Question

Challenge Problems. \begin{equation}(x-y-z)(x+y+z)\end{equation}

Short answer

\(x^2 - y^2 - 2yz - z^2\)

Step-by-step solution

Identify the Formula

Recognize that the given expression is in the form of the difference of squares: \(a^2 - b^2\), where \(a = x\) and \(b = y + z\).

Apply the Difference of Squares Formula

Use the difference of squares formula \(a^2 - b^2 = (a+b)(a-b)\) to expand the expression. Here, \(a = x\) and \(b = y + z\), so we can rewrite the expression as \((x-(y+z))(x+(y+z))\).

Expand the Expression

Expand the expression using the distributive property, also known as FOIL (First, Outer, Inner, Last) for binomials: \((x-(y+z))(x+(y+z)) = x^2 +(x(z+y)) -(x(z+y)) - (y+z)^2\) which simplifies to \(x^2 - (y+z)^2\).

Apply the Distributive Property

The expression \(y+z\) must be squared. Apply the distributive property again: \(x^2 - ((y+z)(y+z))\), resulting in \(x^2 - (y^2 + 2yz + z^2)\).

Subtract the Squared Binomial

Subtract the binomial square from \(x^2\) to get the final simplified expression: \((x^2 - y^2 - 2yz - z^2)\).

Key concepts

Distributive Property

Understanding the distributive property is essential for simplifying algebraic expressions and solving equations. The concept revolves around multiplying a single term by each term within a parentheses. For instance, consider the expression \(3(x + 4)\). Here, we distribute the 3 by multiplying it with both \(x\) and 4, resulting in \(3x + 12\).In our original exercise equation, during Step 3 of the solution, the distributive property is applied in concert with another technique known as the FOIL method. By breaking down \(x-(y+z))(x+(y+z))\) and applying the distributive property to multiply the terms individually, we expand the expression systematically to arrive at a simplified form.

FOIL Method

The FOIL method stands for First, Outer, Inner, Last, which refers to a technique for multiplying two binomials. To apply FOIL, you take the first terms in each binomial, multiply them together, then move to the outer terms, then the inner terms, and finally the last terms. All intermediate products are then added together.For example, when multiplying \(x+2)(x-3)\), using FOIL we get: \(x * x\) (First), \(x * -3\) (Outer), \(2 * x\) (Inner), \(2 * -3\) (Last). Combining these, our expanded form is \(x^2 - 3x + 2x - 6\), simplified to \(x^2 - x - 6\).In the context of our difference of squares problem, FOIL helps us expand \(x-(y+z))(x+(y+z))\) into \(x^2 +(x(z+y)) -(x(z+y)) - (y+z)^2\), which demonstrates the overlapping usage of the distributive property and the FOIL method to break down complex expressions.

Binomial Multiplication

Multiplying binomials involves combining two binomials to produce a polynomial. The distributive property and the FOIL method are two sides of the same coin when it comes to binomial multiplication. However, if binomials contain more than one term, the process requires careful application of these principles to ensure accuracy.In the given exercise, we encounter a slightly more complex scenario as we need to square the binomial \(y+z\), which is effectively a binomial multiplication of \(y+z)(y+z)\). Here, we multiply each term in the first binomial by each term in the second binomial, leading to \(y^2 + 2yz + z^2\). This step demonstrates how binomial multiplication naturally incorporates the FOIL method to handle larger expressions and how it's a fundamental component of algebra.By practicing these concepts repeatedly, the techniques become second nature, allowing for quick simplification and solving of algebraic expressions involving binomials and more complex polynomials.