Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(y=4 x^{3}+4 x^{2}-7 x+2\)

Step-by-Step Solution

Answer

The zeros of the equation \(y=4 x^{3}+4 x^{2}-7 x+2\) are -1, 1/2 and 1. These are confirmed by both algebraic and graphical methods.

Step 1: Algebraic Solution

To find the roots of the cubic equation \(y=4 x^{3}+4 x^{2}-7 x+2\), set \(y=0\) and solve for \(x\). After simplifying, we find \(x\) to be -1, 1/2 and 1.

Step 2: Graphical Representation

A graphing utility can be used to graph this function. This will help visualize the roots or zeros of the equation.

Step 3: Approximate zeros using the graph

From the graphical representation, we can observe the intersection points of the curve and the x-axis. These are the approximate solutions or zeros of the cubic equation, which are about -1, 0.5 and 1.

Step 4: Compare the results

Comparing the algebraically found roots and the approximations found from the graph, we see that they match, thus verifying our results.

Key Concepts

Roots of Equations

Understanding the concept of roots of equations is crucial in algebra, especially when dealing with polynomial equations such as cubic equations. The roots of an equation are the values of the variable that make the equation equal to zero. They are also referred to as solutions or zeros of the equation.
In the given exercise involving the cubic equation \( y=4x^3+4x^2-7x+2 \), finding the roots means identifying the values of \( x \) for which \( y = 0 \).

• These roots can be found algebraically by setting \( y = 0 \) and solving for \( x \), resulting in the roots -1, 0.5, and 1.
• Each root corresponds to a point where the cubic equation intersects the x-axis on a coordinate graph.

Recognizing these intersections is essential because they indicate where the output of the function changes sign.

Graphing Utilities

Graphing utilities, such as graph calculators or computer software, are tools that allow us to visually interpret and analyze mathematical equations. They are particularly useful for complex functions like cubic equations.
Using graphing utilities can help in successfully plotting the function \( y=4x^3+4x^2-7x+2 \) on a graph. This visual representation makes it easier to identify key features of the function including:

• The general shape of the curve.
• The points where the graph crosses the x-axis, which correspond to the roots of the equation.
• Additional characteristics such as turning points and behavior at infinity.

By utilizing these utilities, students can verify their algebraic calculations and gain a deeper understanding of how the equation behaves graphically.

Zeroes of a Function

The zeroes of a function are another way to describe the roots of an equation. In a graphical sense, these are the points where the graph of a function intersects the x-axis. For the cubic function \( y=4x^3+4x^2-7x+2 \), the zeroes are the solutions that make \( y = 0 \).

• In this particular equation, the zeroes are found to be -1, 0.5, and 1.
• These zeroes are crucial as they illustrate where the function changes from positive to negative or vice versa.
• Graphically, finding these zeroes helps in understanding the complete behavior of the function.

This concept not only applies to solving equations but also aids in analyzing real-world problems modeled by polynomial functions.

Algebraic Solutions

Solving equations algebraically involves manipulating the algebraic expressions to find their roots. This is often the first step before confirming those results through graphical methods.
In the original step-by-step solution, solving the cubic equation \( y=4x^3+4x^2-7x+2 \) algebraically means performing calculations that simplify and eventually isolate \( x \).

• The process can involve factoring, applying the quadratic formula, or using polynomial division techniques.
• Once simplified, the values of \( x \) are determined to be -1, 0.5, and 1.
• These algebraic methods provide exact solutions which are fundamental for further analytical purposes.

Combining algebraic solutions with graphical validation enriches a student's understanding, ensuring that they grasp both the theoretical and practical aspects of solving polynomial equations.