Questions: Sketch the graph of the polynomial function P(x)=2x^3+7x^2+3x. You must show your work (i.e. find zeros, find intercepts, determine end behavior, set up a table).

Solution Steps

To find the zeros of the polynomial \( P(x) = 2x^3 + 7x^2 + 3x \), we set the polynomial equal to zero and solve for \( x \): \[ 2x^3 + 7x^2 + 3x = 0 \] Factor out the common term \( x \): \[ x(2x^2 + 7x + 3) = 0 \] This gives us one zero at \( x = 0 \).

Next, solve the quadratic equation \( 2x^2 + 7x + 3 = 0 \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 2 \), \( b = 7 \), and \( c = 3 \).

Calculate the discriminant: \[ b^2 - 4ac = 7^2 - 4 \cdot 2 \cdot 3 = 49 - 24 = 25 \]

The roots are: \[ x = \frac{-7 \pm \sqrt{25}}{4} = \frac{-7 \pm 5}{4} \]

Thus, the zeros are: \[ x = \frac{-7 + 5}{4} = -\frac{1}{2}, \quad x = \frac{-7 - 5}{4} = -3 \]

The y-intercept occurs when \( x = 0 \): \[ P(0) = 2(0)^3 + 7(0)^2 + 3(0) = 0 \] Thus, the y-intercept is at \( (0, 0) \).

The x-intercepts are the zeros found in Step 1: \( x = 0 \), \( x = -\frac{1}{2} \), and \( x = -3 \).

The end behavior of the polynomial is determined by the leading term \( 2x^3 \). As \( x \to \infty \), \( P(x) \to \infty \) and as \( x \to -\infty \), \( P(x) \to -\infty \).

Final Answer