Question

1. on the grid below, sketch a cubic polynomial whose zeros are 1, 3, and - 2.

Explanation:

Step1: Recall polynomial - zero relationship

If \(x = a\) is a zero of a polynomial, then \((x - a)\) is a factor of the polynomial. Given zeros \(x = 1\), \(x=3\), and \(x=- 2\), the cubic polynomial can be written in factored form as \(y=(x - 1)(x - 3)(x + 2)\).

Step2: Find the \(y\) - intercept

Set \(x = 0\) in \(y=(x - 1)(x - 3)(x + 2)\). Then \(y=(0 - 1)(0 - 3)(0 + 2)=(-1)\times(-3)\times2 = 6\).

Step3: Analyze end - behavior

The leading term of \(y=(x - 1)(x - 3)(x + 2)\) is obtained by multiplying the leading terms of each factor. \((x)(x)(x)=x^{3}\). Since the leading coefficient is \(1\) (positive) and the degree is \(3\) (odd), as \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to+\infty\).

Step4: Sketch the graph

1. Plot the \(x\) - intercepts at \(x=-2\), \(x = 1\), and \(x = 3\).
2. Plot the \(y\) - intercept at \((0,6)\).
3. Use the end - behavior (falls to the left and rises to the right) and the fact that a cubic function is a smooth curve to sketch the graph passing through the plotted points.

Answer:

Sketch a smooth curve passing through \((-2,0)\), \((0,6)\), \((1,0)\), and \((3,0)\) with end - behavior \(y\to-\infty\) as \(x\to-\infty\) and \(y\to+\infty\) as \(x\to+\infty\).