How do you factor a perfect square trinomial? This is a common question in algebra, as understanding how to factor these expressions is essential for solving various algebraic problems. A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. In this article, we will explore the steps and techniques to factor a perfect square trinomial effectively.
A perfect square trinomial is of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. The key characteristic of a perfect square trinomial is that it can be expressed as \((dx + e)^2\), where \(d\) and \(e\) are also constants. By recognizing this pattern, we can factor the trinomial more easily.
To factor a perfect square trinomial, follow these steps:
1. Identify the first term: The first term of the trinomial, \(ax^2\), represents the square of the binomial’s first term. In this case, the first term is \(x^2\), which suggests that the binomial’s first term is \(x\).
2. Identify the last term: The last term of the trinomial, \(c\), represents the square of the binomial’s second term. Since the square of a real number is always positive, the last term must be a perfect square. Determine the square root of the last term, which will be the second term of the binomial.
3. Determine the middle term: The middle term, \(bx\), is twice the product of the binomial’s first and second terms. To find the second term of the binomial, multiply the first term by the square root of the last term. Then, multiply this product by 2 to get the middle term.
4. Write the factored form: Once you have identified the binomial’s first and second terms, write the trinomial as the square of the binomial: \((dx + e)^2\).
Let’s apply these steps to an example:
Example: Factor the perfect square trinomial \(x^2 + 6x + 9\).
1. Identify the first term: The first term is \(x^2\), which suggests that the binomial’s first term is \(x\).
2. Identify the last term: The last term is \(9\), which is the square of \(3\). Thus, the second term of the binomial is \(3\).
3. Determine the middle term: Multiply the first term (\(x\)) by the square root of the last term (\(3\)), which gives \(3x\). Multiply this product by 2 to get the middle term, \(6x\).
4. Write the factored form: The trinomial can be factored as \((x + 3)^2\).
In conclusion, factoring a perfect square trinomial involves identifying the binomial’s first and second terms and then writing the trinomial as the square of the binomial. By following these steps, you can effectively factor any perfect square trinomial and solve various algebraic problems.
