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Chapter 0: Problem 10

Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$\left(-7 x^{3}+6 x^{2}-11 x+13\right)+\left(19 x^{3}-11 x^{2}+7 x-17\right)$$

Short Answer

Expert verified
The resulting polynomial after addition and when represented in standard form is \(12x^{3} - 5x^{2} - 4x - 4\). The degree of this polynomial is 3.

Step by step solution

01

Arrange the Polynomials

Arrange the given polynomials in decreasing powers of x, such that like terms are grouped together.\[\begin{align*}-7x^{3} + 19x^{3} \ 6x^{2} - 11x^{2} \ -11x + 7x \ 13 - 17 \end{align*}\]
02

Add Like Terms

Add the coefficients for each group of like terms:\[\begin{align*}(-7 + 19)x^{3} = 12x^{3} \ (6 - 11)x^{2} = -5x^{2} \ (-11 + 7)x = -4x \ (13 - 17) = -4\end{align*}\]
03

Standard Form and Degree of Polynomial

After performing the addition, put the polynomial in standard form and determine its degree. The standard form for a polynomial is writing the terms from highest degree to lowest, which in this case already is. So, the final polynomial becomes \(12x^{3} - 5x^{2} - 4x - 4\). The degree of the polynomial is the highest power of x, which in this geometric sequence is 3.

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