Problem 14 - College Algebra

Question

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

Step-by-Step Solution

Answer

The resulting polynomial in standard form is \(-6x^{3}+10x^{2}+11x-2\) and its degree is 3.

Step 1: Distributing Negatives

First, distribute the negative sign (where necessary) to each term inside the parentheses. This gives: \(8x^{2}+7x-5 - 3x^{2}+4x - (-6x^{3}+5x^{2}-3)\).

Step 2: Simplify the Expression

Next, simplify the expression by combining like terms, it becomes: \(-6x^{3}+(8-3+5)x^{2}+(7+4)x-5+3\). This simplifies to \(-6x^{3}+10x^{2}+11x-2\).

Step 3: Write in Standard Form and Indicate Degree

Finally, rearrange the resulting polynomial in standard form and indicate its degree, that is: \(-6x^{3}+10x^{2}+11x-2\). The degree of the polynomial is 3, since the maximum exponent of x is 3.