Answered

Which function represents a function with zeros at -3, -1, 0, and 6?


A$y=(x-6)(x+1)(x+3)$
B$y=x(x-3)(x-1)(x+6)$
C$y=x(x-6)(x+1)(x+3)$
D$y=(x-3)(x-1)(x+6)$

Verified Answer :

  • Identify that for each zero \(r\), the expression \((x - r)\) is a factor of the polynomial.
  • Construct the polynomial by multiplying the factors associated with each zero: \(x\), \((x+3)\), \((x+1)\), and \((x-6)\).
  • Observe that the resulting polynomial is \(y = x(x-6)(x+1)(x+3)\).
  • Match this result with the provided options, concluding the third option is correct.
  • Final Answer: \(\boxed{y = x(x-6)(x+1)(x+3)}\)

Explanation

  1. Applying the Factor Theorem

To determine which polynomial function has zeros at \(-3, -1, 0,\) and \(6,\) we apply the Factor Theorem. The Factor Theorem states that if \(r\) is a zero of a polynomial \(f(x),\) then \((x - r)\) must be a factor of the polynomial.

  • For the zero \(r = -3,\) the corresponding factor is \((x - (-3)) = (x + 3)\).
  • For the zero \(r = -1,\) the corresponding factor is \((x - (-1)) = (x + 1)\).
  • For the zero \(r = 0,\) the corresponding factor is \((x - 0) = x\).
  • For the zero \(r = 6,\) the corresponding factor is \((x - 6)\).
  1. Constructing the Polynomial Expression

Now that we have identified the necessary factors for each zero, we can construct the polynomial by multiplying these factors together. The polynomial function is given by:
\[f(x) = x(x + 3)(x + 1)(x - 6)\]
By the commutative property of multiplication, the order in which we write these factors does not change the function. Thus, this expression represents a polynomial with the specified zeros.

  1. Comparing with Given Options and Conclusion

Comparing our derived expression \(y = x(x + 3)(x + 1)(x - 6)\) to the provided options:

  • \(y = (x-6)(x+1)(x+3)\) is missing the factor \(x\).
  • \(y = x(x-3)(x-1)(x+6)\) has zeros at \(0, 3, 1, -6\).
  • \(y = x(x-6)(x+1)(x+3)\) matches our derived expression exactly.
  • \(y = (x-3)(x-1)(x+6)\) is missing the factor \(x\) and has incorrect zeros.

Therefore, the third option is the correct representation.

Examples

Polynomial functions with specific zeros are frequently used in engineering and physics to model phenomena such as wave patterns or resonance. For example, if a signal-processing system needs to filter out frequencies at specific points (like 0 Hz, -3 Hz, -1 Hz, and 6 Hz relative to a baseline), an engineer would construct a transfer function using these roots to ensure the signal is nullified at those exact values. Understanding how to build these functions helps in designing stable and efficient electronic components.