Problem 100 - Beginning and Intermediate Algebra

Question

Find the following special products. $$(x-8)^{2}$$

Step-by-Step Solution

Answer

The special product of \((x-8)^2\) is: \[(x-8)^2 = x^2 - 16x + 64\]

Step 1: Identify the values of a and b in the binomial

In the given expression \((x-8)^2\), we can see that \(a = x\) and \(b = 8\).

Step 2: Apply the binomial square formula

Use the formula for the square of a binomial, which is \((a-b)^2 = a^2 - 2ab + b^2\). Plug in the values of \(a\) and \(b\) from Step 1: \[(x-8)^2 = (x)^2 - 2(x)(8) + (8)^2\]

Step 3: Simplify the expression

Carry out the arithmetic and simplify the expression: \[(x-8)^2 = x^2 - 16x + 64\] The special product of \((x-8)^2\) is given by: \[(x-8)^2 = x^2 - 16x + 64\]

Key Concepts

Binomial Square Formula

The binomial square formula is a specific case of the broader binomial theorem, which offers a convenient shortcut for expanding expressions that involve squaring binomials. When you have an expression like \((a-b)^2\), the binomial square formula helps in saving time and reducing errors in computation. The formula is:\[ (a-b)^2 = a^2 - 2ab + b^2 \]This formula allows you to expand the squared binomial into a quadratic expression easily. It involves three simple steps:

• Square the first term: \(a^2\)
• Multiply the first and second term together, then double it: \(-2ab\)
• Square the second term: \(b^2\)

These steps collectively help us conclude that \((x-8)^2 = x^2 - 16x + 64\), showing the usefulness of this formula.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematic operations such as addition, subtraction, and multiplication. Expressions are an integral part of algebra, allowing us to represent general relationships and make calculations more seamlessly.In the expression\((x-8)^2\), algebraic elements consist of:

• Variables, like \(x\), which stand for unknown quantities that can vary.
• Constants, such as \(-8\), which remain unchanged during simplification.
• Operations, including squaring, which modify how terms are combined and converted into expressions.

Working with algebraic expressions can involve expanding them, factoring, or simplifying to solve problems or understand relationships between quantities.

Special Products

Special products in algebra provide a set of formulas to quickly and accurately expand expressions that follow certain patterns. They simplify the process of multiplying specific types of binomials without traditional multiplication methods. In the given expression \((x-8)^2\), a special product formula is applied.Typical forms of special products include:

• Square of a binomial: \((a+b)^2\) or \((a-b)^2\)
• Product of sum and difference: \((a+b)(a-b)\)
• Square of a trinomial: \((x^2 + 2xy + y^2)\)

By recognizing these patterns, you can use formulas to convert complex expressions directly into a simpler form. For example, knowing the binomial square formula allows us to transform \((x-8)^2\) into \(x^2 - 16x + 64\) without lengthy calculations.