Problem 10 - College Algebra

Question

Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$\left(-7 x^{3}+6 x^{2}-11 x+13\right)+\left(19 x^{3}-11 x^{2}+7 x-17\right)$$

Step-by-Step Solution

Answer

The resulting polynomial in standard form is \(12x^{3} - 5x^{2} - 4x - 4\) and its degree is 3.

Step 1: Distribute the Sign

First distribute the positive sign into the second polynomial. This will not change the polynomial, but it's an important step to remember. The equation now is: \(-7x^{3} + 6x^{2} - 11x + 13 + 19x^{3} - 11x^{2} + 7x - 17\).

Step 2: Group Like Terms

The goal is to group like terms together, which means identifying the terms with the same variable and degree. That that results in: \(-7x^{3} + 19x^{3} + 6x^{2} - 11x^{2} - 11x + 7x + 13 - 17\).

Step 3: Simplify

Now, add and subtract those like terms: \ \[-7x^{3} + 19x^{3} = 12x^{3}\] \[6x^{2} - 11x^{2} = -5x^{2}\] \[-11x + 7x = -4x\] \[13 - 17 = -4\] Combining these results in: \(12x^{3} - 5x^{2} - 4x - 4\)

Step 4: Determine the Degree

Finally, determine the degree of the polynomial. The degree of a polynomial is the highest power of the term with the highest degree. In this case, the degree is 3, from the term \(12x^{3}\)