Which of the following polynomial functions has zeros [tex]x=1,-3[/tex]?
A. [tex]f(x)=x^3-7 x+6[/tex]
B. [tex]f(x)=x^3+x+6[/tex]
C. [tex]f(x)=x^3-2 x^2-3 x[/tex]
D. [tex]f(x)=x^3-2 x^2-7 x_6[/tex]
Answer (2)
Check each polynomial by substituting x = 1 and x = − 3 into the function.
For f ( x ) = x 3 − 7 x + 6 , f ( 1 ) = 0 and f ( − 3 ) = 0 .
For the other polynomials, at least one of f ( 1 ) or f ( − 3 ) is not equal to 0.
Therefore, the polynomial with zeros at x = 1 and x = − 3 is f ( x ) = x 3 − 7 x + 6 .
Explanation
Understanding the Problem We are given four polynomial functions and asked to identify which one has zeros at x = 1 and x = − 3 . A zero of a polynomial is a value of x for which the polynomial evaluates to zero. Thus, we need to check each polynomial to see if f ( 1 ) = 0 and f ( − 3 ) = 0 .
Checking the First Polynomial For the first polynomial, f ( x ) = x 3 − 7 x + 6 , we evaluate f ( 1 ) and f ( − 3 ) .
f ( 1 ) = ( 1 ) 3 − 7 ( 1 ) + 6 = 1 − 7 + 6 = 0 f ( − 3 ) = ( − 3 ) 3 − 7 ( − 3 ) + 6 = − 27 + 21 + 6 = 0
Since f ( 1 ) = 0 and f ( − 3 ) = 0 , this polynomial has zeros at x = 1 and x = − 3 .
Checking the Second Polynomial For the second polynomial, f ( x ) = x 3 + x + 6 , we evaluate f ( 1 ) and f ( − 3 ) .
f ( 1 ) = ( 1 ) 3 + ( 1 ) + 6 = 1 + 1 + 6 = 8 f ( − 3 ) = ( − 3 ) 3 + ( − 3 ) + 6 = − 27 − 3 + 6 = − 24
Since f ( 1 ) e q 0 and f ( − 3 ) e q 0 , this polynomial does not have zeros at x = 1 and x = − 3 .
Checking the Third Polynomial For the third polynomial, f ( x ) = x 3 − 2 x 2 − 3 x , we evaluate f ( 1 ) and f ( − 3 ) .
f ( 1 ) = ( 1 ) 3 − 2 ( 1 ) 2 − 3 ( 1 ) = 1 − 2 − 3 = − 4 f ( − 3 ) = ( − 3 ) 3 − 2 ( − 3 ) 2 − 3 ( − 3 ) = − 27 − 18 + 9 = − 36
Since f ( 1 ) e q 0 and f ( − 3 ) e q 0 , this polynomial does not have zeros at x = 1 and x = − 3 .
Checking the Fourth Polynomial For the fourth polynomial, f ( x ) = x 3 − 2 x 2 − 7 x + 6 , we evaluate f ( 1 ) and f ( − 3 ) .
f ( 1 ) = ( 1 ) 3 − 2 ( 1 ) 2 − 7 ( 1 ) + 6 = 1 − 2 − 7 + 6 = − 2 f ( − 3 ) = ( − 3 ) 3 − 2 ( − 3 ) 2 − 7 ( − 3 ) + 6 = − 27 − 18 + 21 + 6 = − 18
Since f ( 1 ) e q 0 and f ( − 3 ) e q 0 , this polynomial does not have zeros at x = 1 and x = − 3 .
Final Answer The first polynomial, f ( x ) = x 3 − 7 x + 6 , is the only one that has zeros at both x = 1 and x = − 3 .
Examples
Understanding polynomial zeros is crucial in many engineering applications. For example, when designing a bridge, engineers use polynomials to model the load distribution and ensure stability. The zeros of these polynomials represent critical points where the structure experiences maximum stress or deflection. By identifying these points, engineers can reinforce the structure at those locations, preventing potential failures and ensuring the bridge's safety and longevity. This ensures that the bridge can withstand various loads and environmental conditions, making it safe for public use.
The polynomial function that has zeros at x = 1 and x = − 3 is f ( x ) = x 3 − 7 x + 6 . We confirmed this by evaluating each polynomial at these points. The first polynomial yielded the required zeros, while the others did not.
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