Question

1. write an equation for a polynomial with x -intercepts at x = 5,x = -2,x = 7 and a y -intercept of 35.

Explanation:

Step1: Write the factored - form of the polynomial

If a polynomial has \(x\) - intercepts at \(x = a\), \(x = b\), \(x = c\), then the factored - form of the polynomial is \(y = k(x - a)(x - b)(x - c)\). Here, \(a = 5\), \(b=-2\), \(c = 7\), so \(y=k(x - 5)(x + 2)(x - 7)\).

Step2: Find the value of \(k\) using the \(y\) - intercept

The \(y\) - intercept occurs when \(x = 0\). Substitute \(x = 0\) and \(y = 35\) into the equation \(y=k(x - 5)(x + 2)(x - 7)\).
\[

$$\begin{align*} 35&=k(0 - 5)(0 + 2)(0 - 7)\\ 35&=k(-5)\times2\times(-7)\\ 35&=k(70)\\ k&=\frac{35}{70}=\frac{1}{2} \end{align*}$$

\]

Step3: Write the final polynomial equation

Substitute \(k=\frac{1}{2}\) back into \(y = k(x - 5)(x + 2)(x - 7)\).
\[y=\frac{1}{2}(x - 5)(x + 2)(x - 7)\]

Answer:

\(y=\frac{1}{2}(x - 5)(x + 2)(x - 7)\)