Definition
To factor a polynomial completely means to write it as a product of polynomials of lower degree, such that none of the factors can be factored further using real numbers (or complex numbers, if specified). This process involves finding common factors, applying factoring formulas (like difference of squares, trinomials, or grouping), and using the Rational Root Theorem when necessary.
Worked Example
Factor the polynomial completely:
$$ f(x) = x^3 - 6x^2 + 11x - 6 $$
Step 1: Look for Rational Roots
Possible rational roots (by Rational Root Theorem): $pm1, pm2, pm3, pm6$
Test $x = 1$:
$$ f(1) = 1 - 6 + 11 - 6 = 0 $$
So, $x = 1$ is a root. Therefore, $(x - 1)$ is a factor.
Step 2: Divide by $(x - 1)$
Use synthetic or long division:
Divide $x^3 - 6x^2 + 11x - 6$ by $(x - 1)$:
Result: $x^2 - 5x + 6$
Step 3: Factor the Quadratic
$$ x^2 - 5x + 6 = (x - 2)(x - 3) $$
Step 4: Write the Complete Factorization
$$ x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3) $$
Takeaways
- Factoring polynomials involves finding roots and applying factoring techniques until all factors are irreducible.
- Always check for common factors and use formulas like difference of squares or quadratic factoring.
- The Rational Root Theorem helps identify possible rational roots for higher-degree polynomials.