Problem 57 - A Graphical Approach to College Algebra

Question

RELATING CONCEPTS For individual or group investigation (Exercises \(55-60\) ) The close relationships among \(x\) -intercepts of a graph of a function, real zeros of the function, and real solutions of the associated equation should, by now, be apparent to you. Consider the graph of the polynomial function \(P(x)=x^{3}-2 x^{2}-11 x+12,\). What are the zeros of the function \(P ?\)

Step-by-Step Solution

Answer

The zeros of the function are 1, 4, and -3.

Step 1: Understand the Problem

We need to find the zeros of the polynomial function \(P(x) = x^3 - 2x^2 - 11x + 12\). A zero of the function corresponds to the \(x\)-intercepts of its graph, which are the values of \(x\) where the function equals zero.

Step 2: Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational solution, or zero, of \(P(x)=0\) is a factor of the constant term (12) divided by a factor of 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 Rational Roots Using Substitution

Substitute the possible rational roots found in Step 2 into the polynomial to see which one makes \(P(x) = 0\). We start with \(x = 1\): \(P(1) = 1^3 - 2 \times 1^2 - 11 \times 1 + 12 = 0\). Hence, \(x = 1\) is a root.

Step 4: Perform Polynomial Division

We now perform synthetic division using the root \(x = 1\) to divide \(P(x)\) and factor out \((x - 1)\). After division, we get a quotient of \(x^2 - x - 12\). Thus \(P(x) = (x - 1)(x^2 - x - 12)\).

Step 5: Factor the Quadratic Polynomial

Next, we factor the quadratic \(x^2 - x - 12\). We look for two numbers that multiply to \(-12\) and add to \(-1\). These numbers are \(-4\) and \(3\). Therefore, \(x^2 - x - 12 = (x - 4)(x + 3)\).

Step 6: Write the Complete Factored Form

The complete factorization of the polynomial is \(P(x) = (x - 1)(x - 4)(x + 3)\).

Step 7: Identify the Zeros

From the factored form, the zeros of the function \(P\) are \(x = 1\), \(x = 4\), and \(x = -3\), as each factor set to zero will yield the solutions for \(P(x) = 0\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool used to determine the possible rational zeros of a polynomial function. It states that if a polynomial has a rational zero or root, it must be a fraction \( \frac{p}{q} \) where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.
This theorem greatly narrows down the number of potential rational roots we need to test, making it an invaluable first step in solving polynomial equations. To utilize the theorem:

• Identify the constant term in the polynomial. In this case, it's \(12\).
• Find factors of the constant term. For \(12\), the factors are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\).
• Identify the leading coefficient. Here, the leading coefficient is \(1\).
• List all possible rational roots as fractions, although if the leading coefficient is 1, these become whole numbers.

With Rational Root Theorem, we quickly identified possible rational roots for the polynomial \(P(x) = x^3 - 2x^2 - 11x + 12\). Testing these values will determine the actual zeros of the polynomial.

Synthetic Division

Synthetic Division is a simplified version of polynomial division, especially useful for dividing by linear factors of the form \(x - c\). This method is faster and less cumbersome compared to long division.
To perform Synthetic Division:

• Write down the coefficients of the polynomial in descending order of powers.
• Select a potential root, here we use \(x = 1\), which was determined as an actual root using the Rational Root Theorem and substitution.
• Place this root in a "box" and set up the synthetic division step.
• Bring down the leading coefficient as it is.
• Multiply this value by the root in the box and write the result under the next coefficient.
• Add vertically and repeat the multiplication step with each new sum until you reach the end.

In this scenario, after applying Synthetic Division to \(x^3 - 2x^2 - 11x + 12\) by \(x = 1\), we obtained the quadratic \(x^2 - x - 12\). This polynomial was subsequently factored further to find all roots.

Zeros of a Function

The zeros of a function are the values of \(x\) where the polynomial equals zero. These zeros are critical as they are also the \(x\)-intercepts of the graph of the polynomial.
To find zeros, one must set the polynomial equal to zero and solve for \(x\). There are different methods to find zeros:

• Testing possible roots identified by the Rational Root Theorem.
• Using Synthetic Division to simplify the polynomial.
• Factoring the polynomial completely.

In the exercise, once we identified that \(x-1\) was a factor, we continued to factor the remaining quadratic \(x^2 - x - 12\). This yielded the solutions \(x = 1\), \(x = 4\), and \(x = -3\), meaning that the polynomial \(P(x) = x^3 - 2x^2 - 11x + 12\) has these zeros.

Graphing Polynomials

Graphing polynomial functions provides a visual representation of their behavior, specifically around the intercepts and end behavior. For the polynomial \(P(x) = (x - 1)(x - 4)(x + 3)\), the graph intersects the x-axis at the zeros of the function: \(x = 1\), \(x = 4\), and \(x = -3\).
Important aspects to remember when graphing polynomials:

• End Behavior: Determined by the degree and leading coefficient of the polynomial. A third-degree polynomial like \(P(x)\) generally has different behavior at each end of the graph.
• Turning Points: A cubic polynomial can have up to two turning points. Examining these points can help predict the general shape of the graph.
• X-intercepts (zeros): Points where the graph crosses or touches the \(x\)-axis, corresponding to the zeros of the polynomial.

Graphing these functions not only verifies algebraic solutions, but also helps in understanding the overall shape and orientation of the polynomial curve.