Problem 57 - Precalculus Mathematics for Calculus

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

Zeros: \(x = -2, x = 2, x = 3\). Graph is a cubic curve crossing x-axis at these points.

Step 1: Identify the Polynomial

The given polynomial is \(P(x) = x^3 - 3x^2 - 4x + 12\). This is a cubic polynomial.

Step 2: Apply Rational Root Theorem

According to the Rational Root Theorem, possible rational roots of the polynomial are the factors of the constant term (12) over the leading coefficient (1). Thus, the possible rational roots are: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

Step 3: Test Possible Roots with Synthetic Division

Test the potential roots using synthetic division to find a root. After testing, we find that \(x = 2\) is a root of the polynomial as the remainder is 0, confirming that \(x - 2\) is a factor.

Step 4: Factor Polynomial by Division

Since \( x = 2 \) is a root, divide the original polynomial by \( (x - 2) \) using synthetic or long division. This results in \(P(x) = (x - 2)(x^2 - x - 6)\).

Step 5: Factor the Quadratic Expression

Next, factor the quadratic \(x^2 - x - 6\). It can be factored further into \((x - 3)(x + 2)\) because \(-3 \times 2 = -6\) and \(-3 + 2 = -1\).

Step 6: List Real Zeros

The polynomial thus factors as \(P(x) = (x - 2)(x - 3)(x + 2)\). The real zeros of the polynomial are \(x = 2, x = 3,\) and \(x = -2\).

Step 7: Sketch the Graph

Using the zeros found, we can sketch the graph. The cubic polynomial has zeros at -2, 2, and 3 and will cross the x-axis at these points. As the leading term \(x^3\) is positive, the graph will start from the bottom left and end at the top right.

Key Concepts

Cubic Polynomial

A cubic polynomial is a type of polynomial that has a degree of three, which means the highest power of the variable is three. In general, a cubic polynomial can be written as: \[ P(x) = ax^3 + bx^2 + cx + d \] where \( a \), \( b \), \( c \), and \( d \) are constants and \( a eq 0 \). Because the degree is three, a cubic polynomial can have up to three real roots or zeros. These roots are the values of \( x \) that make the polynomial equal to zero.
When graphed, a cubic polynomial will have a distinctive "S" or "N" shaped curve known as an inflection point. Understanding the general shape of the graph of a cubic polynomial can be helpful when identifying roots and sketching the graph.

Rational Root Theorem

The Rational Root Theorem is a useful tool for finding potential rational roots of a polynomial. This theorem states that any rational solution of the polynomial equation \[ P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0 \]can be expressed as a fraction \( \frac{p}{q} \), where \( p \) is a factor of the constant term \( a_0 \), and \( q \) is a factor of the leading coefficient \( a_n \).

• For our polynomial \( P(x) = x^3 - 3x^2 - 4x + 12 \), the constant term is 12, and the leading coefficient is 1.
• Thus, the possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

These are the numbers you can test to find rational solutions. Remember, these are just possibilities, and actual roots need to be confirmed through additional methods like synthetic division.

Synthetic Division

Synthetic division is an easier alternative to traditional long division for dividing polynomials, particularly when dealing with a linear divisor. This method is especially handy for testing potential roots of a polynomial. It involves fewer steps and is less error-prone than long division, making it favoured for its simplicity.
To use synthetic division with a cubic polynomial, you follow these steps:

• Write down all coefficients of the polynomial in descending order of degree.
• Choose a potential root to test (e.g., \( x = 2 \)).
• Bring down the first coefficient.
• Multiply this value by the potential root and write the result under the next coefficient.
• Add the value to the coefficient above.
• Repeat these steps for all coefficients.

If your final value (the remainder) is zero, then you have verified that the root is valid. For \( P(x) = x^3 - 3x^2 - 4x + 12 \), testing \( x = 2 \) shows zero remainder, confirming \( x - 2 \) as a factor.

Graphing Polynomials

Graphing polynomials involves plotting the curve representing the polynomial equation on the Cartesian plane. This helps to visually identify the behavior and roots of the polynomial.
For a cubic polynomial like \( P(x) = (x - 2)(x - 3)(x + 2) \), graphing begins with:

• Identifying the zeros: In this case, the zeros are \( x = 2, x = -2, x = 3 \). The graph will intersect the x-axis at these points.
• Considering the end behavior: Since the leading term \( x^3 \) is positive, the graph will start in the negative y-direction (bottom left) and end in the positive y-direction (top right).

The graph will have a smooth curve that only changes direction at its zeros, with an inflection point often appearing when plotted over an interval. By using both analytical and graphical techniques, understanding polynomials becomes more intuitive.