Problem 74 - College Algebra with Modeling and Visualization

Question

Factor the expression completely, if possible. \((y+2)^{2}-1\)

Step-by-Step Solution

Answer

The factored expression is \((y+1)(y+3)\).

Step 1: Recognize the Expression Form

The given expression, \((y+2)^2 - 1\), is a difference of squares. Recall that a difference of squares is of the form \(a^2 - b^2\), which factors into \((a - b)(a + b)\).

Step 2: Identify the Square Terms

Identify \(a\) and \(b\) in the expression. Here, \(a = (y+2)\) and \(b = 1\), as \((y+2)^2\) is a perfect square and 1 is also a perfect square since \(1^2 = 1\).

Step 3: Apply Difference of Squares Formula

Apply the difference of squares formula, \((a - b)(a + b)\), to the expression \((y+2)^2 - 1\). Substitute \(a = (y+2)\) and \(b = 1\) into the formula: \((y+2-1)(y+2+1)\).

Step 4: Simplify the Expression

Simplify the products from Step 3. \((y+2-1)\) simplifies to \((y+1)\) and \((y+2+1)\) simplifies to \((y+3)\).

Step 5: Write the Factored Form

The fully factored form of the given expression \((y+2)^2 - 1\) is \((y+1)(y+3)\).

Key Concepts

Difference of squares

The difference of squares is a specific algebraic identity that comes in the form of \(a^2 - b^2\). This identity is one of the easiest to spot and use, especially when factoring polynomials.

When you have an expression like \((y+2)^2 - 1\), it fits the pattern of a difference of squares quite nicely. Let's break down why:

• The expression \((y+2)^2\) is a perfect square because it results from squaring \((y+2)\).
• The number \(1\) is also a perfect square since \(1 = 1^2\).
• Therefore, our expression is in the form \(a^2 - b^2\), where \(a = y+2\) and \(b = 1\).

To factor the difference of squares, you use the identity: \((a - b)(a + b)\). This means you take the square root of each component of the expression, forming two binomials that multiply together to reproduce the original expression. In this specific problem, it simplifies the factoring process effectively.

Polynomial factoring

Polynomial factoring is an essential skill in algebra that involves breaking down polynomials into simpler, more manageable pieces called factors. Recognizing different patterns and using algebraic identities, such as the difference of squares, can greatly simplify this process.

In our exercise, the expression \((y+2)^2 - 1\) was identified as a candidate for the difference of squares, making the factoring relatively straightforward.

• The expression was simplified first by identifying the perfect squares: \((y+2)^2\) and \(1\).
• Applying the difference of squares identity lead to the intermediate expression \((y+2-1)(y+2+1)\).
• Finally, simplifying these binomials provided the factors: \((y+1)\) and \((y+3)\).

Each step in polynomial factoring relies on correctly identifying the form the polynomial fits, then applying the corresponding factoring techniques to simplify.

Algebraic identities

Algebraic identities are equations that are true for all values of the variables involved. They provide powerful shortcuts for algebraic expressions, enabling quick simplification and factoring.

One of the most useful identities in algebra is the difference of squares, which is part of a broader set of identities used frequently in polynomial factoring.

• Besides the difference of squares, other well-known identities include the square of a binomial \((a + b)^2 = a^2 + 2ab + b^2\) and \((a - b)^2 = a^2 - 2ab + b^2\).
• Recognizing these patterns can help quickly break down complex algebraic expressions into simpler forms.
• These identities help in factoring, solving equations, and simplifying expressions efficiently.

Understanding and memorizing algebraic identities such as the difference of squares can significantly ease the process of working through algebra problems like the one in our exercise.