Given the polynomial:
[tex]$-x^2-8 x-7+7 x^3+2 x^4$[/tex]
Evaluate the polynomial for [tex]$x=5$[/tex]
Answer (2)
To evaluate the polynomial at x = 5 , we substitute and simplify, resulting in a final answer of 2053 . This involves calculating each term and combining the results systematically. The steps include substitution, simplification, and addition of the calculated values.
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Substitute x = 5 into the polynomial: P ( 5 ) = − ( 5 ) 2 − 8 ( 5 ) − 7 + 7 ( 5 ) 3 + 2 ( 5 ) 4 .
Simplify the expression: P ( 5 ) = − 25 − 40 − 7 + 875 + 1250 .
Combine the terms: P ( 5 ) = − 72 + 2125 .
Calculate the final value: P ( 5 ) = 2053 .
Explanation
Understanding the Problem We are given the polynomial P ( x ) = − x 2 − 8 x − 7 + 7 x 3 + 2 x 4 and we want to evaluate it at x = 5 . This means we need to substitute x = 5 into the polynomial and simplify the expression.
Substitution Substitute x = 5 into the polynomial: P ( 5 ) = − ( 5 ) 2 − 8 ( 5 ) − 7 + 7 ( 5 ) 3 + 2 ( 5 ) 4
Simplification Now, we simplify the expression: P ( 5 ) = − 25 − 40 − 7 + 7 ( 125 ) + 2 ( 625 ) P ( 5 ) = − 25 − 40 − 7 + 875 + 1250 P ( 5 ) = − 72 + 875 + 1250 P ( 5 ) = 803 + 1250 P ( 5 ) = 2053
Final Answer Therefore, the value of the polynomial at x = 5 is 2053.
Examples
Polynomials are used in various fields such as physics, engineering, and economics. For example, in physics, polynomials can be used to describe the trajectory of a projectile. In economics, they can be used to model cost and revenue functions. Evaluating a polynomial at a specific value helps us understand the behavior of the model at that point. For instance, if the polynomial represents the profit of a company, evaluating it at a certain value of 'x' (e.g., number of units sold) gives us the profit at that sales level.
