Finding a Polynomial Function from Its Real Zeros

Q1. Find a polynomial function of degree 3 whose real zeros are –1, 2, and 3.

Background

Topic: Polynomial Functions and Real Zeros

This question tests your understanding of how to construct a polynomial function given its real zeros. You are expected to use the relationship between zeros and factors of a polynomial.

Key Terms and Formulas

• Real Zero: If $f(r) = 0$, then $r$ is a real zero of $f$.

• Factor Theorem: If $r$ is a real zero, then $(x - r)$ is a factor of the polynomial.

• General Form: For zeros $r_1, r_2, r_3$, the polynomial can be written as $f(x) = a(x - r_1)(x - r_2)(x - r_3)$, where $a$ is a nonzero real number.

Step-by-Step Guidance

1. List the given real zeros: $-1$, $2$, and $3$.

2. Write the corresponding factors using the Factor Theorem: $(x + 1)$, $(x - 2)$, and $(x - 3)$.

3. Construct the general form of the polynomial function: $f(x) = a(x + 1)(x - 2)(x - 3)$, where $a$ is a nonzero real number.

4. Expand the expression if required, but keep $a$ as a parameter (do not substitute a value yet).

Try solving on your own before revealing the answer!

Final Answer: $f(x) = a(x + 1)(x - 2)(x - 3)$

Any nonzero value for $a$ will give a degree 3 polynomial with the specified real zeros. If $a = 1$, the expanded form is $f(x) = (x + 1)(x - 2)(x - 3)$.

This polynomial has zeros at $x = -1$, $x = 2$, and $x = 3$ as required.