Problem 9 - College Algebra

Question

Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$\left(-6 x^{3}+5 x^{2}-8 x+9\right)+\left(17 x^{3}+2 x^{2}-4 x-13\right)$$

Step-by-Step Solution

Answer

The added polynomial in standard form is \(11x^3 + 7x^2 - 12x - 4\) and its degree is 3.

Step 1: Addition of Polynomial terms

Combine 'like terms'. This refers to terms that have the exact variable part. Start with the terms containing \(x^3\): \(-6x^3\) from the first polynomial and \(+17x^3\) from the second. Their addition results in \(+11x^3\). Then, add the terms with \(x^2\), \(+5x^2\) and \(+2x^2\), which sums to \(+7x^2\). Proceed with the terms with \(x\), \(-8x\) and \(-4x\), which sums to \(-12x\). Lastly, add the constants \(+9\) and \(-13\), this gives \(-4\).

Step 2: Writing in standard form

Arrange the terms in descending order of the exponent’s values. The standard form of a polynomial sorts the terms by degree in descending order, from highest degree to lowest degree. This results in: \(+11x^3 + 7x^2 - 12x - 4\)

Step 3: Degree of the polynomial

The degree is the highest power of the variable. In this case, the highest power of the \(x\) variable is 3. Hence, the degree of the polynomial is 3.