14)\)[/tex] is [tex]\(3x^3 + 7x^2 - 5x - 10\)[/tex].
