Difference of Squares

Definition

The difference of squares is a special type of polynomial expression where the terms are the difference between two perfect squares. This concept is particularly important in the context of factoring polynomials, working with rational expressions, solving quadratic equations, and understanding the properties of power functions and polynomial functions.

5 Must Know Facts For Your Next Test

1. The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$, which can be used to factor such expressions.
2. Recognizing the difference of squares pattern is a key strategy in factoring polynomials, as it allows for efficient factorization.
3. The difference of squares can be used to simplify rational expressions by factoring the numerator and/or denominator.
4. Solving quadratic equations by factoring is often facilitated by identifying the difference of squares form.
5. Power functions and polynomial functions can exhibit the difference of squares pattern, which influences their graphical behavior.

Review Questions

• Explain how the difference of squares formula can be used to factor polynomial expressions.
  • The difference of squares formula, $a^2 - b^2 = (a + b)(a - b)$, can be used to factor polynomial expressions that are the difference between two perfect squares. By identifying the two perfect squares within the expression, you can apply this formula to factor the polynomial into a product of two binomial factors. This factorization strategy is particularly useful in the context of 1.5 Factoring Polynomials, as it provides an efficient way to break down certain types of polynomial expressions.
• Describe how the difference of squares concept can be applied in the context of rational expressions (1.6 Rational Expressions).
  • The difference of squares can be used to simplify rational expressions by factoring the numerator and/or denominator. If the numerator or denominator of a rational expression can be written as the difference of two perfect squares, the expression can be factored using the difference of squares formula. This factorization can then be used to cancel common factors in the numerator and denominator, resulting in a simpler, equivalent rational expression.
• Analyze how the difference of squares pattern can influence the solutions of quadratic equations (2.5 Quadratic Equations) and the graphs of polynomial functions (5.3 Graphs of Polynomial Functions).
  • When a quadratic equation can be written in the difference of squares form, $a^2 - b^2 = 0$, it can be easily solved by factoring the expression and setting each factor equal to zero. This factorization approach is often more efficient than using the quadratic formula. Additionally, the difference of squares pattern in polynomial functions can affect the shape and behavior of their graphs, as the factors $(a + b)$ and $(a - b)$ can influence the number and location of the function's zeros, as well as the overall concavity and end behavior of the graph.

Related terms

Perfect Square: A number or expression that can be written as the product of two equal factors, such as $x^2$ or $9$.

Factoring:

The process of breaking down a polynomial expression into a product of simpler expressions, such as factors.

Quadratic Equation:

A polynomial equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$.