How to factor a cubic trinomial?
Understand the Problem
The question is asking for the steps or methods to factor a cubic trinomial, which typically involves finding the roots of the polynomial or using specific algebraic techniques to simplify it. We can approach this problem by looking for a common factor or using techniques like synthetic division or grouping.
Answer
The cubic trinomial factors to $(x - 1)(x - 2)(x - 3)$.
Answer for screen readers
The cubic trinomial $x^3 - 6x^2 + 11x - 6$ factors to $(x - 1)(x - 2)(x - 3)$.
Steps to Solve
- Identify the cubic trinomial
Determine the form of the cubic trinomial, which typically looks like $ax^3 + bx^2 + cx + d$. Let's say we have a specific trinomial such as $x^3 - 6x^2 + 11x - 6$.
- Look for common factors
Check if there are any common factors among the terms. In our example, there are no common factors to factor out.
- Use the Rational Root Theorem
List possible rational roots which are the factors of the constant term divided by the factors of the leading coefficient. For $x^3 - 6x^2 + 11x - 6$, possible rational roots are $\pm 1, \pm 2, \pm 3, \pm 6$.
- Test possible roots
Substitute the possible roots into the polynomial to find actual roots. For instance, testing $x = 1$: $$ 1^3 - 6(1^2) + 11(1) - 6 = 0 $$ This shows $x = 1$ is a root.
- Use synthetic division
Divide the cubic trinomial by the factor $(x - 1)$ since $x = 1$ is a root. Set up synthetic division:
1 | 1 -6 11 -6
| 1 -5 6
-----------------
1 -5 6 0
This results in $x^2 - 5x + 6$.
- Factor the resulting quadratic
Now we can factor $x^2 - 5x + 6$. This factors to $(x - 2)(x - 3)$.
- Combine the factors
Therefore, the cubic trinomial $x^3 - 6x^2 + 11x - 6$ can be factored as: $$ (x - 1)(x - 2)(x - 3) $$
The cubic trinomial $x^3 - 6x^2 + 11x - 6$ factors to $(x - 1)(x - 2)(x - 3)$.
More Information
Factoring cubic trinomials often involves finding one root first, then breaking down the remaining quadratic polynomial. The Rational Root Theorem can be particularly helpful in identifying potential roots.
Tips
- Confusing the terms and incorrectly identifying the factors of the leading coefficient and the constant term.
- Failing to confirm that a candidate root is indeed a root by not substituting it back into the original polynomial.
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