Problem 53 - Algebra 2

Question

The graph of the polynomial function \(f(x)=a x(x-4)(x+1)\) goes through thepoint at \((5,15) .\) For what value(s) of \(x\) will \(f(x)=0 ?\)

Step-by-Step Solution

Answer

The solutions for \( f(x)=0 \) are \( x = 0, x = 4, \text{ and } x = -1 \).

Step 1: Understand the Problem

The polynomial function given is in factored form: \( f(x) = a(x)(x-4)(x+1) \). The problem asks for the values of \(x\) where \( f(x) = 0 \). Values of \(x\) where a polynomial equals zero are called the roots or solutions of the polynomial.

Step 2: Identify Roots from Factors

In a factored polynomial such as \( f(x) = a(x)(x-4)(x+1) \), each factor set equal to zero gives a root: \(x = 0\), \(x - 4 = 0\), and \(x + 1 = 0\). Solve each: \( x = 0 \), \( x = 4 \), \( x = -1 \). These are the values of \( x \) for which \( f(x) = 0 \).

Step 3: Verification Using Point (Optional for Understanding)

Given that the graph passes through \((5,15)\), the value of \(a\) can be determined for additional verification if needed: \(15 = a(5)(5-4)(5+1)\), leading to \(15 = 30a\), so \(a = \frac{1}{2}\). This confirms \( f(x) = \frac{1}{2}(x)(x-4)(x+1) \), still with the same roots \( x = 0, 4, -1 \). Spot checking doesn't change the solution roots but confirms consistency if needed.

Key Concepts

Factored Form

The factored form of a polynomial provides a way to write a polynomial as the product of its factors. This form is very handy in understanding the key features of a polynomial function, especially when solving for roots or zeros. In factored form, a polynomial is expressed as a multiplication of its linear factors, like in the example: \(f(x) = a(x)(x-4)(x+1)\).

This makes it easier to find roots because each linear factor set to zero can quickly reveal the values of \(x\) that make the entire function zero. It's simpler and more direct than other forms. Essentially, when a polynomial is in factored form, you can see the roots or solutions just by looking at the individual factors.

Utilizing factored form can help you quickly determine crucial characteristics of polynomial functions, which is useful for both solving equations and graphing.

Roots and Zeros

Roots, often referred to as zeros or solutions of a polynomial function, are the values of \(x\) where the function equals zero. They are also the points where the graph of the polynomial crosses or touches the x-axis.

In the given example \(f(x) = a(x)(x-4)(x+1)\), setting each factor equal to zero reveals the roots: \(x = 0\), \(x = 4\), and \(x = -1\). These roots indicate where the function intersects the x-axis. Generally, a polynomial will have as many roots as its degree, counting multiplicities, which are instances where the graph touches but may not cross the axis, indicating repeated roots.

Understanding roots and their corresponding factors in a polynomial function is fundamental for solving equations and understanding the shape and behavior of the graph.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to positive integer powers, combined with coefficients. A typical polynomial function looks like: \(f(x) = ax^n + bx^{n-1} + \, ... \, + k\).

The degree of a polynomial is the highest power of the variable present. This degree often gives insight into the number of roots the polynomial will have and the general shape of its graph. For example, a quadratic polynomial (degree 2) has a maximum of two roots. A cubic polynomial (degree 3) might have up to three roots.

Polynomials can be manipulated into various forms such as standard, factored, or vertex forms. Each form provides different insights or conveniences, whether you are solving for roots, factoring, or graphing. Understanding these functions is key for algebra and calculus, as they are foundational to many cases of mathematical problem-solving.