Question

write a polynomial function of least degree with integral coefficients that has the given zero.

1. 3, 2, -2
2. 3, 1, -2, -4

Explanation:

Step1: Use the zero - root form

If \(x = a\) is a root of a polynomial, then \((x - a)\) is a factor of the polynomial.
For the roots \(x = 3\), \(x = 2\), and \(x=-2\), the polynomial \(f(x)\) can be written as \(f(x)=(x - 3)(x - 2)(x + 2)\)

Step2: Expand \((x - 2)(x + 2)\)

Using the difference of squares formula \((a - b)(a + b)=a^{2}-b^{2}\), where \(a = x\) and \(b = 2\). So \((x - 2)(x + 2)=x^{2}-4\)

Step3: Expand \((x - 3)(x^{2}-4)\)

Use the distributive property (FOIL for binomial - polynomial):

$$
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$$

Answer:

\(f(x)=x^{3}-3x^{2}-4x + 12\)