Problem 7 - College Algebra and Calculus: An Applied Approach

Question

Factor out the common factor.Factor the difference of two squares.\(x^{2}-36\)

Step-by-Step Solution

Answer

The factored form of the expression \(x^{2}-36\) is \((x-6)(x+6)\).

Step 1: Identify the terms

We can identify our a and b terms as \(a = x\), \(b = 6\).

Step 2: Apply the difference of two squares pattern

We substitute a and b into the difference of two squares pattern \((a-b)(a+b)\) we get \((x-6)(x+6)\).

Key Concepts

Factoring

Factoring is like simplifying an expression to uncover what "ingredients" it's made of. Think of it like finding the prime numbers for a larger number. In algebra, factoring means breaking down a polynomial into simpler, multiplied forms.

• This process helps to solve equations, as the factored form makes roots more visible.
• In expressions, look for a common factor in each term.

Factoring can involve taking out a common number, as in integers, or algebraic expressions in cases like polynomials. The most basic type might involve numbers or variables being taken out of multiple terms.
For instance, if you have an expression like:\[ ax + ay \]You can factor it to:\[ a(x + y) \]This is very helpful when dealing with quadratic equations, ultimately making it easier to solve or simplify the problem at hand.

Difference of Squares

The "difference of squares" is a specific type of polynomial that has a unique factorization pattern. Recognizing this pattern helps quickly simplify expressions, as seen in the exercise.Here's the general pattern:\[ a^2 - b^2 = (a - b)(a + b) \]The expression must be made of two square terms with a subtraction between them, hence the name 'difference of squares'.

• Each term must be a perfect square.
• The subtraction (difference) is important for the formula to apply.

This method speeds up the factoring process immensely. In our exercise, with terms \(x^2\) and \(36\), both are perfect squares. Therefore, using this rule gets us from:\[ x^2 - 36 \]To its factored form:\[ (x - 6)(x + 6) \]

Polynomials

Polynomials are expressions consisting of variables and coefficients, joined together through addition, subtraction, and multiplication. They can look complicated but breaking them down is manageable.In general, a polynomial can have:

• Multiple terms
• Varying degrees, based on the powers of the variable(s)
• Constant numbers

They can be expressed in the form:\[ a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \]Where each \(a\) is a coefficient, and \(x\) is a variable with the degree \(n\) determining the polynomial's "level". Understanding polynomials and how to manipulate them is crucial in algebra because they appear everywhere: in equations, when modeling real-life situations, and more.The example exercise deals with a polynomial of degree 2 (quadratic), and through methods like factoring, we understand its roots and solutions better.