Problem 113 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find all the intercepts of the graph of $$ f(x)=x^{3}+2 x^{2}-9 x-18 $$

Step-by-Step Solution

Answer

The intercepts are: y-intercept (0, -18) and x-intercepts (3, 0), (-3, 0), (-2, 0).

Step 1: Understand the Problem

To find the intercepts of the function, figure out where the graph crosses the x-axis (x-intercepts) and y-axis (y-intercept).

Step 2: Find the Y-Intercept

The y-intercept occurs where the graph crosses the y-axis, which means we set x to 0. Substitute 0 into the function: \[ f(0) = (0)^3 + 2(0)^2 - 9(0) - 18 = -18\] Thus, the y-intercept is (0, -18).

Step 3: Factor the Function to Find X-Intercepts

The x-intercepts occur where the function equals zero. Therefore, solve the equation \[ x^3 + 2x^2 - 9x - 18 = 0 \] Try factoring by grouping:

Step 4: Group Terms and Factor

Group the terms as follows: \[ (x^3 + 2x^2) - (9x + 18) = 0 \] Factor out the common terms: \[ x^2(x + 2) - 9(x + 2) = 0 \] Now factor out \( (x + 2) \): \[ (x^2 - 9)(x + 2) = 0 \]

Step 5: Solve the Factored Equation

The equation \( (x^2 - 9)(x + 2) = 0 \) implies \[ x^2 - 9 = 0 \text{ or } x + 2 = 0 \] Solve each equation separately:1. \( x^2 - 9 = 0 \) gives \( x = \pm 3 \)2. \( x + 2 = 0 \) gives \( x = -2 \) Thus, the x-intercepts are at (3, 0), (-3, 0), and (-2, 0).

Key Concepts

x-intercepts

To find the x-intercepts of a polynomial function, we need to determine where the function's graph crosses the x-axis. This happens when the function's value, or \(f(x)\), equals zero. Thus, we set the polynomial equation equal to zero and solve for \(x\).
For the given function, \(f(x) = x^3 + 2x^2 - 9x - 18\), we solve \(x^3 + 2x^2 - 9x - 18 = 0\).
Once we factorize the polynomial, we find values of \(x\) that make the equation zero. For example, if we have factors \((x-3)\), \((x+3)\), and \((x+2)\), solving \(x-3=0\), \(x+3=0\), and \(x+2=0\) gives us \(x=3\), \(x=-3\), and \(x=-2\), respectively. Hence, the x-intercepts for this function are at points \((3, 0)\), \((-3, 0)\), and \((-2, 0)\).
Remember, finding the x-intercepts requires factoring the polynomial and setting each factor to zero.

y-intercepts

To determine the y-intercept of a polynomial function, we need to find the point where the graph crosses the y-axis. This occurs when the value of \(x\) is \(0\) since the y-axis is the vertical line at \(x = 0\).
For the given function, \(f(x) = x^3 + 2x^2 - 9x - 18\), we substitute \(x = 0\) into the equation and solve for \(f(0)\).
Substituting \(0\) for \(x\), we get:
\(f(0) = (0)^3 + 2(0)^2 - 9(0) - 18 = -18\)
Hence, the y-intercept is at point \((0, -18)\). This means the graph of the function crosses the y-axis at \(-18\).

factoring polynomials

Factoring polynomials is a crucial step in finding the x-intercepts of a function. It involves rewriting the polynomial as a product of simpler polynomials or factors.
Steps to factor a polynomial:

• Look for common factors in all terms and factor them out.
• Group terms to create binomials that can be factored further.
• Apply factoring techniques like factoring by grouping, using the difference of squares, or recognizing special products.

For the given function, \(f(x) = x^3 + 2x^2 - 9x - 18\), we group terms:
\((x^3 + 2x^2) - (9x + 18)\)
Then, factor out the common terms in each group:
\(x^2(x + 2) - 9(x + 2)\)
Notice \((x + 2)\) is common, so we factor them out:
\((x + 2)(x^2 - 9)\)
Recognize \((x^2 - 9)\) as a difference of squares:
\((x + 2)(x - 3)(x + 3)\)
Therefore, \(x^3 + 2x^2 - 9x - 18\) factors into \((x + 2)(x - 3)(x + 3)\).
Successfully factoring the polynomial helps you then solve for the roots or x-intercepts.

solving equations

Solving equations is the final step after factoring to find the intercepts. Once you have factored the polynomial, you set each factor equal to zero and solve for \(x\).
For a factored polynomial \((x + 2)(x - 3)(x + 3) = 0\), solving each factor for zero gives:
\(x + 2 = 0\)
\(x = -2\)
\(x - 3 = 0\)
\(x = 3\)
\(x + 3 = 0\)
\(x = -3\)
Thus, the solutions \(x = -2\), \(x = 3\), and \(x = -3\) give us the x-intercepts.
Solving equations after factoring is essential because it gives the precise points at which the polynomial function crosses the x-axis.