Problem 115 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) determine the real zeros of \(f\) and (b) sketch the graph of \(f\). $$f(x)=x^{3}+x^{2}-12 x$$

Step-by-Step Solution

Answer

The real zeros of the function \(f(x)=x^{3}+x^{2}-12 x\) are \(x=0\), \(x=3\), and \(x=-4\). A graph of the function shows this cubic function passes through these points and changes direction at each of the roots, becoming increasingly positive as \(x\) increases beyond 3 and increasingly negative as \(x\) decreases beyond -4.

Step 1: Find the real zeros of the function

To find real zeros for the function \(f(x)=x^{3}+x^{2}-12 x\), the function should be set equal to zero and solve for \(x\). Utilizing the fact that \(a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)\), the function can be rewritten in a factored form and solving for x: \[f(x) = x^{3}+x^{2}-12 x = x(x^{2}+x-12) =x(x-3)(x+4)=0\] This gives the possible values as \(x=0\), \(x=3\), and \(x=-4\), which are the real zeros of the function.

Step 2: Identify the behavior of the function at the zeros and at large x

Knowing the roots, the next step is to examine the behavior of the function at these points. As the highest power of \(x\) is 3 and has positive coefficinet, the function will tend to negative infinity as \(x\) goes to negative infinity and will tend to positive infinity as \(x\) goes to positive infinity. At \(x=0\), the function is zero. Furthermore, when \(x\) is smaller than -4, the result will be positive; when x is between -4 and 0, the result will be negative; when x is larger than 3, the result will be positive again. In between the zeros, the function should be negative.

Step 3: Sketch the graph

Now, it's time to sketch the graph using the information gathered above. The x-intercepts are at -4, 0, and 3. As \(x\) approaches infinity, \(f(x)\) approaches infinity; as \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity. Using this information, as well as the changes of the signs, it is possible to make a rough sketch of the function.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial method to determine the real zeros of a polynomial function. For the function \(f(x) = x^3 + x^2 - 12x\), we begin by setting the function equal to zero: \[ x^3 + x^2 - 12x = 0 \]The first step is to factor out the common term, which is \(x\) in this case:\[ x(x^2 + x - 12) = 0 \] Next, the quadratic \(x^2 + x - 12\) needs to be factored. We look for two numbers that multiply to \(-12\) and add up to \(1\). The numbers \(3\) and \(-4\) work:\[ x(x - 3)(x + 4) = 0 \]Now, we have factored the polynomial completely, revealing the real zeros at \(x = 0\), \(x = 3\), and \(x = -4\). Each zero represents where the graph of the function crosses the x-axis. Understanding how to factor polynomials is essential as it simplifies the process of finding these intersections.

Graphing Polynomial Functions

When graphing polynomial functions, we use the real zeros found from the factoring process as key points where the graph intersects the x-axis. For \(f(x) = x(x-3)(x+4)\), the zeros are \(x = 0\), \(x = 3\), and \(x = -4\).

• The x-intercepts are at these points: where the graph touches or crosses the x-axis.
• Since this polynomial is of degree 3 (cubic), the graph will exhibit a curve that may have turns.

To sketch the graph, consider the sign of the polynomial in intervals around these zeros:- Between \(-\infty\) and \(-4\), the graph is positive.- Between \(-4\) and \(0\), it is negative.- Between \(0\) and \(3\), the graph is positive.- Beyond \(3\), the graph returns to positive.Understanding these intervals helps you draw the curve correctly as it turns through these intercepts.

End Behavior of Polynomials

The end behavior of polynomials gives us insight into how the function behaves as \(x\) approaches infinity (both positive and negative). This aspect is influenced by the leading term of the polynomial.For the polynomial function \(f(x) = x^3 + x^2 - 12x\), the leading term is \(x^3\). Since the coefficient of \(x^3\) is positive, the graph's end behavior is as follows:

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

Knowing this end behavior helps in sketching the function accurately, ensuring it moves in the correct direction on both sides of the graph. Observing these trends is important, especially for graphing higher-degree polynomials, as it guides how the tails of the graph extend. End behavior is particularly helpful when combined with the zeros of the polynomial to make an informed sketch of the entire function.