Problem 27 - Algebra and Trigonometry

Question

In Exercises 25–32, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. \(n=3 ;-5\) and \(4+3 i\) are zeros; \(f(2)=91\)

Step-by-Step Solution

Answer

The nth-degree polynomial function is \(f(x) = x^3 - 2x^2 - 31x + 155\).

Step 1 : Write the polynomial

Since the roots (zeros) of the polynomial function are -5, \(4+3 i\) and \(4-3 i\), we can write the polynomial as : \(f(x) = k(x + 5)(x - (4 + 3i))(x - (4 - 3i))\), where \(k\) is a real constant.

Step 2 : Simplify the polynomial

If we simplify this equation we get: \(f(x) = k(x + 5)[(x-4)^2 - (3i)^2]\).\nSince \(- (3i)^2 = -(- 9)\) by the property \(i^2 = - 1\), we simplify further to get \(f(x) = k(x + 5)((x - 4)^2 + 9)\).\nDistributing the terms yields: \(f(x) = k(x^3 -2x^2 -31x + 155)\).

Step 3 : Determine the constant \(k\)

We can find the constant \(k\) using the given condition \(f(2) = 91\). So substitute \(x = 2\) in the function equation to get: \(91 = k(2^3 - 2 * 2 ^2 - 31 * 2 + 155)\).\nSolving for \(k\), we get \(k=1\).

Step 4: step 4 : Final function form

Now that we have \(k=1\), we substitute in the equation to get the final form of the function: \(f(x) = x^3 - 2x^2 - 31x + 155\).

Key Concepts

Polynomial Function with Real Coefficients

A polynomial function with real coefficients is a mathematical expression of the form f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0, where each coefficient a_n, a_{n-1}, ..., a_1, a_0 is a real number, and n is a non-negative integer.
In the context of our exercise, we are dealing with a third-degree polynomial, which is indicated by the highest power of x being three (cubed). Real coefficients provide certain symmetry for the polynomial's zeros. If a polynomial has a complex zero (not a real number), such as 4+3i, then its complex conjugate, 4-3i, will also be a zero of the polynomial. In our example, the polynomial has complex zeros 4 + 3i and 4 - 3i, as well as a real zero, -5.
When crafting a polynomial from given zeros, remember to set the factors equal to zero and solve for x, resulting in the zeros of the polynomial. Subsequently, you multiply those factors together to construct the polynomial. The constant k is a real number that serves as a leading coefficient, which we determine by using additional information provided by the function's value at a specific point.

Complex Zeros of Polynomials

In polynomials, a zero or root can be understood as the value of x at which the polynomial evaluates to zero. Complex zeros of polynomials, which include an imaginary component (the sqrt(-1), denoted as 'i'), come in conjugate pairs if the polynomial has real coefficients.
Complex numbers are in the form a + bi, where a and b are real numbers. In the given exercise, 4 + 3i is one of the zeros, which means by the Conjugate Root Theorem that its conjugate, 4 - 3i, must also be a zero. This theorem is crucial because it helps us predict and confirm the zeros of polynomials that might otherwise be difficult to visualize.

• If a polynomial has real coefficients, any non-real complex zero will always have its conjugate also as a zero.
• Identifying one complex zero allows us to immediately write down its conjugate.

These properties are essential to remember since they streamline the process of finding polynomial functions from given zeros and help in their factorization.

Polynomial Function Graphing

Graphing is a powerful way to visualize polynomial functions and their characteristics, such as intercepts, turning points, and end behavior. The degree of the polynomial provides information about the possible number of turns and the behavior of the function as x approaches positive or negative infinity. For a third-degree polynomial, like the one in our exercise, we can expect up to two turns in its graph.
When plotting a polynomial graph, here are some steps:

• Identify and plot the real zeros (x-intercepts) of the polynomial.
• Determine the multiplicity of the zeros, which affects how the graph behaves as it crosses the x-axis at those points.
• Evaluate the polynomial at several points, especially around the zeros, to determine its shape in those regions.
• Use the end behavior of the polynomial, based on its leading coefficient and degree, to sketch the tails of the graph.

Graphical tools, such as graphing calculators or software, are often used to plot the polynomial to confirm the real zeros and the behavior of the function. They provide instantaneous visual feedback and validate the algebraically derived zeros, thus enhancing somatic understanding. The graph of the function also serves as a check against any analytical mistakes in finding the zeros and constructing the polynomial.