Perform the indicated operations on the following polynomials.
Add: [tex]3 x^3+4 x^2-x+8[/tex] and [tex]x^3-7 x^2+2 x-16[/tex]
Answer (2)
Add the coefficients of the x 3 terms: 3 x 3 + x 3 = 4 x 3 .
Add the coefficients of the x 2 terms: 4 x 2 − 7 x 2 = − 3 x 2 .
Add the coefficients of the x terms: − x + 2 x = x .
Add the constant terms: 8 − 16 = − 8 . The final answer is 4 x 3 − 3 x 2 + x − 8 .
Explanation
Understanding the Problem We are asked to add two polynomials: 3 x 3 + 4 x 2 − x + 8 and x 3 − 7 x 2 + 2 x − 16 . Our goal is to combine like terms to find the sum of these polynomials.
Adding Like Terms To add the polynomials, we add the coefficients of the corresponding terms. This means we add the coefficients of the x 3 terms, the x 2 terms, the x terms, and the constant terms.
Combining x 3 Terms Combining the x 3 terms: 3 x 3 + x 3 = ( 3 + 1 ) x 3 = 4 x 3 .
Combining x 2 Terms Combining the x 2 terms: 4 x 2 + ( − 7 x 2 ) = ( 4 − 7 ) x 2 = − 3 x 2 .
Combining x Terms Combining the x terms: − x + 2 x = ( − 1 + 2 ) x = 1 x = x .
Combining Constant Terms Combining the constant terms: 8 + ( − 16 ) = 8 − 16 = − 8 .
Writing the Resulting Polynomial Now, we write the resulting polynomial by combining the results from the previous steps: 4 x 3 − 3 x 2 + x − 8 .
Final Answer The sum of the polynomials 3 x 3 + 4 x 2 − x + 8 and x 3 − 7 x 2 + 2 x − 16 is 4 x 3 − 3 x 2 + x − 8 .
Examples
Polynomial addition is used in various fields such as engineering, physics, and computer science. For example, in computer graphics, polynomials can represent curves and surfaces. Adding polynomials can be used to combine different curves or surfaces to create more complex shapes. In physics, polynomials can describe the motion of objects, and adding polynomials can be used to analyze the combined motion of multiple objects.
To add the polynomials 3 x 3 + 4 x 2 − x + 8 and x 3 − 7 x 2 + 2 x − 16 , we combine like terms. The result is 4 x 3 − 3 x 2 + x − 8 .
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