Question

Algebraic and Graphical Approaches In Exercises \(31-46\), find all real zeros of the function algebraically. Then use a graphing utility to confirm your results. $$f(t)=t^{3}-4 t^{2}+4 t$$

Short answer

The real zeros of the function are \(t=0\) and \(t=2\).

Step-by-step solution

Identify the Function

The function to analyze is \(f(t)=t^{3}-4 t^{2}+4 t\).

Set the Function Equal to Zero

To find the zeros of the function, set \(f(t)\) equal to zero, so the equation to solve is \(0=t^{3}-4 t^{2}+4 t\).

Factor Out the Common Factor

It's observed that all of the terms on the right side of the equation include \(t\), so factoring out \(t\) gives \(0=t(t^{2}-4 t+4)\)

Simplify the Quadratic Term

The simplified form of \(t^{2}-4 t+4\) is \((t-2)^2\), using the formula \(a^2 - 2ab + b^2 = (a-b)^2\). So the equation turns into \(0=t(t-2)^2\).

Set Each Factor Equal to Zero

Setting each factor equal to zero gives \(t=0\) and \(t-2=0\), which result in the solutions \(t=0\) and \(t=2\).

Verify the Solutions Graphically

Plot the function \(f(t)=t^{3}-4 t^{2}+4 t\). It will be seen that the graph intersects the x-axis at \(t=0\) and \(t=2\), confirming that those are indeed the zeros of the function.

Key concepts

Algebraic Methods

Finding the zeros of a polynomial function using algebraic methods involves a systematic approach. First, you set the given function equal to zero. This is because zeros of a function are the points where the function's value is equal to zero. To find these, we need to solve the equation obtained by setting the function to zero. Next, you can apply factoring techniques to simplify the equation, making it easier to solve for the zeros.

This process often includes selecting a factoring method, such as factoring out the greatest common factor or applying special formulas (like the difference of squares). Once the function is factored completely, set each factor equal to zero. Solving these simple equations will give you the zeros of the original polynomial function. This algebraic approach is both systematic and reliable, providing a methodical way to find all possible real zeros of a function.

• Set the function equal to zero.
• Factor the polynomial if possible.
• Use algebraic equations to find zeros.

Graphical Analysis

Graphical analysis is a visual approach to verifying the zeros of a function. By plotting the polynomial function on a graph, you can observe where the curve intersects the x-axis. The points of intersection are the zeros of the function.

Graphical methods provide a way to confirm algebraically derived zeros and detect any graphical behavior or characteristics that were not immediately evident in the algebraic form, such as multiplicity of roots or turning points. It is important to have a precise graphing tool, often found in graphing calculators or computer software, to accurately visualize the function's behavior.

• Plot the function on a graph.
• Identify where the curve crosses the x-axis.
• Verify algebraic solutions with intersection points.

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables raised to various powers and multiplied by coefficients. The highest power of the variable is the degree of the polynomial, which can help determine the number of potential zeros or roots.

For instance, a cubic polynomial, like the one in our example, could have up to three real zeros. However, some roots could be repeated, indicating that the polynomial might intersect or just touch the x-axis at fewer points. Understanding the structure and degree of polynomial functions is crucial when seeking to find all real zeros.

• Recognize the degree of the polynomial.
• Understand the significance of each term.
• Identify possible real zeros based on the polynomial's degree.

Factoring Quadratics

Factoring quadratics is a common method for solving polynomial equations. Quadratics are polynomials of degree two, often taking the form of \( ax^{2} + bx + c \). To factor a quadratic, you seek two binomials whose product yields the original quadratic expression.

The process usually starts with finding two numbers that multiply to the constant term, \( c \), and add up to the linear coefficient, \( b \). This technique, often called "factoring by inspection," is efficient for simple quadratics. More complicated quadratics might require completing the square or using the quadratic formula. When factoring polynomials like the one given, recognizing special binomial squares, such as \( (a-b)^2 = a^2 - 2ab + b^2 \), can simplify the process considerably.

• Identify and use known factoring patterns.
• Ensure all terms are simplified correctly.
• Verify factors by multiplication.