Problem 11 - Algebra A Combined Function

Question

Factor each trinomial completely. $$ x^{2}+22 x+121 $$

Step-by-Step Solution

Answer

The trinomial factors to \((x + 11)^2\).

Step 1: Identify the trinomial form

The given expression \(x^2 + 22x + 121\) is a quadratic trinomial written in the standard form \(ax^2 + bx + c\). Here, \(a = 1\), \(b = 22\), and \(c = 121\).

Step 2: Check for perfect square trinomial

Check if the trinomial is a perfect square trinomial. A perfect square trinomial is of the form \((ax + b)^2 = a^2x^2 + 2abx + b^2\). Here, notice that \(c = 121\) is a perfect square (\(11^2\)), and similarly, the middle term \(22x\) can be expressed as \(2 imes 11x\). This indicates that the expression might be \((x + 11)^2\).

Step 3: Verify the factorization

To verify that \((x + 11)^2\) is the correct factorization, expand it: \((x + 11)(x + 11) = x^2 + 11x + 11x + 121\). Combine like terms: \(x^2 + 22x + 121\). This matches the original trinomial.

Step 4: Write the factorized expression

Since the expanded form matches the original trinomial, we conclude that the correct factorization of \(x^2 + 22x + 121\) is \((x + 11)^2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic trinomial that can be expressed as the square of a binomial. This means you can write it in the form \((ax + b)^2\).
In general, when expanding \((ax + b)^2\), it results in:

• \(a^2x^2\): The square of the first term \((ax)\).
• \(2abx\): Twice the product of the two terms \((2\times ax \times b)\).
• \(b^2\): The square of the second term \((b\).

Identifying a perfect square trinomial is a matter of recognizing these patterns. For example, the trinomial \(x^2 + 22x + 121\) is a perfect square because:

• The constant term, 121, is a perfect square (11 squared).
• The middle term, 22x, equals twice the product of the square root of the first term \(x\) and the square root of the third term, 11 (i.e., \(2 \times x \times 11\)).

Thus, this trinomial can be factored as \((x + 11)^2\). Recognizing these characteristics makes it much easier to factor such trinomials quickly.

Quadratic Trinomial

A quadratic trinomial is a polynomial with three terms, where the highest degree of the variable is 2. It generally takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. When factoring a quadratic trinomial, the goal is to express it as a product of two binomials.
In this particular exercise, the quadratic trinomial is \(x^2 + 22x + 121\). Here, the coefficients are:

• \(a = 1\)
• \(b = 22\)
• \(c = 121\)

Since \(a = 1\), this simplifies the process because we can directly look for two numbers that multiply to \(c\) and add to \(b\). However, recognizing if \(c\) is a perfect square (like 121 in this case), can direct you to check for a perfect square trinomial. Don’t forget that quadratic trinomials can sometimes have more complex factorization methods when \(a\) is not 1 or when it's not a perfect square.

Trinomial Factorization

Trinomial factorization is a method used to rewrite a trinomial in a product form, typically involving binomials. The goal of factoring is to break down a complex expression into simpler factors that, when multiplied, produce the original expression. This is particularly useful in solving equations and simplifying expressions.

For the trinomial \(x^2 + 22x + 121\), recognizing it as a perfect square was essential because it allowed for straightforward factorization into \((x + 11)^2\).
To factor trinomials:

• Examine the signs and terms to determine if it aligns with special cases such as a perfect square.
• Search for two numbers whose product is \(c\) and whose sum is \(b\) when \(a = 1\). If \(a\) is not 1, more complex methods are used, such as the "ac method" or factoring by grouping.
• Once the factors are identified, rewrite the trinomial as a product of two binomials or, if possible, a squared binomial as in the perfect square trinomial.

Understanding these patterns and techniques will help in comprehensively factoring any quadratic trinomials, whether perfect squares or requiring more detailed methods.