Question

simplify and give the degree of the polynomial. al\\(\frac{1}{8}x^{5} + \frac{1}{2}x^{5} - 6x - 19\\)\\(\frac{1}{8}x^{5} + \frac{1}{2}x^{5} - 6x - 19 = \square\\)\\((\text{simplify your answer.})\\)

Explanation:

Step1: Combine like terms

The like terms here are \(\frac{1}{8}x^5\) and \(\frac{1}{2}x^5\). To combine them, we need a common denominator for the coefficients. The common denominator of 8 and 2 is 8. So we rewrite \(\frac{1}{2}\) as \(\frac{4}{8}\). Then we add the coefficients: \(\frac{1}{8} + \frac{4}{8} = \frac{5}{8}\). So the combined \(x^5\) term is \(\frac{5}{8}x^5\). The other terms, \(-6x\) and \(-19\), remain unchanged as there are no like terms to combine with them.

$$ \frac{1}{8}x^5 + \frac{1}{2}x^5 - 6x - 19 = \frac{1}{8}x^5 + \frac{4}{8}x^5 - 6x - 19 = \frac{5}{8}x^5 - 6x - 19 $$

Step2: Determine the degree

The degree of a polynomial is the highest power of the variable. In the simplified polynomial \(\frac{5}{8}x^5 - 6x - 19\), the highest power of \(x\) is 5. So the degree of the polynomial is 5.

Answer:

The simplified polynomial is \(\frac{5}{8}x^5 - 6x - 19\) and its degree is 5.