Problem 12 - College Algebra

Question

Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$\left(18 x^{4}-2 x^{3}-7 x+8\right)-\left(9 x^{4}-6 x^{3}-5 x+7\right)$$

Step-by-Step Solution

Answer

The resulting polynomial in standard form after subtraction is \(9 x^{4} + 4 x^{3} - 2x + 1\). The degree of this polynomial is 4.

Step 1: Gather Like Terms

Start by arranging the polynomial in such a way that like terms are together. In this case, match the terms with the same exponents like this: \(18 x^{4} - 9 x^{4}\), \(-2 x^{3} - (-6 x^{3})\), \(-7 x - (-5 x)\) and \(8 - 7\).

Step 2: Perform Subtraction

Now perform the subtraction of like terms. \(18 x^{4} - 9 x^{4} = 9 x^{4}\), \(-2 x^{3} - (-6 x^{3}) = 4 x^{3}\), \(-7 x - (-5 x) =-2x\) and \(8 - 7 = 1\). So you get the result \(9 x^{4} + 4 x^{3} - 2x + 1\).

Step 3: Identify the Degree

The degree of a polynomial is the highest power of its variable. In the resulting polynomial \(9 x^{4} + 4 x^{3} -2x + 1\), the highest power is 4, so the degree of the polynomial is 4.