Question

Graph each polynomial function. $$ f(x)=x^{3}+2 x^{2}-5 x-6 $$

Short answer

Graph has zeros at \(-3\), \(-1\), and \(2\), and y-intercept at \(-6\).

Step-by-step solution

- Identify Key Features

Examine the polynomial function and identify key features such as the degree, leading coefficient, and constant term. The function is given by \(f(x) = x^3 + 2x^2 - 5x - 6\). It is a cubic polynomial.

- Determine End Behavior

Use the leading term to determine the end behavior. Since it is a cubic polynomial with a positive leading coefficient, as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to -\infty\).

- Find the Zeros and Factorize

Find the zeros of the polynomial to determine the x-intercepts. This can involve factoring or using the Rational Root Theorem. The polynomial can be factored as \((x+3)(x+1)(x-2)\), yielding zeros at \(x = -3\), \(x = -1\), and \(x = 2\).

- Calculate the Y-Intercept

The y-intercept is found by evaluating \(f(0)\). Substituting 0 in place of \(x\) gives \(f(0) = -6\). So, the y-intercept is \( (0, -6) \).

- Plot Key Points

Plot the zeros \( (-3,0) \), \( (-1,0) \), \( (2,0) \), and the y-intercept \((0, -6)\) on the graph.

- Sketch the Graph

Using the plotted points, the end behavior, and the nature of a cubic polynomial (which will cross the x-axis at each zero), sketch the graph smoothly connecting these points, ensuring that the graph correctly represents the behavior derived in the previous steps.

Key concepts

Cubic Polynomial

A cubic polynomial is a type of polynomial where the highest power of the variable is three. The general form is given by: \[ f(x) = ax^3 + bx^2 + cx + d \] In the given exercise, the cubic polynomial is: \[ f(x) = x^3 + 2x^2 - 5x - 6 \] Here, the coefficients are:

• a: 1 (leading coefficient, associated with \(x^3\))
• b: 2 (associated with \(x^2\))
• c: -5 (associated with \(x\))
• d: -6 (constant term)

In graphing, the degree (here 3) tells us about the end behavior and the general shape of the graph. Cubic polynomials typically have up to three roots (or zeros) and will intersect the x-axis as it changes direction.

Zeros of Polynomials

The zeros of a polynomial are the values of \(x\) that make the polynomial equal to zero. These are also the points where the graph of the polynomial crosses the x-axis. To find the zeros, we solve the equation: \[ x^3 + 2x^2 - 5x - 6 = 0 \] In this case, factoring the polynomial helps: \[ (x+3)(x+1)(x-2) = 0 \] From this factorization, we can find the zeros:

• x = -3
• x = -1
• x = 2

These points, \((-3,0), (-1,0), (2,0)\), are crucial when plotting the graph because they indicate where the polynomial touches or crosses the x-axis.

End Behavior

The end behavior of a polynomial describes how the values of \(f(x)\) behave as \(x\) approaches positive or negative infinity. For the given cubic polynomial \(f(x) = x^3 + 2x^2 - 5x - 6\), we look at the term with the highest power to determine end behavior. Since the leading coefficient (the coefficient of \(x^3\)) is positive:

• As \( x \to \infty \), \( f(x) \to \infty \)
• As \( x \to -\infty \), \( f(x) \to -\infty \)

This means that the polynomial will go up to infinity on the right side of the graph and down to negative infinity on the left side of the graph. This behavior is typical of cubic polynomials with a positive leading coefficient.

Y-Intercept

The y-intercept of a polynomial is where the graph crosses the y-axis. This occurs when \(x = 0\). To find the y-intercept, we substitute \(x = 0\) into the polynomial function: \[ f(0) = 0^3 + 2(0)^2 - 5(0) - 6 = -6 \] Therefore, the y-intercept is at the point \((0, -6)\). This point is important when sketching the polynomial graph, as it provides a specific starting point on the y-axis where the graph cuts across. Plotting this alongside zeros gives a clear idea of how the polynomial behaves near the origin.