Solve by Rational Root Theorem - Algebra-2

1. Fundamental Concepts

Definition: The Rational Root Theorem states that any rational solution, expressed in lowest terms, of the polynomial equation $$a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0$$, where all coefficients are integers, must be a factor of the constant term $$a_0$$ divided by a factor of the leading coefficient $$a_n$$.
Factors: Factors of the constant term and the leading coefficient are used to list possible rational roots.
Testing Roots: Each possible rational root is tested by substituting it into the polynomial to see if it yields zero.

2. Key Concepts

Rational Root Theorem: If $$\frac{p}{q}$$ is a root, then $$p$$ is a factor of $$a_0$$ and $$q$$ is a factor of $$a_n$$.

Factorization: Once a root is found, the polynomial can be factored using synthetic division or polynomial long division.

Application: Used to find rational roots of polynomials, which can help in solving higher-degree equations.

3. Examples

Example 1 (Basic)

Problem: Find the rational roots of the polynomial $$f(x) = 2x^3 - 5x^2 - 4x + 3$$.

Step-by-Step Solution:

List factors of the constant term $$a_0 = 3$$: $$\pm 1, \pm 3$$.
List factors of the leading coefficient $$a_n = 2$$: $$\pm 1, \pm 2$$.
Possible rational roots: $$\pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2}$$.
Test each possible root:
- $$f(1) = 2(1)^3 - 5(1)^2 - 4(1) + 3 = 2 - 5 - 4 + 3 = -4 \neq 0$$
- $$f(-1) = 2(-1)^3 - 5(-1)^2 - 4(-1) + 3 = -2 - 5 + 4 + 3 = 0$$
- $$f(\frac{1}{2}) = 2(\frac{1}{2})^3 - 5(\frac{1}{2})^2 - 4(\frac{1}{2}) + 3 = \frac{1}{4} - \frac{5}{4} - 2 + 3 = 0$$
- $$f(-\frac{1}{2}) = 2(-\frac{1}{2})^3 - 5(-\frac{1}{2})^2 - 4(-\frac{1}{2}) + 3 = -\frac{1}{4} - \frac{5}{4} + 2 + 3 = 4 \neq 0$$
- $$f(3) = 2(3)^3 - 5(3)^2 - 4(3) + 3 = 54 - 45 - 12 + 3 = 0$$
- $$f(-3) = 2(-3)^3 - 5(-3)^2 - 4(-3) + 3 = -54 - 45 + 12 + 3 = -84 \neq 0$$
- $$f(\frac{3}{2}) = 2(\frac{3}{2})^3 - 5(\frac{3}{2})^2 - 4(\frac{3}{2}) + 3 = \frac{27}{4} - \frac{45}{4} - 6 + 3 = -\frac{27}{4} \neq 0$$
- $$f(-\frac{3}{2}) = 2(-\frac{3}{2})^3 - 5(-\frac{3}{2})^2 - 4(-\frac{3}{2}) + 3 = -\frac{27}{4} - \frac{45}{4} + 6 + 3 = -\frac{45}{4} \neq 0$$
The rational roots are $$-1, \frac{1}{2}, 3$$.

Validation: Substitute the roots back into the polynomial to confirm they yield zero.

Example 2 (Intermediate)

Problem: Find the rational roots of the polynomial $$g(x) = 3x^4 - 2x^3 - 11x^2 + 12x - 4$$.

Step-by-Step Solution:

List factors of the constant term $$a_0 = -4$$: $$\pm 1, \pm 2, \pm 4$$.
List factors of the leading coefficient $$a_n = 3$$: $$\pm 1, \pm 3$$.
Possible rational roots: $$\pm 1, \pm 2, \pm 4, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}$$.
Test each possible root:
- $$g(1) = 3(1)^4 - 2(1)^3 - 11(1)^2 + 12(1) - 4 = 3 - 2 - 11 + 12 - 4 = -2 \neq 0$$
- $$g(-1) = 3(-1)^4 - 2(-1)^3 - 11(-1)^2 + 12(-1) - 4 = 3 + 2 - 11 - 12 - 4 = -22 \neq 0$$
- $$g(2) = 3(2)^4 - 2(2)^3 - 11(2)^2 + 12(2) - 4 = 48 - 16 - 44 + 24 - 4 = 8 \neq 0$$
- $$g(-2) = 3(-2)^4 - 2(-2)^3 - 11(-2)^2 + 12(-2) - 4 = 48 + 16 - 44 - 24 - 4 = -8 \neq 0$$
- $$g(4) = 3(4)^4 - 2(4)^3 - 11(4)^2 + 12(4) - 4 = 768 - 128 - 176 + 48 - 4 = 508 \neq 0$$
- $$g(-4) = 3(-4)^4 - 2(-4)^3 - 11(-4)^2 + 12(-4) - 4 = 768 + 128 - 176 - 48 - 4 = 672 \neq 0$$
- $$g(\frac{1}{3}) = 3(\frac{1}{3})^4 - 2(\frac{1}{3})^3 - 11(\frac{1}{3})^2 + 12(\frac{1}{3}) - 4 = \frac{1}{27} - \frac{2}{27} - \frac{11}{9} + 4 - 4 = -\frac{10}{27} \neq 0$$
- $$g(-\frac{1}{3}) = 3(-\frac{1}{3})^4 - 2(-\frac{1}{3})^3 - 11(-\frac{1}{3})^2 + 12(-\frac{1}{3}) - 4 = \frac{1}{27} + \frac{2}{27} - \frac{11}{9} - 4 - 4 = -\frac{100}{27} \neq 0$$
- $$g(\frac{2}{3}) = 3(\frac{2}{3})^4 - 2(\frac{2}{3})^3 - 11(\frac{2}{3})^2 + 12(\frac{2}{3}) - 4 = \frac{16}{27} - \frac{16}{27} - \frac{44}{9} + 8 - 4 = 0$$
- $$g(-\frac{2}{3}) = 3(-\frac{2}{3})^4 - 2(-\frac{2}{3})^3 - 11(-\frac{2}{3})^2 + 12(-\frac{2}{3}) - 4 = \frac{16}{27} + \frac{16}{27} - \frac{44}{9} - 8 - 4 = -\frac{100}{27} \neq 0$$
- $$g(\frac{4}{3}) = 3(\frac{4}{3})^4 - 2(\frac{4}{3})^3 - 11(\frac{4}{3})^2 + 12(\frac{4}{3}) - 4 = \frac{256}{27} - \frac{128}{27} - \frac{176}{9} + 16 - 4 = 0$$
- $$g(-\frac{4}{3}) = 3(-\frac{4}{3})^4 - 2(-\frac{4}{3})^3 - 11(-\frac{4}{3})^2 + 12(-\frac{4}{3}) - 4 = \frac{256}{27} + \frac{128}{27} - \frac{176}{9} - 16 - 4 = -\frac{100}{27} \neq 0$$
The rational roots are $$\frac{2}{3}, \frac{4}{3}$$.

Validation: Substitute the roots back into the polynomial to confirm they yield zero.

4. Problem-Solving Techniques

List Possible Roots: List all possible rational roots by finding factors of the constant term and the leading coefficient.
Systematic Testing: Test each possible root systematically by substituting it into the polynomial.
Synthetic Division: Use synthetic division to factor the polynomial once a root is found.
Graphical Verification: Use graphing tools to verify the roots visually.