Question

Factor the perfect squares. $$ 9 x^{2}+6 x+1 $$

Short answer

(3x + 1)^2

Step-by-step solution

Identify the perfect square terms

Recognize that the expression is a trinomial in the form of \(a^2 + 2ab + b^2\). In this case, the first and last terms are perfect squares: \(9x^2\) is \((3x)^2\) and \(1\) is \(1^2\).

Recognize the middle term

Check the middle term to ensure it fits the formula \(2ab\). The middle term is \(6x\). Calculate \(2 \times 3x \times 1\) and confirm it equals \(6x\).

Write the factored form

Since the trinomial \(9x^2 + 6x + 1\) corresponds to the format \((3x + 1)^2\), write the factored form as \((3x + 1)^2\).

Key concepts

Trinomial

A trinomial is a polynomial with three terms. In the exercise, we deal with the trinomial equation: 9x^2 + 6x + 1.
This trinomial can be factored by leveraging its perfect square properties.
To quickly identify if a trinomial is in perfect square form, look for:

• The first and last terms being perfect squares—here, 9x^2 and 1.
• The middle term fitting the formula 2ab—6x being equal to 2 * 3x * 1.

If these conditions are met, the trinomial is a perfect square trinomial.

Factored Form

In algebra, the factored form of an expression breaks it down into a product of its factors. For the trinomial 9x^2 + 6x + 1, we first check if it can be represented as (a^2 + 2ab + b^2).
Here:

• a² = (3x)²
• 2ab = 6x (verified as 2 × 3x × 1)
• b² = 1²

If these match, we can write the trinomial in its factored form as: (3x + 1)^2.
Factoring helps simplify and solve higher-degree equations.

Algebra

Algebra involves working with variables and constants to solve equations.
In this exercise, we applied algebraic techniques to factor a perfect square trinomial.
Recognizing patterns like (a^2 + 2ab + b^2) allows us to simplify expressions and solve for unknown variables efficiently.
Mastering techniques such as factoring equips students with tools to tackle more complex algebraic problems.