Problem 13 - College Algebra

Question

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

Step-by-Step Solution

Answer

The resulting polynomial in standard form is \( 6x^{2} -6x +2 \), and its degree is 2.

Step 1: Distribute the negative sign

First, distribute the negative sign to the third polynomial to change the signs of all its terms: \[ -\left(x^{2}-4x-3\right) = -x^{2} + 4x + 3\]

Step 2: Combine like terms

Now, combine the like terms. These are terms with the same variable(s) raised to the same power: \[ \begin{align*} &(5x^{2}+2x^{2}-x^{2}) + (-7x-3x+4x) + (-8+7+3) \ =& 6x^{2} -6x +2 \end{align*} \]

Step 3: Write the polynomial in standard form

The resulting polynomial after combining the like terms is already in standard form, where the terms are written in descending order of their degrees (powers of x). \[ 6x^{2} -6x +2 \]

Step 4: Indicate the degree

The degree of a polynomial is the highest power of the variable. Here, the highest power of the variable x is 2. Therefore, the degree of this polynomial is 2.