Problem 38 - Algebra 2 and Trigonometry

Question

In \(27-39,\) factor each polynomial completely. $$ 4-(y-1)^{2} $$

Step-by-Step Solution

Answer

The polynomial factors as \((3-y)(y+1)\.\)

Step 1: Recognize the Pattern

The polynomial is of the form \(4 - (y-1)^2\). This resembles the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\). Identify \(a\) and \(b\) such that \(a^2 = 4\) and \(b^2 = (y-1)^2\).

Step 2: Identify 'a' and 'b'

For the expression \(4 - (y-1)^2\), we identify \(a^2 = 4\), which means \(a = 2\) (since \(2^2 = 4\)), and \(b = (y-1)\) since \((y-1)^2 = (y-1)^2\).

Step 3: Apply the Difference of Squares Formula

Using the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\), substitute \(a = 2\) and \(b = (y-1)\) into the formula. This yields: \(4 - (y-1)^2 = (2 - (y-1))(2 + (y-1))\).

Step 4: Simplify the Factors

Simplify the expressions within the parentheses: - For \( (2 - (y-1)) \), simplify to \( (2 - y + 1) = (3 - y) \). - For \( (2 + (y-1)) \), simplify to \( (2 + y - 1) = (y + 1) \).

Step 5: Write the Final Factored Form

The fully factored form of \(4 - (y-1)^2\) is \((3-y)(y+1)\).

Key Concepts

difference of squares

The "difference of squares" is a powerful algebraic identity that simplifies expressions where one square is subtracted from another. It’s given by the formula \[ a^2 - b^2 = (a-b)(a+b) \] This formula tells us that a difference of two squares is equivalent to the product of two binomials. Here, both binomials include the square roots of the original squares, distinguished merely by their signs.

• The first binomial is the difference \(a-b\),
• and the second is the sum \(a+b\).

Such identities can turn complicated expressions into much simpler ones. For instance, if you have an expression like \(4 - (y-1)^2\), identifying it as a difference of squares can swiftly transform it into \( (3 - y)(y + 1) \). Remember, spotting this pattern can save time and simplify your solving process.

algebra 2

Algebra 2 is a branch of mathematics that extends the concepts taught in Algebra 1. It dives deeper into equations, functions, and more complex mathematical relationships. Here’s what you commonly explore in Algebra 2:

• Quadratic expressions and equations, which represent one of the most introduced pieces of algebraic thinking.
• Factoring techniques, like difference of squares, which are integral in simplifying and solving these expressions.
• Advanced polynomials, systems of equations, and other algebraic structures.

This level of algebra requires one to not only solve equations but also appreciate underlying structures and patterns. Recognizing identities like the difference of squares becomes a crucial skill. It assists in efficiently solving problems and simplifying expressions, making Algebra 2 both challenging and rewarding.

factoring techniques

Factoring is one of the fundamental techniques used in algebra to simplify expressions. It involves rewriting a polynomial as a product of simpler polynomials. Key factoring techniques you might encounter include:

• Difference of squares, where expressions in the form of \(a^2 - b^2\) are factored as \( (a-b)(a+b) \).
• Factoring trinomials, which splits a three-term polynomial into two binomials.
• Recognizing perfect square trinomials, using the square of a binomial pattern.

Each technique is designed to make polynomials easier to handle, making them solvable or simplified for further operations. With consistent practice, recognizing the appropriate factoring method becomes intuitive, streamlining your problem-solving approach.

quadratic expressions

Quadratic expressions are polynomials of degree 2, usually in the form \(ax^2 + bx + c\). They hold a significant place in algebra because they appear in various scenarios, from physics to economics. Common properties of quadratic expressions include:

• They take on a parabolic shape when graphed, opening either upwards or downwards depending on the sign of \(a\).
• Factorization is a critical method for solving quadratic equations, often involving techniques like completing the square or using the quadratic formula.
• Roots or solutions of these expressions can indicate real-world quantities, like time or distance, depending on context.

Understanding these expressions’ structure and methods of manipulation, like recognizing when they represent a difference of squares, allows you to tackle a range of problems effectively. Mastery of quadratic expressions is essential for advancing further in mathematics and other sciences.