Question

which polynomial represents the sum below?
$\

$$\begin{array}{r}\\ 2x^{6} + 8x^{2} - x + 1 \\\\ +\\ \\ \\ 3x^{6} + 5x^{5} + 1 \\\\\\hline\\end{array}$$

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\\(\bigcirc\\) a. \\(5x^{6} + 5x^{5} + 8x^{2} - x + 2\\)
\\(\bigcirc\\) b. \\(5x^{6} + 5x^{5} - 8x^{2} + x - 2\\)
\\(\bigcirc\\) c. \\(5x^{12} + 7x^{7} + 8x^{3} - x + 2\\)
\\(\bigcirc\\) d. \\(5x^{12} + 13x^{10} - 8x^{3} - x + 2\\)

Explanation:

Step1: Combine like terms for \(x^6\)

We have \(2x^6 + 3x^6\). When adding like terms, we add the coefficients. So \(2 + 3 = 5\), thus \(2x^6 + 3x^6 = 5x^6\).

Step2: Handle the \(x^5\) term

There is only \(5x^5\) from the second polynomial, so we keep it as \(5x^5\).

Step3: Handle the \(x^2\) term

We have \(8x^2\) from the first polynomial, and no other \(x^2\) terms, so we keep it as \(8x^2\).

Step4: Handle the \(x\) term

We have \(-x\) from the first polynomial, and no other \(x\) terms, so we keep it as \(-x\).

Step5: Combine the constant terms

We have \(1 + 1 = 2\).

Step6: Combine all terms

Putting it all together, we get \(5x^6 + 5x^5 + 8x^2 - x + 2\).

Answer:

A. \(5x^6 + 5x^5 + 8x^2 - x + 2\)