Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Find the product.\([(x+1)-y]^{2}\)

Step-by-Step Solution

Answer

The product is \(x^2-2xy+y^2+2x-2y+1\).

Step 1: Identify a and b

We simplify to a form that resembles the Perfect Square formula and identify \(a=x+1\) and \(b=y\)

Step 2: Apply the Perfect Square formula

Following the Perfect Square formula \(a^2 - 2ab + b^2\), we substitute \(a\) and \(b\) to get \([(x+1)^2 - 2*(x+1)*y+y^2]\)

Step 3: Simplify the expression

We simplify the expression by expanding \((x+1)^2\) to get \([x^2+2x+1-2xy-2y+y^2]\). Then we group the like terms to get \(x^2-2xy+y^2+2x-2y+1\)

Key Concepts

Perfect Square Formula

The Perfect Square Formula is a handy tool when expanding expressions that are squares of binomials. A binomial is just a mathematical expression with two terms. For example, \[(a + b)^2\] or \[(a - b)^2\] are both binomials in perfect square form. Using the formula, we can expand these expressions without multiplying the whole expression manually. The formula for the square of a binomial is given as: \[(a \,\pm \, b)^2 = a^2 \,\pm\, 2ab + b^2\]. - This means that you first square the first term - then add or subtract twice the product of both terms - and finally, add the square of the second termIn our exercise, we recognized that \((x+1) - y\) was in a form that could use the Perfect Square Formula. By identifying \(a\) as \((x+1)\) and \(b\) as \(y\), we were able to apply the formula easily.

Polynomial Expansion

Expanding polynomials is the process of multiplying out expressions and writing them as a sum of terms. When you have a squared binomial, such as the one in our example, polynomial expansion helps to express it as a polynomial.Here are a few things to keep in mind when expanding a binomial:

• First, make sure to apply formulas, such as the Perfect Square Formula, whenever applicable. This saves time and reduces errors.
• Secondly, distribute each term completely, ensuring that every term in the first bracket multiplies with every term in the second bracket.

In our exercise, we expanded \((x+1)^2\) to get\(x^2 + 2x + 1\). This was step one in using polynomial expansion to tackle the problem effectively.This simplification allows for easier combination and identification of like terms in advanced algebraic operations.

Simplification

Simplification in algebra involves combining like terms and reducing an expression to its simplest form. Like terms are terms that have identical variable components and can be combined simply by adding or subtracting their coefficients.During simplification, you follow these steps:

• Identify similar terms. Variables with the same power can be combined.
• Combine these terms by simple addition or subtraction of their coefficients.
• Rearrange to write in standard polynomial form (usually with decreasing powers).

In the original solution, once the polynomial was expanded, the expression was as follows:\[x^2 + 2x + 1 - 2xy - 2y + y^2\]. By combining like terms, we get the simplified form:\[x^2 - 2xy + y^2 + 2x - 2y + 1\].Simplifying expressions makes them easier to read, interpret and solve, especially as you progress to more complex algebraic tasks.