Problem 14 - College Algebra Essentials

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 \(6x^{3} +10x^{2} +11x -8\) and its degree is 3.

Step 1: Distribute the Negative Operator

The negative operator should be distributed into the bracket. This negates each term within the bracket: \[ (8x^{2}+7x-5) - (3x^{2}-4x) - (-6x^{3}-5x^{2}+3) = 8x^{2} +7x -5 -3x^{2} +4x +6x^{3} +5x^{2} -3 \]

Step 2: Combine Like Terms

After distributing, the polynomials should be combined. This simply means to add together all terms that have the same variable of the same degree: \[ 8x^{2} -3x^{2} +5x^{2} +7x +4x +6x^{3} -5 -3 =6x^{3} +10x^{2} +11x -8\]

Step 3: Write in Standard Form and Determine Degree

Write the result in standard form by ordering the terms in descending order by degree. The highest power of x is the degree of the polynomial: \[ 6x^{3} +10x^{2} +11x -8\] The degree of this polynomial is 3.