Problem 34 - Precalculus Mathematics for Calculus

Question

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$P(x)=x^{3}+3 x^{2}-4 x-12$$

Step-by-Step Solution

Answer

The zeros of the polynomial are \(-3, -2,\) and \(2\).

Step 1: Identify the Polynomial Type

The given polynomial is \(P(x) = x^3 + 3x^2 - 4x - 12\). It is a cubic polynomial because the highest power of \(x\) is 3.

Step 2: Use the Rational Root Theorem

The Rational Root Theorem helps find potential rational roots by considering the factors of the constant term \(-12\) and the leading coefficient \(1\) which gives possible roots as \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

Step 3: Test Possible Roots

Test the possible rational roots by substituting them into the polynomial until you find a root. Substituting \(x = -2\) gives \(P(-2) = (-2)^3 + 3(-2)^2 - 4(-2) - 12 = 0\), confirming \(x = -2\) is a root.

Step 4: Perform Synthetic Division

Perform synthetic division of \(P(x)\) by \(x + 2\) since \(x = -2\) is a root. This gives us a quotient polynomial \(x^2 + x - 6\).

Step 5: Factor the Quotient Polynomial

Factor the quadratic polynomial \(x^2 + x - 6\). We find that \(x^2 + x - 6 = (x + 3)(x - 2)\). Thus, the factorization of \(P(x)\) is \((x + 2)(x + 3)(x - 2)\).

Step 6: Find the Zeros of the Polynomial

Using the factored form \((x + 2)(x + 3)(x - 2)\), the zeros are \(x = -2, x = -3,\) and \(x = 2\).

Step 7: Sketch the Graph

The cubic polynomial \(P(x)\) changes sign at each zero. The approximate behavior of the graph shows it starting from negative infinity (as \(x\) approaches negative infinity), crossing the x-axis at \(x = -3, x = -2, x = 2\), and levelling off to positive infinity (as \(x\) approaches positive infinity). The graph has turning points between these zeros.

Key Concepts

Zeros of polynomial

Finding the zeros of a polynomial is a crucial concept in algebra because it helps determine the values for which the polynomial equals zero. The zeros are the points where the graph of the polynomial crosses or touches the x-axis. To find the zeros, the polynomial must be factored first. Let's consider the polynomial given in the exercise: \[ P(x) = (x + 2)(x + 3)(x - 2) \]To find the zeros, set the factored form of the polynomial equal to zero and solve for the variable:

• \(x + 2 = 0\) gives \(x = -2\)
• \(x + 3 = 0\) gives \(x = -3\)
• \(x - 2 = 0\) gives \(x = 2\)

Finding these zeros helps in understanding the behavior of the polynomial and to sketch its graph. When the polynomial is graphed, these zeros will be the x-intercepts of the graph.

Cubic polynomial

A cubic polynomial is a polynomial of degree three, meaning its highest exponent of the variable (let's say \(x\)) is three. Cubic polynomials are of the form:\[ P(x) = ax^3 + bx^2 + cx + d \]where \(a, b, c,\) and \(d\) are constants and \(a eq 0\).The given polynomial in the exercise is:\[ P(x) = x^3 + 3x^2 - 4x - 12 \]Characteristics of cubic polynomials include:

• They can have up to three zeros (or x-intercepts).
• They may have one or two turning points.
• Their end behavior is determined by the leading term \(ax^3\).

Cubic polynomials generally have a characteristic 'S' shape in their graphs, crossing the x-axis at zeros that are found through factorization. Understanding these features allows one to sketch their general shape effectively once the zeros are determined.

Rational Root Theorem

The Rational Root Theorem is a useful tool for finding possible rational roots of a polynomial equation. It stems from the relationship between the possible values of factors of the polynomial's constant term and its leading coefficient.For the polynomial in the exercise:\[ P(x) = x^3 + 3x^2 - 4x - 12 \]The leading coefficient is 1, and the constant term is \(-12\). According to the Rational Root Theorem:

• Possible rational roots are the factors of the constant term divided by the factors of the leading coefficient, which means possible roots could be \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\).

Each of these potential roots is tested in the polynomial until a true root is found. Once a root is discovered using these logical possibilities, factorization can proceed more smoothly.

Synthetic division

Synthetic division is a simplified form of long division used specifically for dividing polynomials. It works well when dividing by linear expressions in the form \(x - r\), where \(r\) is a known root of the polynomial.For instance, in our exercise, after finding that \(x = -2\) is a root of\[ P(x) = x^3 + 3x^2 - 4x - 12 \]we use synthetic division to divide \(P(x)\) by \(x + 2\):1. Write down the coefficients of \(P(x)\): \(1, 3, -4, -12\).2. Use \(-2\) as the number for synthetic division below the line.3. Perform the synthetic division process by bringing down the first coefficient, multiply it by \(-2\), add it to the next coefficient, and continue this cycle.4. The final row of numbers (excluding the remainder) gives the coefficients of the quotient polynomial.This process yields a quotient of \(x^2 + x - 6\), simplifying the factorization process into a more manageable quadratic form.