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

Step 1: Identify Like Terms

In both polynomials, the like terms are the ones that have the same variable and exponent. So, \( -6x^3 \) and \( 17x^3 \) are like terms, \( 5x^2 \) and \( 2x^2 \) are like terms, \( -8x \) and \( -4x \) are like terms and lastly, \( 9 \) and \( -13 \) are like terms.

Step 2: Perform Addition of Like Terms

Now, add the coefficients of like terms: \((-6x^3 + 17x^3)\), \((5x^2 + 2x^2)\), \((-8x + -4x)\), and \((9 + -13)\). The resulting polynomial is \( 11x^3 + 7x^2 -12x -4 \).

Step 3: Write Polynomial in Standard Form and Indicate Degree

The standard form of a polynomial arranges the terms by degree in descending order. Since the terms are already ordered from highest to lowest degree, the standard form is \( 11x^3 + 7x^2 -12x -4 \). The degree of the polynomial is the highest exponent in the polynomial and in our case, the highest exponent is 3. So, the degree of the polynomial is 3.