Problem 11 - College Algebra Essentials

Question

Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$\left(17 x^{3}-5 x^{2}+4 x-3\right)-\left(5 x^{3}-9 x^{2}-8 x+11\right)$$

Step-by-Step Solution

Answer

The resulting polynomial is \(12x^3 + 4x^2 + 12x - 14\), which is a polynomial of degree 3.

Step 1: Perform Subtraction

To subtract polynomials, subtract the corresponding coefficients of the matching power of x. Apply this operation to the given polynomials: \(17x^3 - 5x^3\), \(-5x^2 - -9x^2\), \(4x - -8x\), and \(-3 - 11\). This results in \(12x^3 + 4x^2 + 12x - 14\).

Step 2: Write the polynomial in standard form

The resulting polynomial after subtraction, i.e. \(12x^3 + 4x^2 + 12x - 14\) is already in the standard form which starts with the term of highest degree.

Step 3: Indicate the degree of the polynomial

The degree of the polynomial is the highest degree among all terms. Here, the highest degree in the polynomial \(12x^3 + 4x^2 + 12x - 14\) is 3, thus the polynomial is of degree 3.