A polynomial [tex]$p$[/tex] has zeros when [tex]$x=5, x=-1$[/tex], and [tex]$x=-\frac{1}{4}$[/tex]. What could be the equation of [tex]$p$[/tex]?
A. [tex]$p(x)=(x+5)(x+1)(4 x+1)$[/tex]
B. [tex]$p(x)=(x-5)(x+1)(4 x+1)$[/tex]
C. [tex]$p(x)=(x+5)(x-1)(4 x-1)$[/tex]
D. [tex]$p(x)=(5 x)(-1 x)\left(-\frac{1}{4} x\right)$[/tex]
Answer (2)
The polynomial has zeros at x = 5 , x = − 1 , and x = − 4 1 .
If x = a is a zero, then ( x − a ) is a factor.
The factors are ( x − 5 ) , ( x + 1 ) , and ( 4 x + 1 ) .
Therefore, the equation of the polynomial is p ( x ) = ( x − 5 ) ( x + 1 ) ( 4 x + 1 ) .
Explanation
Understanding the Problem The polynomial p has zeros at x = 5 , x = − 1 , and x = − 4 1 . This means that when x takes these values, the polynomial p ( x ) evaluates to zero. We need to find a possible equation for p ( x ) from the given options.
Zeros and Factors If x = a is a zero of a polynomial, then ( x − a ) is a factor of the polynomial. This is because when x = a , the factor ( x − a ) becomes ( a − a ) = 0 , making the entire polynomial equal to zero.
Finding the Factors Since x = 5 is a zero, ( x − 5 ) is a factor of p ( x ) .
Since x = − 1 is a zero, ( x − ( − 1 )) = ( x + 1 ) is a factor of p ( x ) .
Since x = − 4 1 is a zero, ( x − ( − 4 1 )) = ( x + 4 1 ) is a factor of p ( x ) . To avoid fractions, we can multiply this factor by 4, which gives ( 4 x + 1 ) as a factor.
Constructing the Polynomial Therefore, a possible equation for the polynomial p ( x ) is given by the product of these factors: p ( x ) = ( x − 5 ) ( x + 1 ) ( 4 x + 1 ) Comparing this with the given options, we see that option (B) matches our result.
Examples
Understanding polynomial zeros is crucial in various fields. For instance, in engineering, zeros of a transfer function represent system stability. If a system's transfer function has zeros with positive real parts, the system might be unstable. Similarly, in physics, finding the zeros of a wave function can help determine the possible energy levels of a quantum system. This concept also extends to economics, where polynomial models can be used to analyze market trends, and the zeros can represent equilibrium points.
The polynomial with zeros at x = 5 , x = − 1 , and x = − 4 1 can be expressed as p ( x ) = ( x − 5 ) ( x + 1 ) ( 4 x + 1 ) . Among the provided choices, the correct answer is option B . This matches the identified factors corresponding to the zeros of the polynomial.
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