In Exercises 49–56, factor each perfect square trinomial. 4x^2+4x+1

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1. Identify the structure of the trinomial. A perfect square trinomial takes the form \((a^2 + 2ab + b^2)\), which factors into \((a + b)^2\). Compare the given trinomial \(4x^2 + 4x + 1\) to this form.
2. Recognize that \(4x^2\) is a perfect square because \(4x^2 = (2x)^2\). Similarly, \(1\) is a perfect square because \(1 = 1^2\). This suggests the trinomial might be a perfect square trinomial.
3. Check the middle term \(4x\). The middle term in a perfect square trinomial is \(2ab\), where \(a\) and \(b\) are the square roots of the first and last terms. Here, \(a = 2x\) and \(b = 1\), so \(2ab = 2(2x)(1) = 4x\), which matches the middle term.
4. Since the trinomial matches the form \(a^2 + 2ab + b^2\), factor it as \((a + b)^2\). Substitute \(a = 2x\) and \(b = 1\) to get \((2x + 1)^2\).
5. Write the final factored form of the trinomial as \((2x + 1)^2\). This is the simplified expression for the given perfect square trinomial.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be expressed as the square of a binomial. It takes the form a^2 + 2ab + b^2, which factors to (a + b)^2. Recognizing this pattern is essential for factoring such expressions efficiently.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In the case of perfect square trinomials, this involves identifying the binomial that, when squared, produces the trinomial.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two, typically in the form ax^2 + bx + c. Understanding their structure is crucial for factoring, as it allows one to identify coefficients and apply relevant factoring techniques, such as recognizing perfect squares.