Problem 13 - College Algebra Essentials

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 its standard form is \(6x^{2}-6x+2\) and its degree is 2.

Step 1: Combine like terms

Firstly, consider the terms having the same degree, to combine (add or subtract) them together. The terms that need to be combined here are as follows: the \( x^{2} \) terms ( \(5x^{2}, 2x^{2}, -x^{2}\)), the \(x\) terms (\(-7x, -3x, -4x\)) and the constant terms (\(-8, 7, -3\)). Performing these operations, get \((5+2-1)x^{2} + (-7-3+4)x+(-8+7+3)\). This simplifies to \(6x^{2}-6x+2\).

Step 2: Write in standard form

A polynomial is in standard form when its terms are ordered from highest degree to lowest, or from left to right. Here, the polynomial is already in standard form as \(6x^{2}-6x+2\).

Step 3: Determine the degree of the polynomial

The degree of the polynomial is the largest exponent of any term (only look for the variable not including the coefficients). In this case, the degree is 2, because the largest exponent in the standard form is 2.