Problem 44 - 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. $$f(x)=x^{4}-8 x^{3}+17 x^{2}-8 x+16$$

Step-by-Step Solution

Answer

The zeros of the function \(f(x)=x^{4}-8 x^{3}+17 x^{2}-8 x+16\) are \(x = 2\). The function can be rewritten as a product of linear factors as \(f(x) = (x - 2)^4\). The x-intercepts of the function are at x = 2.

Step 1: Find the Zeros of the Polynomial

Start by setting the function equal to zero and solve for \(x\). That gives the equation \[x^{4}-8 x^{3}+17 x^{2}-8 x+16 = 0\] This is a quartic equation. However, we can notice it is a perfect square trinomial so it can be factored as \[(x^{2}-4x+4)^2 = 0\] We can solve it by taking square root on both sides, resulting in two solutions: \(x^{2}-4x+4 = 0\) and \(x^{2}-4x+4 = 0\], both giving us the same result 'x = 2'.

Step 2: Rewrite the Polynomial as a Product of Linear Factors

We now have two identical linear factors 'x = 2', and since the equation was a quartic (degree 4), the complete factorization of the polynomial is: \(f(x) = (x - 2)^4\)

Step 3: Find the Function's X-Intercepts

The function's x-intercepts are the real values of x that we found such that the function \(f(x)\) equals 0, which is 'x = 2'. Thus, the graph of function \(f(x)\) intersects the x-axis at the point (2, 0).

Step 4: Verify with a Graphing Utility

In this step, plot the function \(f(x)=x^{4}-8 x^{3}+17 x^{2}-8 x+16\). As predicted, the curve should only intersect the x-axis at x = 2. Any graphing utility such as Desmos or GeoGebra can be used to verify this visually.

Key Concepts

zeros of polynomials

Zeros of polynomials are the values of the variable for which the polynomial function equals zero. Identifying these zeros is crucial since they signify where the graph of the polynomial intersects the x-axis.
To find the zeros, you set the polynomial equal to zero and solve for the variable. In our exercise, this involved solving the equation:

• \(x^{4} - 8x^{3} + 17x^{2} - 8x + 16 = 0\)

For our specific polynomial, recognizing a perfect square trinomial allowed for simplification. By factoring as \((x^2 - 4x + 4)^2 = 0\), the solution for both parts gave us the zero \(x = 2\).
Finding zeros helps in understanding the structure of a polynomial and predicts the shape and position of its graph on the coordinate plane.

factoring polynomials

Factoring is the process of breaking down a polynomial into simpler components called factors, usually into products of polynomials of lower degrees. This aids in solving polynomial equations.

• It can reveal the zeros of the polynomial directly.
• It allows easy multiplication and division within algebraic expressions.

To factor our polynomial \(f(x) = x^{4} - 8x^{3} + 17x^{2} - 8x + 16\), we discovered it can be factored into \((x - 2)^4\).
This denotes that our polynomial is expressed as the product of four linear factors, each equal to \(x - 2\). Recognizing familiar patterns like perfect square trinomials can significantly simplify the factoring process.
Factoring is an essential algebraic skill that facilitates solving polynomial equations and understanding the behavior of functions.

x-intercepts

The x-intercepts of a function's graph are points where the graph crosses the x-axis. These occur at the polynomial's zeros.
After finding the zeros of our polynomial \(f(x)\), which is \(x = 2\), this translates into a single x-intercept at the coordinate (2, 0).
Calculating x-intercepts is vital because:

• It provides insight into the root structure of the polynomial.
• The number and location of x-intercepts affect the shape of the graph.

Thus, the x-intercepts are direct reflections of the polynomial's zeros, and this information is invaluable when sketching or analyzing the polynomial's graph.

graphing utilities

Graphing utilities are tools that help visualize functions, examining their behavior in a specific interval. They are powerful aids in confirming algebraic calculations.

• They can verify zeros and x-intercepts.
• They provide a visual representation of the polynomial's behavior.

In the exercise, after identifying the zeros and factoring the polynomial, the use of a graphing utility confirmed that the function \(f(x) = x^{4} - 8x^{3} + 17x^{2} - 8x + 16\) has an x-intercept at (2, 0).
Graphing utilities like Desmos or GeoGebra allow for dynamic interaction with the function. By plotting our polynomial, one can visually observe the intersection with the x-axis occurs solely at \(x = 2\).
Such tools enhance understanding by combining algebraic techniques with the visual insight they provide, strengthening comprehension of polynomial functions and their graphs.