Problem 103 - Introductory Algebra for College Students

Question

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) $$(x+1)^{2}-25$$

Step-by-Step Solution

Answer

The factored form of the expression \((x+1)^{2}-25\) is \((x - 4)(x + 6)\).

Step 1: Identify the Difference of Squares Formula

Look for elements in the expression where it looks like a difference of two squares. In this exercise, \((x+1)^{2}-25\) fits the bill because it can be viewed as \((x+1)^{2} - (5)^{2}\) or, in other words, \(a^2 - b^2\). 'a' here is \(x+1\) and 'b' is 5.

Step 2: Apply the Difference of Squares Formula

Rewrite the expression by factoring it according to the difference of squares formula, \( (a - b)(a + b) \). This gives us \((x+1 - 5)(x+1 + 5)\).

Step 3: Simplify the Expression

Now simplify the factored expression to get the final result. This will give us \((x - 4)(x + 6)\).

Key Concepts

Difference of Squares

The difference of squares is a special algebraic pattern that makes factoring simpler. The general form is given as \(a^2 - b^2\).
This can be factored into \((a - b)(a + b)\).
In this method, you split the original expression into two factors, subtracting and adding the same value.

• Think of it as "something squared minus something else squared".
• The "something" in the context of the general formula represents any expression raised to the power of two.

Take, for example, the expression \((x+1)^2 - 25\).
It clearly shows two squares: \((x+1)^2\) and \(5^2\).
This is why you can apply the difference of squares formula to factor it.
Once factored, you get two more simple expressions, \((x - 4)(x + 6)\), making the task of solving or simplifying easier!

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, like \(\frac{p}{q}\), where \(q eq 0\).
They play an important role in factoring because they help retain an integer or simplified form of expressions.

• In the given exercise, you use rational numbers to clearly express the squares.
• For instance, the number \(25\) was used as a rational number by considering its square root, which is \(5\).

When using rational numbers in the context of factoring, ensure that you identify numbers that are perfect squares.
This helps in creating simpler factors and leads to a straightforward solution.
Recognizing these numbers can greatly simplify the solving process.

Factoring Techniques

Factoring techniques involve strategies to simplify algebraic expressions.
They help break down complex equations into manageable parts.

• The first step is to identify and match the pattern of the expression to a known factoring formula.
• Next, you rearrange the equation into factors, such as in the difference of squares formula.

Beginners often start with simple binomials and work up to more complicated expressions.
Once you identify the type of expression, applying the right formula makes the process faster and more efficient.
In the example of \((x+1)^2 - 25\), applying this knowledge leads directly to the factored form \((x - 4)(x + 6)\).
Keep practicing different techniques, like grouping, using the quadratic formula, or recognizing special patterns, to gain proficiency and confidence.