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 in standard form is \( 12x^3 + 4x^2 + 12x - 14 \) and its degree is 3.

Step 1: Write down the given expressions

The given expressions are: \(17 x^{3}-5 x^{2}+4 x-3\) and \(5 x^{3}-9 x^{2}-8 x+11\). When subtracting the second polynomial from the first, it is important to keep track of the signs.

Step 2: Subtract corresponding terms

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

Step 3: Write in standard form and find its degree

The polynomial \( 12x^3 + 4x^2 + 12x - 14 \) is already in standard form, because the terms are ordered by degree in descending order. The degree of a polynomial is the highest exponent value of its terms. Here, the highest exponent is 3, so the degree of the polynomial is 3.