Problem 26 - Algebra 2 and Trigonometry

Question

In \(19-28 :\) a. Find \(\mathrm{f}(a)\) for each given function. b. Is \(a\) a root of the function? $$ \mathrm{f}(x)=x^{4}-2 x^{2}+x \text { and } a=\sqrt{3} $$

Step-by-Step Solution

Answer

\( \mathrm{f}(\sqrt{3}) = 3 + \sqrt{3} \); \( \sqrt{3} \) is not a root.

Step 1: Substitute the Value of a

To find \( \mathrm{f}(a) \), we need to substitute \( a = \sqrt{3} \) into the function \( \mathrm{f}(x) = x^4 - 2x^2 + x \). This results in: \[ \mathrm{f}(\sqrt{3}) = (\sqrt{3})^4 - 2(\sqrt{3})^2 + \sqrt{3}. \]

Step 2: Simplify Each Term

Calculate each term in the expression separately: \((\sqrt{3})^4 = 9\), \((\sqrt{3})^2 = 3\), so the equation becomes \( 9 - 2 \cdot 3 + \sqrt{3} \).

Step 3: Complete the Calculation

Complete the arithmetic operations: \(9 - 2 \cdot 3 = 9 - 6 = 3\), therefore \( \mathrm{f}(\sqrt{3}) = 3 + \sqrt{3} \).

Step 4: Evaluate if a is a Root

A root of the function occurs when \( \mathrm{f}(a) = 0 \). Since \( \mathrm{f}(\sqrt{3}) = 3 + \sqrt{3} \), which is not zero, \( \sqrt{3} \) is not a root.

Key Concepts

Roots of a Function

Roots of a function are special values where the function equals zero. To determine if a number is a root, simply replace the variable in the polynomial with this number and carry out the arithmetic operations. If the result is zero, then the number is a root.
In the example provided, we replaced the variable in the function \( f(x) = x^4 - 2x^2 + x \) with \( \sqrt{3} \). The outcome was \( 3 + \sqrt{3} \), which is not zero. Therefore, \( \sqrt{3} \) is not a root.
When dealing with polynomial functions:

• Understand that roots represent the intersection points with the x-axis on a graph.
• Polynomials have as many roots as their degree, though some may be complex or repeated.
• Finding roots can also help factor polynomials, making them easier to work with in other problems.

Understanding roots is crucial for solving polynomial equations and for analyzing their graphs.

Function Evaluation

Evaluating a function involves replacing each occurrence of the variable in the expression with a given value and performing the necessary arithmetic operations.
To evaluate \( f(a) \) in a polynomial such as \( f(x) = x^4 - 2x^2 + x \), substitute \( a \) for \( x \). For the provided exercise with \( a = \sqrt{3} \), the function becomes \( f(\sqrt{3}) = (\sqrt{3})^4 - 2(\sqrt{3})^2 + \sqrt{3} \).
The steps:

• Substitute \( a \) directly into the function wherever \( x \) appears.
• Simplify each term by performing powers and products.
• Carry out any addition or subtraction to arrive at the final result.

Evaluating functions this way allows you to explore specific values, crucial for analyzing behavior at certain points and for solving equations.

Simplifying Expressions

Simplifying a polynomial expression involves performing arithmetic operations to reduce the expression to its most basic form. This makes it easier to understand and work with.
In our exercise, after substituting \( \sqrt{3} \), the expression was initially \( (\sqrt{3})^4 - 2(\sqrt{3})^2 + \sqrt{3} \). It simplifies as follows:

• First, compute powers: \( (\sqrt{3})^4 = 9 \) and \( (\sqrt{3})^2 = 3 \).
• Next, incorporate those results: \( 9 - 2 \cdot 3 \) simplifies further by performing the multiplication.
• The final simplification yields \( 3 + \sqrt{3} \).

Practicing simplification strengthens your algebra skills, making it easier to handle complex mathematical expressions.