Question

Factor the perfect squares. $$ 4 x^{2}+4 x+1 $$

Short answer

The factored form is (2x + 1)^2 .

Step-by-step solution

Identify the Perfect Square Trinomial

Recognize that the expression given, 4x^{2} + 4x + 1, is a perfect square trinomial. A perfect square trinomial takes the form a^2 + 2ab + b^2 .

Rewrite the Expression as a Perfect Square

Confirm that the expression can be written in the form ( ax + b)^2 . Compare the given equation with (2x + 1)^2 , where a = 2x and b = 1 .

Factor the Expression

Now that the expression is identified as a perfect square trinomial, factor it. The factored form is (2x + 1)^2 .

Key concepts

perfect square trinomial

In algebra, a perfect square trinomial is a special type of polynomial. It's called 'perfect square' because it can be expressed as the square of a binomial. Generally, it has the form \(a^2 + 2ab + b^2\). This means that when you expand the binomial \( (a + b)^2 \), you will get a perfect square trinomial.

For example, let’s consider \[4x^2 + 4x + 1 \]. We can see this matches the form \(a^2 + 2ab + b^2\), where \(a = 2x\) and \(b = 1\).

Identifying perfect square trinomials is useful because it simplifies factoring. It reduces the problem to recognizing patterns.

factoring trinomials

Factoring trinomials is the process of breaking down a trinomial into the product of two binomials. Let's take a closer look using our example: \[4x^2 + 4x + 1\].

We need to identify if the given trinomial is a perfect square trinomial. We've already confirmed that \( 4x^2 + 4x + 1 \) matches \((a + b)^2\). Now, we can rewrite it as \((2x + 1)^2\).

This means that the expression \[4x^2 + 4x + 1\] can be factored as \( (2x + 1)(2x + 1) \), or simply \((2x + 1)^2\).

This is the final factored form. Factoring makes dealing with complex expressions much easier. It helps simplify polynomials and solve equations efficiently.

algebraic expressions

An algebraic expression is a combination of numbers, variables, and operators (like +, -, *, and /). For example, in the expression \[4x^2 + 4x + 1\], we have:

\cdot 4x^2 \text{ where } 4 \text{ is a coefficient and } x \text{ is a variable squared}
\cdot 4x \text{ where } 4 \text{ is a coefficient and } x \text{ is a variable}
\cdot \text{1 which is a constant term}

Expressions can be simplified by combining like terms and using operations that respect the properties of algebra.

Factoring is a key technique in simplifying and solving equations. For example, recognizing \((2x + 1)^2\) as the factored form of \[4x^2 + 4x + 1\] can make solving algebraic equations much simpler. Remember, recognizing the structure of expressions is crucial to mastering algebra.