Find the sum.
$\begin{array}{c}
(-2 x+3)+\left(-x^5+2 x+4\right) \\
{[?] x^5+\square x+\square}
\end{array}$
Answer (1)
Combine like terms from the polynomials: ( − 2 x + 3 ) + ( − x 5 + 2 x + 4 ) .
Simplify the expression: − x 5 + ( − 2 x + 2 x ) + ( 3 + 4 ) .
The simplified form is: − x 5 + 0 x + 7 .
The final answer is: − 1 x 5 + 0 x + 7
Explanation
Understanding the Problem We are asked to find the sum of two polynomials: ( − 2 x + 3 ) and ( − x 5 + 2 x + 4 ) . The result should be expressed in the form [ ?] x 5 + □ x + □ .
Combining Like Terms To find the sum, we combine like terms from the two polynomials. This means adding the coefficients of the x 5 terms, the x terms, and the constant terms separately.
Performing the Addition The sum of the polynomials is calculated as follows:
( − 2 x + 3 ) + ( − x 5 + 2 x + 4 ) = − x 5 + ( − 2 x + 2 x ) + ( 3 + 4 ) = − x 5 + 0 x + 7 .
Expressing the Result Expressing the result in the required format, we have: − 1 x 5 + 0 x + 7 . Therefore, the coefficient of x 5 is -1, the coefficient of x is 0, and the constant term is 7.
Examples
Polynomials are used to model various real-world phenomena. For example, they can describe the trajectory of a ball thrown in the air, the shape of a suspension bridge, or the growth of a population over time. In economics, polynomials can represent cost and revenue functions, helping businesses analyze their profitability. Understanding how to add and manipulate polynomials is essential for making predictions and optimizing outcomes in these scenarios.
