A polynomial [tex]$p$[/tex] has zeros when [tex]$x=\frac{1}{5}, x=-4$[/tex], and [tex]$x=2$[/tex]. What could be the equation of [tex]$p$[/tex]?
Choose 1 answer:
A. [tex]$p(x)=\left(\frac{1}{5} x\right)(-4 x)(2 x)$[/tex]
B. [tex]$p(x)=\left(-\frac{1}{5} x\right)(4 x)(-2 x)$[/tex]
C. [tex]$p(x)=(5 x+1)(x-4)(x+2)$[/tex]
D. [tex]$p(x)=(5 x-1)(x+4)(x-2)$[/tex]
Answer (1)
The polynomial has zeros at x = 5 1 , x = − 4 , and x = 2 .
The factors corresponding to these zeros are ( x − 5 1 ) , ( x + 4 ) , and ( x − 2 ) .
Multiplying the first factor by 5 gives ( 5 x − 1 ) .
Therefore, the polynomial can be written as p ( x ) = ( 5 x − 1 ) ( x + 4 ) ( x − 2 ) , which corresponds to option (D). ( 5 x − 1 ) ( x + 4 ) ( x − 2 )
Explanation
Understanding the Problem The polynomial p has zeros at x = 5 1 , x = − 4 , and x = 2 . This means that when x takes these values, the polynomial p ( x ) equals zero. Our goal is to find a possible equation for p ( x ) from the given options.
Identifying the Factors If x = a is a zero of a polynomial, then ( x − a ) is a factor of the polynomial. Therefore, the factors corresponding to the zeros x = 5 1 , x = − 4 , and x = 2 are ( x − 5 1 ) , ( x − ( − 4 )) = ( x + 4 ) , and ( x − 2 ) , respectively.
Forming the Polynomial We can write p ( x ) as a product of these factors multiplied by a constant k : p ( x ) = k ( x − 5 1 ) ( x + 4 ) ( x − 2 ) To get rid of the fraction, we can multiply the first factor by 5: 5 ( x − 5 1 ) = ( 5 x − 1 ) . Then p ( x ) can be written as p ( x ) = k ′ ( 5 x − 1 ) ( x + 4 ) ( x − 2 ) where k ′ = 5 k is another constant.
Selecting the Correct Option Now we compare this form to the answer choices. We are looking for an expression that matches ( 5 x − 1 ) ( x + 4 ) ( x − 2 ) . Option (D) is exactly this expression.
Final Answer Therefore, the equation of the polynomial p could be p ( x ) = ( 5 x − 1 ) ( x + 4 ) ( x − 2 ) .
Examples
Understanding polynomial zeros is crucial in many fields. For example, in engineering, zeros of transfer functions determine the stability of a system. If you're designing a bridge, the zeros of a function describing the load distribution can help you identify critical points where the structure is most stressed. Similarly, in economics, finding the zeros of a cost function can help determine break-even points for a business. These applications highlight how finding polynomial zeros is not just an abstract mathematical exercise but a tool for solving real-world problems.
