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Chapter 0: Problem 40

Factor the difference of two squares. $$x^{2}-144$$

Short Answer

Expert verified
The factors of the given expression are \(x - 12\) and \(x + 12\).

Step by step solution

01

Identify a and b

Recognise that \(x^{2} - 144\) is a difference of squares, where \(a = x\) and \(b = 12\) because \(12^{2} = 144\).
02

Apply the Formula

Substitute the values of \(a\) and \(b\) into the formula \(a^{2} - b^{2} = (a - b)(a + b)\). Therefore, \(x^{2} - 144 = (x - 12)(x + 12)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring
Factoring is a method used in solving polynomial equations and expressions. It involves breaking down complex expressions into simpler ones that are multiplied together to give the original expression.
Think of it as finding the pieces that can be multiplied back together to form the original number or expression.

When we talk about the difference of squares, we mean recognizing patterns within an equation, like in our example:
  • Recognize the pattern: A difference of squares is a special binomial (a two-term expression) that takes the form of \(a^2 - b^2\).
  • Split it using its formula: The formula for the difference of squares is \(a^2 - b^2 = (a - b)(a + b)\).
Understanding factoring is essential because it helps simplify expressions and solve equations that involve square numbers.
Polynomial Expressions
Polynomial expressions are mathematical expressions consisting of variables and coefficients combined using addition, subtraction, multiplication, and sometimes division.
They include terms with whole number exponents. For instance, \(x^2 - 144\) is a polynomial expression鈥攎ore specifically, a quadratic polynomial.

Here's what makes up a polynomial:
  • Variables: Symbols like \(x\) or \(y\) used to represent a number that can change.
  • Coefficients: Numbers that multiply the variables, such as 1 in \(x^2\).
  • Exponents: The power to which the variable is raised, such as 2 in \(x^2\).
Recognizing the structure of polynomial expressions is crucial for factoring since it helps identify patterns like the difference of squares, allowing for easier simplification and manipulation.
Algebraic Formulas
Algebraic formulas are mathematical sentences that represent relationships between numbers and variables.
They are used to simplify and solve equations or expressions efficiently.

An essential algebraic formula is the difference of squares formula given by \(a^2 - b^2 = (a - b)(a + b)\).
  • This formula is useful because it transforms a cumbersome subtraction problem into a simpler multiplication problem.
  • By applying this formula, you factor the expression into a product of two binomials.
Understanding and applying algebraic formulas like the difference of squares formula allows one to solve complex polynomial expressions quickly by simplifying them into easier-to-handle terms.

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Most popular questions from this chapter

How is the quadratic formula derived?

Factor completely. $$-x^{2}-4 x+5$$

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