Problem 25 - Algebra 2

Question

Expand each binomial. $$ (3 x-y)^{4} $$

Step-by-Step Solution

Answer

\(81x^4 -108x^3y + 54x^2y^2 -12xy^3 + y^4\)

Step 1: List out each term in the binomial to the power mentioned

Write out the binomial expression \((3x-y)^4\) in expanded form: \((3x-y)^4 = (3x-y)(3x-y)(3x-y)(3x-y)\)

Step 2: Apply the Binomial Theorem

Apply the Binomial Theorem to further expand the expression. The Binomial Theorem states that: \((a-b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4\). Replace \(a\) with \(3x\) and \(b\) with \(y\) to get: \((3x-y)^4 = (3x)^4 - 4(3x)^3*y + 6(3x)^2*y^2 - 4*(3x)*y^3 + y^4\)

Step 3: Simplify the expression

Further simplify the expression by completing the calculations within the exponents and coefficients: \((3x)^4 = 81x^4\), \(4(3x)^3*y = 108x^3y\), \(6(3x)^2*y^2 = 54x^2y^2\), \(4*(3x)*y^3 = 12xy^3\). Therefore, the simplified expression is \(81x^4 -108x^3y + 54x^2y^2 -12xy^3 + y^4\)

Step 4: Write the simplified form

The final step is to present the completed simplification of the binomial: \(81x^4 -108x^3y + 54x^2y^2 -12xy^3 + y^4\)

Key Concepts

Binomial Theorem

The Binomial Theorem provides a convenient way of expanding expressions that are raised to a power. It states that any binomial expression of the form \((a+b)^n\) can be expanded into a sum involving terms of the form \(C(n, k) \, a^{n-k} \, b^k\), where \(C(n, k)\) represents the binomial coefficient. This coefficient can be found by the formula \(\frac{n!}{k!(n-k)!}\).
For example, for \((a-b)^4\), the theorem leads to the expression \(a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4\). This allows us to easily substitute values for \(a\) and \(b\) to expand the binomial expression.
This technique is extremely useful when dealing with higher powers, as manually expanding would otherwise be time-consuming and complex.

Polynomials

Polynomials are expressions that consist of variables and coefficients. Each term in a polynomial has a variable raised to a non-negative integer exponent. They can be classified based on the number of terms:

• Monomial: A single term, e.g., \(7x^2\).
• Binomial: Two terms, e.g., \(3x - y\).
• Trinomial: Three terms, e.g., \(2x^2 + 4x - 5\).

When expanding a binomial expression like \((3x-y)^4\), the result is a polynomial with multiple terms. Each term results from applying the Binomial Theorem, involving various powers of the original expression's components.

Algebraic Expressions

Algebraic expressions are mathematical phrases containing numbers, variables, and operators. They form the foundation for algebraic operations, such as addition and multiplication. Algebraic expressions can represent real-world situations in mathematical terms.
An expression like \((3x-y)\) is an algebraic expression that can be expanded using the Binomial Theorem when raised to a power. By substituting values into an algebraic expression, you can simplify, solve, or analyze complex mathematical problems.

Exponents

Exponents are a vital component of mathematics, representing repeated multiplication of a number. They are expressed in the form \(a^n\), where \(a\) is the base and \(n\) is the exponent.

• Base: The number being multiplied.
• Exponent: Indicates how many times the base is multiplied by itself.

In an expression like \((3x)^4\), 3x is the base, and 4 is the exponent, denoting \((3x)\cdot(3x)\cdot(3x)\cdot(3x)\). Mastery of exponents is crucial for tasks like expanding binomials, simplifying polynomials, and solving complex algebraic equations.