Problem 51 - Introductory Algebra for College Students

Question

Factor any perfect square trinomials, or state that the polynomial is prime. $$x^{2}+22 x+121$$

Step-by-Step Solution

Answer

The factored form of the trinomial given is \((x + 11)^{2}\).

Step 1: Identify 'a' and 'b'

Recognize the trinomial in its standard form \(a^{2} + 2ab + b^{2}\). Here, \(a^{2} = x^{2}\) therefore \(a = x\). And \(b^{2} = 121\) therefore, \(b = 11\) as this is a positive square root.

Step 2: Check for perfect square

Verify if the original expression follows the form \(a^{2} + 2ab + b^{2}\). Substituting value of 'a' as 'x' and 'b' as '11' into the formula, we obtain \(x^{2}+ 2*x*11+ 121\), which simplifies to \(x^{2}+ 22x+ 121\). This matches the original expression, verifying that our trinomial is indeed a perfect square.

Step 3: Factor the trinomial

The perfect square trinomial can be factored as \((a + b)^{2}\). Substituting 'x' for 'a' and '11' for 'b', we get \((x + 11)^{2}\).

Key Concepts

Factoring Trinomials

Factoring trinomials is a process in algebra that involves rewriting a quadratic expression as a product of two simpler expressions. This technique is crucial because it simplifies complex equations and makes solving them easier. A trinomial is a type of polynomial with three terms.When you face a trinomial like \(x^2 + 22x + 121\), one approach is to determine if it's a 'perfect square trinomial'. This special form of trinomial can be factored into the square of a binomial:

• The expression must fit the pattern \(a^2 + 2ab + b^2\).
• This means it factors into the form \((a + b)^2\).

For our example, identifying 'a' and 'b' helps us factor the trinomial. If the trinomial can't be matched to a specific pattern, it may still be factored using other techniques, such as finding a pair of numbers that multiply to give the third term while adding to give the second term.

Polynomial Factorization

Polynomial factorization is a core concept in algebra. It involves breaking down a complex polynomial into simpler factors, much like turning a large number into a product of its prime factors.For a trinomial like \(x^2 + 22x + 121\), polynomial factorization identifies that it can be expressed as \((x + 11)^2\). This process involves:

• Checking if the polynomial is a perfect square trinomial, as shown in the original exercise.
• Recognizing patterns that allow the expression to be rewritten as a squared binomial.

In some cases, you may also have to use the distributive property backward (also known as the reverse FOIL method) to simplify expressions. This makes solving polynomial equations and sketching graphs more manageable.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. These expressions are fundamental in mathematics because they represent real-world situations with general symbols that can be manipulated and solved.Trinomials and polynomials are types of algebraic expressions. Specifically, trinomials like \(x^2 + 22x + 121\) have three distinct terms. When dealing with algebraic expressions, it’s essential to understand operations, such as:

• Adding and subtracting like terms to simplify the expression.
• Multiplying binomials, where practice with FOIL and special products can aid understanding.
• Factoring, which involves rewriting the expression in a way that reveals solutions to equations and zeros of functions.

Handling algebraic expressions effectively forms the basis for further study in algebra, such as solving equations and inequalities.