Problem 127 - Introductory Algebra for College Students

Question

Factor each polynomial. $$(x+3)^{2}-2(x+3)+1$$

Step-by-Step Solution

Answer

The factored form of the polynomial \((x+3)^{2} - 2(x+3)+1\) is \((x+2)^{2}\).

Step 1: Identify the constituents of the polynomial

Looking at the polynomial, we have two perfect squares \((x+3)^{2}\) and \(1\), and a middle term \(-2(x+3)\). With the recognition of these constituents, we can identify them with \(a^{2}\), \(-2ab\) and \(b^{2}\) of the perfect square trinomial formula. Thus, we can say \(a = x+3\) and \(b = 1\).

Step 2: Verify the polynomial is a perfect square trinomial

We have to ensure that \((x+3)^{2}\) equates to \(a^{2}\), \(-2ab\) equates to \(-2(x+3)\) and \(b^{2}\) aligns with \(1\). On substitution of a and b, it becomes evident that the polynomial is indeed a perfect square trinomial. Since \(a^{2} = (x+3)^{2}\), \(-2ab = -2(x+3).1 = -2(x+3)\) and \(b^{2} = 1^{2} = 1\).

Step 3: Factor the polynomial

As we have now established that the polynomial is a perfect square trinomial, we simply factor it using the perfect square trinomial factoring formula \((a-b)^{2}\), where \(a = x+3\) and \(b = 1\). So the factored form of the perfect square trinomial \((a-b)^{2}\) becomes \((x+3-1)^{2}\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of algebraic expression that can be expressed as the square of a binomial.
The standard form of a perfect square trinomial is given by the expression \(a^2 \pm 2ab + b^2\), which can be factored into \((a \pm b)^2\).
Let’s explore the components of this formula:

• \(a^2\) is the square of the first term in the binomial.
• \(b^2\) is the square of the second term.
• \(2ab\) is twice the product of the first term and the second term.

This pattern should be clearly recognized in the polynomial to facilitate easy factoring.For example, in the exercise provided, the expression \((x+3)^2 - 2(x+3) + 1\) is a perfect square trinomial. Here, we defined \(a\) as \(x+3\) and \(b\) as \(1\), which neatly fits the pattern.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition or multiplication) arranged together to represent mathematical situations.
These expressions are foundational in algebra, used to describe relationships and solve problems. They consist of:

• **Terms**: Parts of the expression separated by plus or minus signs, which may include variables, coefficients, or constants.
• **Variables**: Symbols used to represent unknown values, such as \(x\) or \(y\).
• **Coefficients**: Numbers multiplying a variable, exemplified by the '3' in \(3x\).
• **Constants**: Stand-alone numbers without accompanying variables.

Algebraic expressions can often be simplified or factored. Recognizing patterns, such as perfect square trinomials, allows for transformation into more manageable forms.

Introductory Algebra

Introductory algebra lays the groundwork for understanding and manipulating algebraic expressions.
This level of algebra involves learning basic concepts and operations that are crucial for solving equations and understanding mathematical relationships. Key concepts include:

• **Operations**: Addition, subtraction, multiplication, and division used to combine numbers and variables.
• **Equations**: Mathematical statements that express equality, typically solved to find the value of unknowns.
• **Simplification**: The process of reducing expressions to simpler forms, which makes them easier to work with.
• **Factoring**: A method used to rewrite expressions as the product of simpler expressions.

An essential part of introductory algebra is spotting special patterns, like perfect square trinomials. Understanding these foundational concepts makes more advanced topics in algebra much more approachable.