Question

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

Short answer

(x+1)^2

Step-by-step solution

Identify the Perfect Square Form

Recognize that the expression is in the form of a perfect square. A perfect square trinomial has the form a^2 + 2ab + b^2.

Compare with Standard Form

Compare the given expression x^2 + 2x + 1 with (a + b)^2 = a^2 + 2ab + b^2. Identify a and b in the expression.

Find a and b

From x^2 + 2x + 1, identify the value of a as x and the value of b as 1 since 2ab equals 2x.

Write the Factored Form

Substitute a = x and b = 1 into the factored form (a + b)^2. The factored form of x^{2}+2 x+1 is (x+1)^2.

Key concepts

perfect square trinomial

A perfect square trinomial is a special type of polynomial. It takes the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). This trinomial can be factored into a single binomial squared. For example, \(x^2 + 2x + 1\) fits the form \(a^2 + 2ab + b^2\). Let's break it down:
1. The first term \(x^2\) is \(a^2\). Here, \(a = x\).
2. The second term \(2x\) is \(2ab\). Given that \(a = x\), we identify \(b = 1\).
3. The third term \(1\) is \(b^2\), which matches because \(1 = 1^2\).
Once we identify \(a\) and \(b\), the trinomial \(x^2 + 2x + 1\) can be factored to \((x + 1)^2\).
The opposite also works: \(x^2 - 2x + 1\) can be factored to \((x - 1)^2\).
This is useful in simplifying algebraic expressions and solving quadratic equations.

factored form

Factored form is when you express a polynomial as a product of its factors. For a perfect square trinomial, it means writing it as a binomial squared.
Taking our example again:
The trinomial \(x^2 + 2x + 1\) can be expressed in its factored form \((x + 1)^2\).
Here's how you identify the terms for factoring:

• Match the pattern \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\).
• Determine values for \(a\) and \(b\).
• Write the factor as \((a + b)^2\) or \((a - b)^2\).

Factored forms make it easier to solve quadratic equations, simplify expressions, and understand the underlying patterns.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They can represent real-world scenarios and help us solve problems. Examples include monomials, binomials, and polynomials.
A perfect square trinomial is a type of polynomial. When we factor an algebraic expression like \(x^2 + 2x + 1\), we break it down into simpler parts.
Let's decompose it:

• The variable is \(x\).
• The coefficients are the numbers multiplying the variables (e.g., 2 in \(2x\)).
• Constants are the standalone numbers (e.g., 1).

By understanding the structure of algebraic expressions, we can more easily manipulate and solve them. Factoring is a key tool in this process, especially with quadratic equations.

quadratic equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). Solving these equations can involve factoring.
For example, consider:
\(x^2 + 2x + 1 = 0\). This is a quadratic equation in its standard form. To solve for \(x\), we:

• Factor the quadratic equation: \(x^2 + 2x + 1 = (x + 1)^2\).
• Set the factored form equal to zero: \((x + 1)^2 = 0\).
• Solve for \(x\): The solution is \(x = -1\).

Factoring makes solving quadratic equations simpler.
Other methods include using the quadratic formula or completing the square. However, when you recognize a perfect square trinomial, factoring it immediately can save time and effort.