Question

1. find the sum of the following polynomials.

(-3x³ + 8x² - 6x + 5) + (9x³ - 3x² + 7x + 12)
○ 16x³ + 15x² + x + 7
○ 12x³ + 5x² + 13x + 17
○ 6x³ + 5x² - x - 17
○ 6x³ + 5x² + x + 17

Explanation:

Step1: Combine like - terms for \(x^{3}\) terms

We have \(-3x^{3}\) from the first polynomial and \(9x^{3}\) from the second polynomial.
Using the rule of combining like - terms \(ax^{n}+bx^{n}=(a + b)x^{n}\), for \(n = 3\), \(a=-3\) and \(b = 9\).
So, \(-3x^{3}+9x^{3}=(-3 + 9)x^{3}=6x^{3}\).

Step2: Combine like - terms for \(x^{2}\) terms

We have \(8x^{2}\) from the first polynomial and \(-3x^{2}\) from the second polynomial.
For \(n = 2\), \(a = 8\) and \(b=-3\). Then \(8x^{2}-3x^{2}=(8-3)x^{2}=5x^{2}\).

Step3: Combine like - terms for \(x\) terms

We have \(-6x\) from the first polynomial and \(7x\) from the second polynomial.
For \(n = 1\), \(a=-6\) and \(b = 7\). Then \(-6x + 7x=(-6 + 7)x=x\).

Step4: Combine like - terms for constant terms

We have \(5\) from the first polynomial and \(12\) from the second polynomial.
\(5+12 = 17\).

Step5: Combine all the combined terms

Putting it all together, \((-3x^{3}+8x^{2}-6x + 5)+(9x^{3}-3x^{2}+7x + 12)=6x^{3}+5x^{2}+x + 17\).

Answer:

\(6x^{3}+5x^{2}+x + 17\) (the fourth option: \(6x^{3}+5x^{2}+x + 17\))