Question

find a polynomial with the following zeros.
1, 4, 9
f(x) = x³ + ?x² + x +

Explanation:

Step1: Recall the factor theorem

If \( r \) is a zero of a polynomial, then \( (x - r) \) is a factor. So for zeros \( 1, 4, 9 \), the factors are \( (x - 1) \), \( (x - 4) \), and \( (x - 9) \).

Step2: Multiply the factors

First, multiply \( (x - 1)(x - 4) \):
\[

$$\begin{align*} (x - 1)(x - 4)&=x^2 - 4x - x + 4\\ &=x^2 - 5x + 4 \end{align*}$$

\]
Then multiply the result by \( (x - 9) \):
\[

$$\begin{align*} (x^2 - 5x + 4)(x - 9)&=x^3 - 9x^2 - 5x^2 + 45x + 4x - 36\\ &=x^3 - 14x^2 + 49x - 36 \end{align*}$$

\]

Answer:

For the \( x^2 \) term, the coefficient is \(-14\). For the \( x \) term, the coefficient is \(49\). For the constant term, it is \(-36\). So the polynomial is \(f(x)=x^3 - 14x^2 + 49x - 36\).