Rational Zero Theorem
Another use for the remainder theorem is to test whether a rational number is a zero for a given polynomial. But first we need a pool of rational numbers to test. The rational zero theorem helps us narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomial.
Consider a quadratic function with two zeros, [latex]x = \frac{2}{5}[/latex] and [latex]x = \frac{3}{4}[/latex]. By the factor theorem, these zeros have factors associated with them. Let us set each factor equal to [latex]0[/latex], and then construct the original quadratic function absent its stretching factor.
[latex]\begin{array}{l l} x - \frac{2}{5} = 0 \text{ or } x - \frac{3}{4} = 0 & \text{Set each factor equal to 0.} \\ 5x - 2 = 0 \text{ or } 4x - 3 = 0 & \text{Multiply both sides of the equation to eliminate fractions.} \\ f(x) = (5x - 2)(4x - 3) & \text{Create the quadratic function, multiplying the factors.} \\ f(x) = 20x^2 - 23x + 6 & \text{Expand the polynomial.} \\ f(x) = (5 \cdot 4)x^2 - 23x + (2 \cdot 3) & \\ \end{array}[/latex]
Notice that two of the factors of the constant term, [latex]6[/latex], are the two numerators from the original rational roots: [latex]2[/latex] and [latex]3[/latex]. Similarly, two of the factors from the leading coefficient, [latex]20[/latex], are the two denominators from the original rational roots: [latex]5[/latex] and [latex]4[/latex].
We can infer that the numerators of the rational roots will always be factors of the constant term and the denominators will be factors of the leading coefficient. This is the essence of the rational zero theorem; it is a means to give us a pool of possible rational zeros.
the rational zero theorem
The Rational Zero Theorem states that, if the polynomial [latex]f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0[/latex] has integer coefficients, then every rational zero of [latex]f(x)[/latex] has the form [latex]\frac{p}{q}[/latex] where [latex]p[/latex] is a factor of the constant term [latex]a_0[/latex] and [latex]q[/latex] is a factor of the leading coefficient [latex]a_n[/latex].
When the leading coefficient is [latex]1[/latex], the possible rational zeros are the factors of the constant term.
- Determine all factors of the constant term and all factors of the leading coefficient.
- Determine all possible values of [latex]\frac{p}{q}[/latex], where [latex]p[/latex] is a factor of the constant term and [latex]q[/latex] is a factor of the leading coefficient. Be sure to include both positive and negative candidates.
- Determine which possible zeros are actual zeros by evaluating each case of [latex]f\left(\frac{p}{q}\right)[/latex].