Problem 54 - Algebra and Trigonometry

Question

Factor the expression completely. $$ x^{2}+10 x+25 $$

Step-by-Step Solution

Answer

The factored form is \((x+5)^2\).

Step 1: Identify the expression

We have the quadratic expression \(x^2 + 10x + 25\) that needs to be factored completely. Our task is to rewrite it as a product of two binomials if possible.

Step 2: Look for Perfect Square Trinomials

Recognize if the expression is a perfect square trinomial. A perfect square trinomial takes the form \((a+b)^2 = a^2 + 2ab + b^2\). Here, compare \(x^2 + 10x + 25\) with \(a^2 + 2ab + b^2\). We identify \(a^2 = x^2\), so \(a = x\). Now, \(b^2 = 25\), so \(b = 5\), and \(2ab = 10x\). The expression matches the form of a perfect square \((x+5)^2\).

Step 3: Write down the factored form

Since we determined that \(x^2 + 10x + 25\) is a perfect square trinomial, it can be factored as \((x+5)(x+5)\). This is sometimes written as \((x+5)^2\) since both factors are identical.

Key Concepts

Perfect Square Trinomials

Perfect square trinomials are special types of quadratic expressions that can make the process of factoring much easier. They take the form \( (a+b)^2 = a^2 + 2ab + b^2 \). When identifying a perfect square trinomial, you look for three terms:

• First term: a square number, \( a^2 \)
• Middle term: twice the product of two numbers, \( 2ab \)
• Last term: another square number, \( b^2 \)

An example can help clarify this. Consider the trinomial \( x^2 + 10x + 25 \). First, you notice the first term is \( x^2 \), which is a perfect square, \( (x)^2 \). The last term is 25, another perfect square, \( (5)^2 \). The middle term is \( 10x \), which fits because it equals \( 2 \times x \times 5 \), or \( 2ab \). This confirms that \( x^2 + 10x + 25 \) is a perfect square trinomial, allowing us to write it as \( (x+5)^2 \). This method simplifies and shortens the process of polynomial factorization when applicable.

Quadratic Expressions

Quadratic expressions are fundamental in algebra and appear in the standard form of \( ax^2 + bx + c \). Recognizing the parts of a quadratic expression is essential:

• The first term, \( ax^2 \), is the quadratic term.
• The middle term, \( bx \), is the linear term.
• The last term, \( c \), is the constant term.

Factoring quadratic expressions involves rewriting them as a product of simpler binomials. This is a crucial skill as it helps in solving quadratic equations and simplifies complex algebraic manipulations. You often start by searching for factors of \( c \) that add up to \( b \) in non-perfect square trinomials. However, when dealing with perfect square trinomials or ones that match specific patterns, the process simplifies greatly, as seen in expressions like \( x^2 + 10x + 25 \), which neatly factors to \( (x+5)^2 \). Understanding the structure of quadratic expressions helps identify the most efficient factoring method.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into the product of smaller, simpler polynomials. This is vital for simplifying expressions and solving equations. The main goal is to rewrite the original polynomial into a form that’s easier to work with.

• For quadratics, start by looking for patterns or special types, like perfect square trinomials or the difference of squares.
• If no clear pattern emerges, attempt trial and error by testing common factor pairs.

For example, the polynomial \( x^2 + 10x + 25 \) can be recognized as a perfect square trinomial. This allows it to be factored into \( (x+5)^2 \), a simpler form. Such strategies highlight the importance of recognizing patterns. In more complex expressions, other techniques such as grouping or synthetic division might be needed. However, perfect square trinomials offer a quick route to factorization when present. Mastery of polynomial factorization involves practice and familiarity with different types of polynomials.