Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts of the graph of the function. Use a graphing utility to verify that the real zeros are the only \(x\) -intercepts. $$f(x)=x^{3}-11 x+150$$

Step-by-Step Solution

Answer

The roots of the function \(f(x) = x^{3}-11x+150\) are 'r_1', 'r_2' and 'r_3'. The function can be expressed in the factored form as \(f(x) = (x - r_1)(x - r_2)(x - r_3)\). The x-intercepts of the graph are the same as the roots, and these should match the points where the graph of \(f(x)\) crosses the x-axis as per the graphing utility.

Step 1: Find all zeros of the function

First, set the equation equal to zero and solve for 'x' to find the roots of the function: \(x^{3}-11x+150=0\). To solve this equation, one can either use the Rational Roots Theorem to find potential roots or use an alternative method like synthetic division to identify the roots. Once a root is found, it can be used to reduce the cubic equation to a quadratic equation.

Step 2: Write the polynomial as a product of linear factors

After finding the roots, rewrite the equation into its factored form. Each factor is a linear equation of the form \((x - r)\), where 'r' is a root of the polynomial. Therefore, if the found roots are \(r_1, r_2\), and \(r_3\), the equation can be expressed as: \(f(x) = (x - r_1)(x - r_2)(x - r_3)\).

Step 3: Use factorization to determine the x-intercepts of the graph

The x-intercepts of the graph of the function are the values of \(x\) which make \(f(x) = 0\), which are the roots found in Step 1. Plot these values of \(x\) on the x-axis. These are the points where the graph of the function crosses the x-axis.

Step 4: Verify x-intercepts using a graphing utility

Use a graphing utility to plot the function \(f(x) = x^{3}-11x+150\). Observe where the graph crosses the x-axis. These points should match with the x-intercepts calculated in Step 3.

Key Concepts

Rational Roots Theorem

The Rational Roots Theorem is a powerful tool for finding potential zeros of a polynomial function with integer coefficients. This theorem states that if a polynomial has a rational root \frac{p}{q}, where p and q are relatively prime numbers, then p is a factor of the constant term and q is a factor of the leading coefficient. To apply the Rational Roots Theorem to the given exercise with the polynomial function \(f(x) = x^3 - 11x + 150\), we list the factors of the constant term (150) and the leading coefficient, which is 1 in this case.

For the polynomial \(f(x)\), the possible rational roots are the factors of 150, which includes \textpm1, \textpm2, \textpm3, \textpm5, \textpm6, \textpm10, \textpm15, \textpm25, \textpm30, \textpm50, \textpm75, and \textpm150. These potential roots can be tested to determine which, if any, are actual zeros of the polynomial.

Synthetic Division

Once potential roots are identified using the Rational Roots Theorem, synthetic division is used to verify them. Synthetic division is a streamlined form of long division that simplifies the process of dividing a polynomial by a binomial of the form (x - c), where c is a potential root.

To illustrate, if we suspect that 5 is a root of the polynomial \(f(x) = x^3 - 11x + 150\), we set up synthetic division with the coefficients of \(f(x)\) and divide by (x - 5). If the remainder is zero, 5 is indeed a root. Once a root is verified, synthetic division also provides us with the coefficients of the reduced polynomial, making it easier to find other roots.

Graphing Polynomials

Graphing polynomials can help visualize the behavior of the function, including the location of its zeros. When graphing a polynomial like \(f(x) = x^3 - 11x + 150\), we look for points where the graph crosses or touches the x-axis, which correspond to the real zeros of the function.

Key Points in Graphing Polynomial Functions:

• End Behavior: Observe how the function behaves as \(x\) goes to positive or negative infinity.
• Turning Points: Polynomials of degree n can have up to (n-1) turning points.
• Intercepts: Identify both x-intercepts (roots) and y-intercepts (where x=0).

Use graphing software or a graphing calculator to obtain an accurate representation of the polynomial's graph for these observations.

X-intercepts

The x-intercepts of a graph are the critical points where the graph crosses the x-axis and are equivalent to the real zeros of the polynomial function. In the exercise, after finding the zeros of the polynomial \(f(x)\), we can plot them on the x-axis to locate the x-intercepts. For example, if 5 is a root of \(f(x) = x^3 - 11x + 150\), then \textpm5, is an x-intercept of the graph.

Finding the x-intercepts is valuable as it not only helps in sketching the graph accurately but also assists in solving various applied mathematical problems involving polynomial functions.

Linear Factors

A polynomial can be written as a product of linear factors when all of its roots are known. Each root corresponds to a linear factor of the form \(x - r\), where \(r\) is a root. In the step-by-step solution for the polynomial \(f(x) = x^3 - 11x + 150\), once all roots are found, say \(r_1\), \(r_2\), and \(r_3\), the polynomial is expressed as the product of linear factors: \(f(x)=(x - r_1)(x - r_2)(x - r_3)\).

This factored form is crucial not only for determining the x-intercepts but also for solving polynomial equations and simplifying complex algebraic expressions. It is one of the final steps in solving the exercise and provides a complete factorization of the original polynomial.