Problem 57 - Algebra and Trigonometry

Question

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\) (b) Sketch the graph of \(P\) . $$ P(x)=x^{3}-3 x^{2}-4 x+12 $$

Step-by-Step Solution

Answer

The zeros are \(x = -2, 2, 3\). Sketch shows a cubic crossing x-axis at these points.

Step 1: Understand the Problem

We are given a polynomial \(P(x) = x^3 - 3x^2 - 4x + 12\). The task is to find all real zeros of the polynomial \(P\) and sketch its graph. Finding the zeros involves solving \(P(x) = 0\).

Step 2: Try Factoring

Look for common factors or simple zeros. Start with substituting possible rational zeros such as \(x = \pm1, \pm2, \pm3, \pm4, \pm6, \pm12\) into \(P(x)\) to test for roots.

Step 3: Use Synthetic Division

By testing, we find that \(x = 2\) is a root since \(P(2) = 0\). Perform synthetic division with \(x = 2\) to factor the polynomial:\[\begin{array}{r|rrrr}2 & 1 & -3 & -4 & 12 \ & & 2 & -2 & -12 \\hline & 1 & -1 & -6 & 0 \end{array}\]The result shows a quotient \((x^2 - x - 6)\). Thus, the polynomial is factored as \((x - 2)(x^2 - x - 6)\).

Step 4: Factor the Quadratic

Factor the quadratic \(x^2 - x - 6\). We look for two numbers that multiply to \(-6\) and add to \(-1\). These numbers are 2 and -3. Therefore, \(x^2 - x - 6 = (x - 3)(x + 2)\).

Step 5: Find All Real Zeros

The complete factorization of \(P(x)\) is \((x - 2)(x - 3)(x + 2)\). Therefore, the zeros are \(x = 2, x = 3, x = -2\).

Step 6: Sketch the Graph

Using the zeros \(x = -2, 2, 3\), plot these points on the x-axis. The polynomial is a cubic, which implies an end behavior where it falls to \(-\infty\) as \(x \to -\infty\) and rises to \(+\infty\) as \(x \to +\infty\). Since it changes signs at the zeros, sketching starts from the bottom left, crosses the x-axis at \(-2\), \(2\), and \(3\), and ends in the top right.

Key Concepts

Polynomial Factoring

Polynomial factoring is a critical process used to simplify polynomials by expressing them as a product of simpler, or more fundamental, polynomials. This process can help in finding the zeros of the polynomial, which are the values of the variable that make the polynomial equal to zero. In our exercise, we factored the cubic polynomial \(P(x) = x^3 - 3x^2 - 4x + 12\) to find its zeros.
To factor a polynomial:

• Identify and check for the greatest common factor (GCF) first.
• Use techniques such as grouping, using standard factoring formulas, or inspecting possible rational roots using the Rational Root Theorem.
• Test these candidates in the polynomial to determine if they are indeed zeros.

In this case, we found that \(x = 2\) is a zero of the polynomial. Using synthetic division, we were then able to further factor the polynomial. Finding factors expresses the polynomial as \((x - 2)(x - 3)(x + 2)\), and these factors immediately reveal the polynomial's zeros at \(x = 2, 3,\) and \(-2\).

Synthetic Division

Synthetic division is an efficient method for dividing a polynomial by a linear divisor of the form \((x - c)\). Often preferred over long division due to its simplicity, synthetic division can quickly determine if a value is a root of a polynomial by checking if the remainder is zero.
This method involves:

• Setting up a row of coefficients from the polynomial in descending order of powers.
• Writing the candidate root \(c\) to the left.
• Bringing down the leading coefficient, multiplying it by \(c\), and adding it to the next coefficient.
• Repeat the multiplication and addition steps for all coefficients.

In the problem, we used synthetic division with \(x = 2\), verifying it as a zero since the remainder was 0, and obtained the resulting quotient \((x^2 - x - 6)\). This simplified the polynomial, confirming \(x = 2\) as a valid root and allowing us to break down the polynomial into more manageable parts for further factorization.

Cubic Functions

Cubic functions are polynomial functions where the highest degree is three, as seen in the function \(P(x) = x^3 - 3x^2 - 4x + 12\). These functions are characterized by their curve, which can have up to three real roots and two turning points, depending on the nature of their zeros and coefficients.
A cubic function generally:

• Has the end behavior whereby as \(x\) approaches \(-\infty\), the function decreases, and as \(x\) approaches \(+\infty\), it increases, or vice versa depending on the leading coefficient.
• Can have intermediate behavior that involves crossing the x-axis at its zeros, changing direction at its critical points identified by taking derivatives or solving via factoring.

In our example, by finding the real zeros at \(-2, 2,\) and \(3\), the graph was sketched to reflect these roots crossing the x-axis, starting low from the left and rising high to the right, showing how cubic functions depict complex behaviors yet obey predictable end behaviors.