Problem 65 - Elementary Algebra

Question

For the following problems, find the products. Expand \((a-b)^{2}\) to prove it is equal to \(a^{2}-2 a b+b^{2}\).

Step-by-Step Solution

Answer

Question: Prove that \((a-b)^{2} = a^{2}-2ab+b^{2}\). Solution: We expanded the expression \((a-b)^{2}\) using the binomial theorem and simplified the result to obtain the expression \(a^{2}-2ab+b^{2}\). Since the expanded expression is equal to the given expression, we have proved the required result.

Step 1: Expand the given expression using binomial theorem or distribution

Apply the binomial theorem on the expression \((a-b)^{2}\): \((a-b)^{2} = \binom{2}{0}a^{2}(-b)^{0} + \binom{2}{1}a^{1}(-b)^{1} + \binom{2}{2}a^{0}(-b)^{2}\) Binomial coefficients can be calculated as: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) Calculate the binomial coefficients: \(\binom{2}{0} = \frac{2!}{0!(2-0)!} = 1\) \(\binom{2}{1} = \frac{2!}{1!(2-1)!} = 2\) \(\binom{2}{2} = \frac{2!}{2!(2-2)!} = 1\) Replace these coefficients in the expansion and simplify: \((a-b)^2 = 1\cdot a^2\cdot (-b)^0 + 2\cdot a^1\cdot (-b)^1 + 1\cdot a^0\cdot (-b)^2\) \((a-b)^2 = a^2 - 2ab + b^2\)

Step 2: Compare the expanded expression to the given expression

The expanded expression we derived is \((a-b)^2 = a^2 - 2ab + b^2\). The given expression to prove is also \(a^{2}-2ab+b^{2}\). Since the expanded expression and the given expression are the same, we have successfully demonstrated that \((a-b)^{2}\) is equal to \(a^{2}-2ab+b^{2}\).

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental concept in algebra, providing a quick way to expand expressions of the form \((a + b)^n\). In this theorem, the expression is expanded into a sum involving terms of the form \( \binom{n}{k}a^{n-k}b^k \), where \( \binom{n}{k} \) denotes a binomial coefficient that can be calculated as \( \frac{n!}{k!(n-k)!} \). This theorem simplifies the process of expanding larger powers, without multiplying the expression repeatedly.

For instance, in the problem of expanding \((a-b)^2\), the binomial theorem allows you to directly calculate this as:

• \( \binom{2}{0}a^2(-b)^0 + \binom{2}{1}a^1(-b)^1 + \binom{2}{2}a^0(-b)^2 \).
• Here, the coefficients \( \binom{2}{0}, \binom{2}{1}, \text{and } \binom{2}{2} \) are particular to the power 2, calculated using factorials.

This systematic approach makes tackling polynomial expansions orderly and efficient.

Polynomial Expansion

Polynomial expansion refers to the process of expressing a polynomial raised to a power in terms of its individual components, often using rules and formulas like the binomial theorem, aiming to make them manageable and easy to work with.

In the example of expanding \((a-b)^2\), we arrived at \(a^2 - 2ab + b^2\) by applying the pattern provided by the binomial theorem. This process entails multiplying each term in the expression in sequence while respecting the power of each term.

Through polynomial expansion, students learn to break down complex algebraic expressions into simpler forms. It is a tried-and-tested technique for simplifying calculations and finding particular terms when looking at expressions that would otherwise seem daunting.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that group together to represent a quantity or solve a problem. These expressions include constants, coefficients, variables, and operations such as addition, subtraction, multiplication, and division.

When dealing with an expression like \((a-b)^2\), understanding each component helps in simplifying the expression correctly. Here, \(a\) and \(b\) are variables, while operations such as subtraction help define the expression's nature.

• In solving or expanding algebraic expressions, recognizing the role of variables and their corresponding constants is vital.
• Substitution and simplification are common practices used to express these in different forms depending on the operation needed.

Algebraic expressions serve as the foundation for more advanced topics in mathematics, making familiarity with them essential for further study.