Problem 70 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 9 y^{4}-121 x^{2} $$

Step-by-Step Solution

Answer

The polynomial factors to \((3y^2 - 11x)(3y^2 + 11x)\).

Step 1: Recognize the form

Observe that the expression \(9y^4 - 121x^2\) is a difference of squares. This is because it can be expressed as \((3y^2)^2 - (11x)^2\). The formula for the difference of squares is \(a^2 - b^2 = (a - b)(a + b)\).

Step 2: Identify \(a\) and \(b\)

In this case, we have \(a = 3y^2\) and \(b = 11x\). We will use these to apply the difference of squares factorization.

Step 3: Apply the difference of squares formula

Substitute \(a = 3y^2\) and \(b = 11x\) into the difference of squares formula \((a - b)(a + b)\). This gives us the factored form: \((3y^2 - 11x)(3y^2 + 11x)\).

Key Concepts

Factorization

Factorization is the process of breaking down a mathematical expression into simpler components called 'factors' that, when multiplied together, give the original expression. This is a fundamental skill in algebra, often used to simplify or solve equations. The concept is widely applicable, from simplifying fractions to solving complex polynomial equations.

One particularly important method of factorization in algebra is the 'difference of squares'. It is used when an expression can be written in the form of two perfect squares separated by a subtraction sign: \( a^2 - b^2 \). This form factors into \( (a - b)(a + b) \).

Recognizing expressions that fit the difference of squares formula is crucial. In the example **9y^4 - 121x^2**, both terms, \(9y^4\) and \(121x^2\), are perfect squares. \(9y^4\) is \((3y^2)^2\) and \(121x^2\) is \((11x)^2\).

Understanding how to rearrange and rewrite expressions to recognize the form enables the application of straightforward algebraic manipulation techniques, making seemingly complicated problems manageable.

Polynomials

Polynomials are mathematical expressions made up of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. A polynomial can have one or multiple terms, each being a product of a constant and a variable raised to an integer power.

Key characteristics of polynomials include:

• Each term in a polynomial is called a 'monomial.'
• The degree of a polynomial is determined by the highest power of the variable within the polynomial.
• Polynomials can have more than one variable but commonly involve expressions in a single variable.

In the example, **9y^4 - 121x^2**, the expression is a polynomial with two terms, making it a binomial. Each term is a monomial by itself—a squared term demonstrating a simple polynomial structure. The difference of the squared elements signifies that it can be factored using algebraic techniques like the difference of squares.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators that represent a particular value or concept. Unlike equations, expressions do not contain an equal sign and often denote patterns or relationships rather than exact solutions.

The components of an algebraic expression include:

• Variables: symbols (often \(x\), \(y\), etc.) that represent unknown values or quantities.
• Coefficients: numbers multiplying the variables in an expression.
• Operators: symbols that represent mathematical operations such as addition, subtraction, multiplication, and division.

In the expression **9y^4 - 121x^2**, each term consists of:- A coefficient: 9 and -121.- A variable: \(y\) and \(x\).- An exponent representing the power to which the variable is raised—4 for \(y\) and 2 for \(x\).

This framework lets us manipulate and perform operations like factorization efficiently, turning complex expressions into simpler or alternate forms. Recognizing variables and coefficients is fundamental to understanding and working with algebraic expressions, particularly when factorizing or solving them.