How to Factor a Trinomial
A trinomial is a polynomial with three terms. Trinomials often arise in algebra, geometry, and other areas of mathematics. They can be factored into products of two binomials, which can be useful for solving equations, simplifying expressions, and understanding the structure of polynomials.
To factor a trinomial in standard American English, follow these steps:
- Identify the coefficients of the trinomial. The coefficients are the numbers that are multiplied by the variables.
- Find two numbers that add up to the coefficient of the middle term and multiply to the constant term.
- Rewrite the middle term as a sum of two terms using the two numbers you found in step 2.
- Factor the first two terms and the last two terms separately.
- Combine the factored expressions to get the factored trinomial.
For example, to factor the trinomial $x^2 + 5x + 6$, we first identify the coefficients: $1, 5,$ and $6$. We then find two numbers that add up to $5$ and multiply to $6$. These numbers are $2$ and $3$. We rewrite the middle term as $2x + 3x$ and factor the first two terms and the last two terms separately:
$$x^2 + 5x + 6 = x^2 + 2x + 3x + 6 = (x + 2)(x + 3)$$
Therefore, the factored trinomial is $(x + 2)(x + 3)$.
Special Cases
There are a few special cases that can arise when factoring trinomials.
- If the trinomial is a perfect square trinomial, it can be factored as the square of a binomial. For example, the trinomial $x^2 + 6x + 9$ is a perfect square trinomial and can be factored as $(x + 3)^2$.
- If the trinomial has a negative constant term, it can be factored as the negative of a trinomial with a positive constant term. For example, the trinomial $x^2 + 5x – 6$ can be factored as $-(x^2 + 5x + 6)$.
- If the trinomial has a coefficient of 1 for the first term, it can be factored using the difference of squares formula. For example, the trinomial $x^2 – 9$ can be factored as $(x + 3)(x – 3)$.
Frequently Asked Questions
Q: What is a trinomial?
A: A trinomial is a polynomial with three terms.
Q: How do I identify the coefficients of a trinomial?
A: The coefficients are the numbers that are multiplied by the variables.
Q: How do I find two numbers that add up to the coefficient of the middle term and multiply to the constant term?
A: You can use trial and error or use the factoring formula: $ac = bd$, where $a$ is the coefficient of the first term, $b$ is the coefficient of the middle term, $c$ is the constant term, and $d$ is the number you are looking for.
Q: How do I rewrite the middle term as a sum of two terms?
A: You can use the distributive property to rewrite the middle term as a sum of two terms. For example, the middle term $5x$ can be rewritten as $2x + 3x$.
Q: How do I factor the first two terms and the last two terms separately?
A: You can use the greatest common factor to factor the first two terms and the last two terms separately. For example, the first two terms $x^2 + 2x$ can be factored as $x(x + 2)$ and the last two terms $3x + 6$ can be factored as $3(x + 2)$.
Q: How do I combine the factored expressions to get the factored trinomial?
A: You can multiply the factored expressions together to get the factored trinomial. For example, the factored expressions $x(x + 2)$ and $3(x + 2)$ can be multiplied together to get $(x + 2)(x + 3)$.
