Problem 10 - College Algebra Essentials

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 from the given operation is \( 12 x^{3} - 5 x^{2} - 4 x - 4 \), and its degree is \( 3 \).

Step 1: Identify like-terms

Going degree by degree, identify like-terms in the two polynomials. The like-terms in the given polynomials are: \( -7 x^{3} \) and \( 19 x^{3} \), \( 6 x^{2} \) and \( -11 x^{2} \), \( -11 x \) and \( 7 x \), \( 13 \) and \( -17 \).

Step 2: Add like-terms

Add each pair of like-terms from step 1 by adding their coefficients. The sum of \( -7 x^{3} \) and \( 19 x^{3} \) is \( 12 x^{3} \), the sum of \( 6 x^{2} \) and \( -11 x^{2} \) is \( -5 x^{2} \), the sum of \( -11 x \) and \( 7 x \) is \( -4 x \), and finally, the sum of \( 13 \) and \( -17 \) is \( -4 \).

Step 3: Write the result in standard form

Standard form of a polynomial lists the terms in descending order by degree. So, the resulting polynomial in standard form is \( 12 x^{3} - 5 x^{2} - 4 x - 4 \).

Step 4: Identify the degree of the polynomial

The degree of a polynomial is the highest power of the variable, \( x \) in this case. So, the degree of the resulting polynomial \( 12 x^{3} - 5 x^{2} - 4 x - 4 \) is \( 3 \).