Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

The zero(s) of the function are given. Verify the zero(s) both algebraically and graphically. \(Function\) \(f(x)=x^{3}-6 x^{2}+5 x\) \(Z e r o(s)\) \(x=0,5,1\)

Step-by-Step Solution

Answer

The zeros of the function, \(x = 0, 5, 1\), are proven to be correct algebraically, as all inserted into the function equate to zero. By analysing the graph of the function, the given zeros are additionally confirmed graphically.

Step 1: Verify Zeros Algebraically

To verify these zeros of the function, insert each zero into the function \(f(x)\) and see if the result equals 0. For \(x = 0\): \(f(0)=0^3-6*0^2+5*0 = 0\)For \(x = 1\): \(f(1)=1^3-6*1^2+5*1 = 0\)For \(x = 5\): \(f(5)=5^3-6*5^2+5*5 = 0\)As we can see, each value of \(x\) provided does indeed give \(f(x) = 0\), so the zeros are correct algebraically.

Step 2: Verify Zeros Graphically

When graphing the function \(f(x) = x^3 - 6x^2 + 5x\), the points where the graph intersects the x-axis are the zeros of the function. The x-coordinates of these intersection points should correspond to the provided zeros \(x = 0, 1, 5\). If the graph intersects the x-axis at these points, then the zeros are correct graphically. You'll use graphing software or a graphing calculator to create the graph to verify this.

Key Concepts

Algebraic Verification

Algebraic verification of polynomial zeros is a straightforward process that involves substituting potential zeros into the polynomial function to see if they make the function equal zero. This method checks each zero individually:

For the polynomial function given, \( f(x) = x^3 - 6x^2 + 5x \), the zeros provided are \( x = 0, 1, \text{ and } 5 \). Let's verify these algebraically by substituting these values one by one.

• For \( x = 0 \): Substitute into the function, \( f(0) = 0^3 - 6 \cdot 0^2 + 5 \cdot 0 = 0 \). Sure enough, \( f(0) = 0 \), meaning \( x = 0 \) is a zero.
• For \( x = 1 \): Substitute into the function, \( f(1) = 1^3 - 6 \cdot 1^2 + 5 \cdot 1 = 1 - 6 + 5 = 0 \). Again, \( f(1) = 0 \), so \( x = 1 \) is also a zero.
• For \( x = 5 \): Substitute into the function, \( f(5) = 5^3 - 6 \cdot 5^2 + 5 \cdot 5 = 125 - 150 + 25 = 0 \). Finally, \( f(5) = 0 \), confirming \( x = 5 \) as a zero.

By substituting each zero into the polynomial and obtaining zero each time, we can confidently say the zeros \( x = 0, 1, 5 \) are algebraically verified.

Graphical Verification

Graphical verification is another powerful way to confirm the zeros of a polynomial function. It involves plotting the polynomial on a graph and observing where the curve intersects the x-axis. The x-coordinates of these intersections are the real zeros of the function.

In the case of the given polynomial \( f(x) = x^3 - 6x^2 + 5x \), upon plotting, you should notice the graph touches the x-axis at \( x = 0, 1, \text{ and } 5 \).

Using a graphing calculator or online software can greatly assist in this process. The steps are as follows:

• Input the function \( f(x) = x^3 - 6x^2 + 5x \) into the graphing tool.
• Generate the plot and focus on where it crosses the x-axis.
• Identify that the x-intercepts are at \( x = 0, 1, \text{ and } 5 \).

This concrete visual confirmation complements the algebraic findings, providing a holistic check on the polynomial’s zeros.

Polynomial Functions

A polynomial function is an essential mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. They play a crucial role in various areas of mathematics and practical applications.

In the given exercise, the polynomial function is \( f(x) = x^3 - 6x^2 + 5x \), which is a cubic polynomial. Its general form can be written as \( ax^3 + bx^2 + cx + d \), where the degree of the polynomial (the highest power of x) is 3.

• The term \( x^3 \) is the leading term, setting the degree of the polynomial.
• The coefficients are \( a = 1 \), \( b = -6 \), and \( c = 5 \).
• A cubic polynomial like this can have up to three real roots or zeros, which are solved using algebraic and graphical methods as shown.

Understanding the structure and behavior of polynomial functions is fundamental to mastering various mathematical concepts and can be applied in real-world situations such as physics, engineering, and economics. They help model relationships and predict trends within data.