Problem 28 - Elementary and Intermediate Algebra

Question

Factor. $$ 121+4 x^{2}+44 x $$

Step-by-Step Solution

Answer

The expression factors into \((2x + 11)^2\).

Step 1: Rearrange the Terms

First, let's rearrange the terms in the expression in decreasing order of the powers of the variable. The given expression is: \(4x^2 + 44x + 121\).

Step 2: Recognize the Expression as a Perfect Square Trinomial

We identify that the expression is a perfect square trinomial of the form \((ax + b)^2 = a^2x^2 + 2abx + b^2\). Look for numbers \(a\) and \(b\) such that: - \(a^2 = 4\), so \(a = 2\).- \(b^2 = 121\), so \(b = 11\).- \(2ab = 44\), which checks out since \(2 \times 2 \times 11 = 44\).

Step 3: Write the Expression as a Perfect Square

Based on the findings from Step 2, we can write the expression \(4x^2 + 44x + 121\) as a perfect square trinomial: \((2x + 11)^2\).

Step 4: Verify the Factorization

Expand \((2x + 11)^2\) to verify the factorization:- \((2x + 11)(2x + 11) = 4x^2 + 22x + 22x + 121 = 4x^2 + 44x + 121\).- The expansion matches the original expression, confirming the factorization is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial expression that can be written as the square of a binomial. This means that if you have a trinomial (an expression with three terms), it can be expressed in the form

• \((ax + b)^2 = a^2x^2 + 2abx + b^2\)

This form is characterized by having three specific components:

• The first term is a perfect square, \(a^2x^2\).
• The last term is a perfect square, \(b^2\).
• The middle term is twice the product of the square roots of the first and last terms, \(2abx\).

To identify a perfect square trinomial, you'll check if these conditions are met by your expression. Recognizing these patterns helps simplify problems by allowing you to factor the trinomial more easily.

Polynomial Expressions

Polynomial expressions are mathematical phrases involving a sum of powers of a variable or variables. Each term in a polynomial consists of:

• A coefficient (a constant multiplier of the variables in the term).
• Variables raised to whole number powers.

A typical polynomial expression looks like \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where each \(a\) is a coefficient. The term with the highest power of \(x\) defines the degree of the polynomial.
Polynomials can include various types, such as monomials (single term), binomials (two terms), and trinomials (three terms). Each operates under the same arithmetic rules. Recognizing different Polynomial types helps in applying the right factoring or simplification techniques.

Algebraic Factorization

Algebraic factorization is the process of breaking down a complex expression into a product of simpler factors. The goal is to express a polynomial as a multiplication of polynomials of lesser degrees, possibly as linear binomials.
This technique leverages patterns (like those found in perfect square trinomials) to simplify expressions. It involves looking for common factors in terms or using identities to convert a sum (or difference) into a product.

• For example, recognizing a perfect square trinomial helps in writing it as \[(ax + b)(ax + b)\].
• Checking expressions for common factors can similarly lead to simplification.

Factorization is a valuable tool in algebra as it aids in solving equations, simplifying expressions, and understanding mathematics better. Comprehending these principles enables effective manipulation and solving of higher-level problems.