Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ (5 x-4 y)^{2} $$

Step-by-Step Solution

Answer

\((5x - 4y)^2 = 25x^2 - 40xy + 16y^2\).

Step 1: Understand the Problem

We are given the expression \((5x - 4y)^2\). Our task is to express this as a polynomial, which involves expanding the square of a binomial.

Step 2: Recall the Binomial Square Formula

The square of a binomial \((a - b)^2\) is given by the formula: \(a^2 - 2ab + b^2\). In this problem, \(a = 5x\) and \(b = 4y\).

Step 3: Apply the Formula - Calculate \(a^2\)

Calculate \(a^2 = (5x)^2 = 25x^2\).

Step 4: Apply the Formula - Calculate \(-2ab\)

Calculate \(-2ab = -2 \times (5x) \times (4y) = -40xy\).

Step 5: Apply the Formula - Calculate \(b^2\)

Calculate \(b^2 = (4y)^2 = 16y^2\).

Step 6: Write the Expanded Polynomial

Combine all terms: \((5x - 4y)^2 = 25x^2 - 40xy + 16y^2\). This is the expression written as a polynomial.

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool used in algebra to expand expressions that are raised to a power. This theorem allows us to express binomials, which are expressions containing two terms, such as \[(a + b)^n\]into expanded polynomial expressions.

The key to using the Binomial Theorem is understanding it can help in predicting the resulting expanded form without direct multiplication. The coefficients of each term in the expanded form align with values from Pascal's Triangle, which makes calculating large powers much simpler. For instance, the expansion of \[(a - b)^2\]can be calculated as:

• The coefficient of each term comes from the second row of Pascal’s Triangle: 1, 2, 1.
• The powers of the first term \(a\) will decrease from \(n\) to 0.
• The powers of the second term \(b\) increase from 0 to \(n\).

This setup makes expanding expressions systematic and straightforward.

Binomial Squares

In algebra, binomial squares are specific instances of binomial expansion where the power is 2. The formula for a binomial square, \((a - b)^2\)or \((a + b)^2\),allows calculation of squares of two-term expressions easily.

The formula \((a - b)^2 = a^2 - 2ab + b^2\)is derived from multiplying the binomial by itself. It follows the pattern:

• Square the first term: \(a^2\).
• Double the product of both terms with a negative sign if the binomial is a difference: \(-2ab\).
• Square the second term: \(b^2\).

This foundational formula can be quickly recalled to expand binomials by substituting the appropriate values. For example, in \((5x - 4y)^2\),we apply \(a = 5x\)and \(b = 4y\) to achieve the expanded form: \(25x^2 - 40xy + 16y^2\).
Recognizing and applying this formula can save time and reduce errors when working with algebraic expressions.

Polynomial Expression

A polynomial expression is an algebraic expression made up of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. Polynomials can be parts of equations or standalone expressions. The expansion of \((5x - 4y)^2\)is an example of transforming a binomial into a polynomial.

The term 'polynomial' itself indicates multiple terms. Typical polynomial expressions could include terms like \(x^2 + 3x + 2\).Here, each term in the polynomial is a power of a variable, often accompanied by a coefficient. The powers are non-negative integers.
Key characteristics of a polynomial expression include:

• The degree of the polynomial: The highest power of the variable in the expression.
• The leading coefficient: The coefficient of the term with the highest degree.
• The constant term: A term that does not contain any variables.

Expanding binomials into polynomial forms, as demonstrated by the earlier example, helps to simplify expressions and make them more manageable. It's a crucial skill in algebra that aids in solving equations and understanding complex functions.