Problem 11 - College Algebra

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 after the subtraction is \(12x^3 + 4x^2 + 12x - 14\), which is in standard form. The degree of this polynomial is 3.

Step 1: Set up the Subtraction

Start by writing down the two polynomial functions that need to be subtracted from each other. In this case, these are \(17x^3 - 5x^2 + 4x - 3\) and \(5x^3 - 9x^2 - 8x + 11\). Now set up the subtraction by arranging them one on top of the other as you would do in a normal subtraction operation.

Step 2: Perform the Subtraction

Subtract the terms in the second polynomial from the corresponding terms in the first polynomial. This gives a new polynomial: \((17x^3 - 5x^2 + 4x - 3) - (5x^3 - 9x^2 - 8x + 11) = 12x^3 + 4x^2 + 12x - 14\).

Step 3: Write the Resulting Polynomial in Standard Form and Determine Its Degree

The polynomial that results from the subtraction is \(12x^3 + 4x^2 + 12x - 14\). This is already in standard form. The standard form of a polynomial lists its terms in descending order of power. The degree of the polynomial is the highest power of the polynomial terms. In this case, the degree is 3, which is the power in the term \(12x^3\).