Verified Answer :

  • The factor theorem implies that if a polynomial has roots at \(r_1, r_2, \dots\), it contains factors \((x - r_1), (x - r_2), \dots\).
  • Given the zeros are \(-8\), \(1\), and \(3\), the required factors are \((x + 8)\), \((x - 1)\), and \((x - 3)\).
  • Among the provided options, \(y = (x - 3)(x + 8)(x - 1)\) is the only one that matches these factors.
  • The final answer is \(\boxed{y = (x - 3)(x + 8)(x - 1)}\).

Explanation

  1. Analyzing the roots and finding the required factors

To determine which polynomial function has zeros at \(x = -8\), \(x = 1\), and \(x = 3\), we use the factor theorem. The factor theorem states that if a polynomial \(p(x)\) has a zero at \(x = r\), then \((x - r)\) must be a factor of the polynomial.
Based on the given zeros:

  • For the zero at \(x = -8\), the factor is \((x - (-8)) = (x + 8)\).
  • For the zero at \(x = 1\), the factor is \((x - 1)\).
  • For the zero at \(x = 3\), the factor is \((x - 3)\).

A polynomial with these specific roots will have the general form \(y = C(x + 8)(x - 1)(x - 3)\), where \(C\) is a non-zero constant.

  1. Evaluating the options

Now, we compare our required factors \((x + 8)\), \((x - 1)\), and \((x - 3)\) against the given options:

  • Option 1: \(y = (x - 3)(x + 8)(x - 1)\). This expression contains the factors \((x - 3)\), \((x + 8)\), and \((x - 1)\), which match our derived requirements exactly.
  • Option 2: \(y = (x + 1)(x - 8)(x + 3)\). This corresponds to zeros at \(-1\), \(8\), and \(-3\).
  • Option 3: \(y = x(x + 8)(x - 3)\). This corresponds to zeros at \(0\), \(-8\), and \(3\).
  • Option 4: \(y = x(x + 3)(x - 8)\). This corresponds to zeros at \(0\), \(-3\), and \(8\).

Comparing the options, we see that the first function perfectly matches our identified factors.

  1. Conclusion

Since the first option, \(y = (x - 3)(x + 8)(x - 1)\), includes exactly the factors needed for zeros at \(-8\), \(1\), and \(3\), it is the correct function. Therefore, the final answer is \(y = (x - 3)(x + 8)(x - 1)\).

Examples

Understanding how roots relate to polynomial factors is essential in fields like physics and engineering, where we often model systems using polynomials. For instance, if you are designing a trajectory for a projectile and you know it needs to cross the ground (height = 0) at specific horizontal distances (e.g., 10 meters, 50 meters, and 100 meters), knowing the factors \((x - 10)\), \((x - 50)\), and \((x - 100)\) allows you to construct the exact equation for the path of the projectile. This helps in predicting behavior and ensuring the object reaches its destination accurately.

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