Question

1. the roots of a quadratic equation are 4 and \\(\frac{1}{5}\\). what are the two factors of the function?

Explanation:

Step1: Recall root - factor relationship

For a quadratic equation, if \(r\) is a root of the quadratic function \(f(x)\), then \((x - r)\) is a factor of \(f(x)\). This is based on the factor theorem, which states that a polynomial \(f(x)\) has a factor \((x - r)\) if and only if \(f(r)=0\) (i.e., \(r\) is a root of the polynomial).

Step2: Find the factor for root 4

Given that one root is \(r = 4\). Using the root - factor relationship, the corresponding factor is \((x - 4)\) because when we substitute \(x = 4\) into \((x - 4)\), we get \(4-4=0\).

Step3: Find the factor for root \(\frac{1}{5}\)

Given that the other root is \(r=\frac{1}{5}\). Using the root - factor relationship, the corresponding factor is \((x-\frac{1}{5})\) because when we substitute \(x = \frac{1}{5}\) into \((x-\frac{1}{5})\), we get \(\frac{1}{5}-\frac{1}{5}=0\). We can also rewrite \((x - \frac{1}{5})\) as \((5x - 1)\) by multiplying both the numerator and denominator (or in this case, multiplying the binomial by 5 to clear the fraction: \(x-\frac{1}{5}=\frac{5x - 1}{5}\), and since we are looking for factors of the quadratic function (up to a non - zero constant multiple), both \((x-\frac{1}{5})\) and \((5x - 1)\) are valid factors corresponding to the root \(\frac{1}{5}\). However, the most direct factors from the root - factor relationship are \((x - 4)\) and \((x-\frac{1}{5})\) (or \((5x - 1)\) if we want to eliminate the fraction in the factor).

Answer:

The two factors of the function are \((x - 4)\) and \((x-\frac{1}{5})\) (or \((x - 4)\) and \((5x - 1)\)).