Problem 23 - 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? $$ \mathrm{f}(x)=x^{4}+x^{2}+x+1 \text { and } a=0 $$

Step-by-Step Solution

Answer

f(0) = 1; hence, 0 is not a root of the function.

Step 1: Substitute $a$ into the Function

To find \( \mathrm{f}(a) \), substitute \( a = 0 \) into the function \( \mathrm{f}(x) = x^{4} + x^{2} + x + 1 \). So we calculate \( \mathrm{f}(0) \).

Step 2: Evaluate the Function

Substitute \( x = 0 \) into the function: \[ \mathrm{f}(0) = 0^{4} + 0^{2} + 0 + 1 \]. Simplify it to find \( \mathrm{f}(0) = 1 \).

Step 3: Determine if $a$ is a Root

A value \( a \) is a root of the function if \( \mathrm{f}(a) = 0 \). Since \( \mathrm{f}(0) = 1 \), \( a = 0 \) is not a root of the function.

Key Concepts

Roots of Polynomial

The roots of a polynomial function are the values of the variable that make the entire expression equal to zero. In simpler terms, they are the input values that, when plugged into the function, yield an output of zero. These values are significant because they represent the points where the graph of the function intersects the x-axis. For polynomial equations like

• quadratic (degree 2) polynomials, there can be up to two roots.
• cubic (degree 3) polynomials, there can be up to three roots.
• quartic (degree 4) polynomials, like the one in the exercise, there can be up to four roots.

To find a root, you can use various techniques such as factoring, synthetic division, or applying the quadratic formula for lower degree polynomials. However, for higher degree polynomials, numerical methods or graphing might be required to find approximate roots. If you find that substituting any specific value of the variable into the polynomial results in a zero output, then this particular value is indeed a root of the polynomial.

Function Evaluation

Evaluating a function means finding the output value for a particular input value. This involves substituting the given input into the function and performing the necessary arithmetic operations. In the context of the example in the exercise,

• the polynomial function is given as: \( \mathrm{f}(x) = x^{4} + x^{2} + x + 1 \).
• We need to evaluate this function at \( a = 0 \). This means substituting 0 wherever there is an \( x \) in the expression.

So, when you substitute \( x = 0 \), you calculate \[ \mathrm{f}(0) = 0^{4} + 0^{2} + 0 + 1 \], which simplifies to 1. Thus, the function value at zero is 1, indicating that the point (0,1) lies on the graph of this polynomial. Function evaluation is a fundamental part of mathematics, as it helps in analyzing and understanding the behavior of functions at specific points.

Zero of a Function

The zero of a function is another term for the root of the function. It refers to the input value that makes the function output zero. If you find that substituting a value for \( x \) into the function yields a zero, then that value is called a "zero" of the function. In the exercise, we are checking whether \( a = 0 \) is a zero of the function \( \mathrm{f}(x) = x^{4} + x^{2} + x + 1 \). Upon substituting, we find that \[ \mathrm{f}(0) = 1 \]. Since the output is not zero, \( a = 0 \) is not a zero of the function. Zeros of a function are crucial in finding solutions to equations and understanding the intersection points of the function with the horizontal axis on the graph. Determining zeros helps in solving polynomial equations and is a central concept in algebra and calculus.