Difference of Squares

Definition

The difference of squares is a special product in algebra where the result of subtracting two perfect squares is itself a perfect square. This concept is crucial in understanding various factorization techniques and strategies for solving polynomial equations.

5 Must Know Facts For Your Next Test

1. The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$, where $a$ and $b$ are any real numbers.
2. Difference of squares is a special case of factoring trinomials of the form $x^2 + bx + c$, where $b = 0$ and $c = -b^2$.
3. Difference of squares can be used to factor special products, such as $x^2 - 4$ or $(x + 3)^2 - (x - 2)^2$.
4. Recognizing difference of squares is an important step in the general strategy for factoring polynomials, as it allows for efficient factorization.
5. Difference of squares is a powerful tool in solving polynomial equations, as it can simplify complex expressions and make them easier to solve.

Review Questions

• Explain how the difference of squares formula, $a^2 - b^2 = (a + b)(a - b)$, can be used to factor a polynomial expression.
  • The difference of squares formula is useful for factoring polynomial expressions that can be written in the form $a^2 - b^2$. By recognizing this pattern, you can factor the expression into the product of two binomials: $(a + b)$ and $(a - b)$. This factorization technique is particularly helpful in simplifying complex polynomial expressions and solving polynomial equations.
• Describe the connection between the difference of squares and factoring trinomials of the form $x^2 + bx + c$, where $b = 0$ and $c = -b^2$.
  • When a trinomial of the form $x^2 + bx + c$ has $b = 0$ and $c = -b^2$, it can be factored using the difference of squares method. In this case, the trinomial can be rewritten as $x^2 - b^2$, which is a difference of squares. By applying the difference of squares formula, $x^2 - b^2 = (x + b)(x - b)$, the trinomial can be factored into a product of two binomials.
• Explain how recognizing the difference of squares can be a crucial step in the general strategy for factoring polynomials.
  • The general strategy for factoring polynomials involves identifying common factors, recognizing special product forms (such as the difference of squares), and applying appropriate factorization techniques. When a polynomial expression can be written in the form of a difference of squares, $a^2 - b^2$, the difference of squares formula can be used to factor the expression into the product of two binomials, $(a + b)$ and $(a - b)$. This efficient factorization method is an important step in the overall strategy for factoring polynomials, as it simplifies complex expressions and makes them easier to solve.

Related terms

Perfect Square: A perfect square is a number that can be expressed as the product of two equal integers, such as 4 (2 x 2), 9 (3 x 3), or 16 (4 x 4).

Factorization: Factorization is the process of breaking down a polynomial or expression into a product of simpler factors.

Polynomial Equation: A polynomial equation is an equation that involves one or more polynomials, which are expressions with variables and coefficients.