Problem 85 - College Algebra Essentials

Question

Factor completely, or state that the polynomial is prime. $$x^{2}-12 x+36-49 y^{2}$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(x^{2}-12 x+36-49 y^{2}\) is \((x - 6 + 7y)(x - 6 - 7y)\).

Step 1: Identify the terms

The given polynomial is \(x^{2}-12 x+36-49 y^{2}\). This can be rewritten as \((x-6)^2 - (7y)^2\) as \(x^{2}-12 x+36 = (x-6)^2\) and \(49y^2 = (7y)^2\).

Step 2: Apply the difference of squares formula

The difference of squares formula can be used to factor expressions of the form \(a^2 - b^2\). Replacing \(a\) with \((x-6)\) and \(b\) with \((7y)\), we can transform \((x-6)^2 - (7y)^2\) into \((x - 6 + 7y)(x - 6 - 7y)\).

Step 3: Final Solution

The complete factorized form of the given polynomial is \((x - 6 + 7y)(x - 6 - 7y)\).

Key Concepts

Difference of Squares

The concept of "difference of squares" is essential in polynomial factoring. It is a special pattern that simplifies expressions like the one given in the exercise. The formula for the difference of squares is generally represented as \[ a^2 - b^2 = (a + b)(a - b) \].
This formula can be applied whenever you spot something like \[ x^2 - y^2 \], where both terms are perfect squares.
In our exercise, the polynomial can be rewritten using this concept. The original polynomial \[ x^2 - 12x + 36 - 49y^2 \] is transformed into \[ (x-6)^2 - (7y)^2 \]. Here, \[ (x-6)^2 \] and \[ (7y)^2 \] represent the squared terms. This makes it easier to apply the difference of squares formula.

• Recognize the pattern: identify terms that are perfect squares.
• Use the formula: \[ a^2 - b^2 = (a + b)(a - b) \].
• Apply to transform expressions into a product of simpler binomials.

Understanding and recognizing this pattern is key as it helps simplify and solve complex algebraic problems quickly and efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are the building blocks of most equations in algebra. In the given problem, we have:\[ x^2 - 12x + 36 - 49y^2 \].
Understanding the components of an algebraic expression is crucial.
Algebraic expressions consist of:

• **Terms**: individual parts of an expression, separated by addition or subtraction signs.
• **Coefficients**: numerical values attached to variables, like \(-12\) in \(-12x\).
• **Variables**: symbols like \(x\) and \(y\) that represent unknown values.
• **Constants**: fixed numbers, such as \(36\).

For factoring, it's often helpful to first look at how these components interact. In our example, by recognizing that \[ x^2-12x+36 \] takes the shape of \[ (x-6)^2 \], we converted an algebraic expression into a simpler, manageable form.
This restructuring into recognizable patterns, like squares or other factorizable expressions, makes algebraic expressions more straightforward to work with.

Math Problem Solving

Math problem solving involves a series of logical steps that help us tackle and resolve mathematical problems effectively. This is a crucial skill in algebra. Breaking down problems into smaller, more manageable parts can make complex problems simpler.
In our exercise, we approach the polynomial by first identifying the terms that we can simplify or restructure. Recognizing patterns like perfect squares helps simplify the factoring process.

• **Understand the problem**: comprehend what is asked. In this case, factor the polynomial.
• **Identify patterns**: look for clues such as perfect squares or like terms.
• **Apply methods**: use known formulas or algorithms, like the difference of squares, to simplify the problem.

These steps required us to rewrite \[ x^2-12x+36 \] as \[ (x-6)^2 \], thus making it easier to apply the difference of squares formula. It illustrates critical thinking and strategic application of mathematical concepts. Efficient problem-solving in math builds confidence and enhances one's ability to tackle increasingly complex problems over time.