What are the steps in factoring perfect square trinomial?
Factoring perfect square trinomials is a fundamental skill in algebra that helps simplify quadratic expressions and solve equations. A perfect square trinomial is a quadratic expression that can be expressed as the square of a binomial. This type of trinomial follows a specific pattern, making it easier to factorize compared to other quadratic expressions. In this article, we will discuss the steps involved in factoring perfect square trinomials.
Step 1: Identify the pattern
The first step in factoring a perfect square trinomial is to identify the pattern. A perfect square trinomial always has the form:
ax^2 + bx + c
where a, b, and c are constants, and a is not equal to 0. The pattern for a perfect square trinomial is:
(a x)^2 + 2 a x b + b^2
If the trinomial matches this pattern, it can be factored as a perfect square.
Step 2: Find the square root of the first term
The next step is to find the square root of the first term, which is ax^2. The square root of a^2 is simply a, so we have:
√(ax^2) = √(a^2 x^2) = a √x
Step 3: Find the square root of the last term
Next, find the square root of the last term, which is c. Since the last term is a perfect square, its square root will be a constant:
√c = c^(1/2)
Step 4: Write the factored form
Now, write the factored form of the perfect square trinomial using the square roots found in steps 2 and 3. The factored form will be:
(a √x – c^(1/2))(a √x + c^(1/2))
Step 5: Simplify the factored form
Finally, simplify the factored form by combining like terms, if necessary. In most cases, the factored form will be in its simplest form.
To summarize, the steps in factoring a perfect square trinomial are:
1. Identify the pattern.
2. Find the square root of the first term.
3. Find the square root of the last term.
4. Write the factored form.
5. Simplify the factored form.
By following these steps, you can factorize perfect square trinomials efficiently and accurately. This skill is essential for solving quadratic equations and simplifying quadratic expressions in various algebraic problems.
