The difference of squares is a special product in algebra where the result of subtracting one perfect square from another perfect square can be factored. This concept is fundamental to understanding polynomial multiplication, factoring trinomials, factoring special products, and solving polynomial equations.
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The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$.
Difference of squares can be used to factor quadratic expressions with a leading coefficient of 1.
Recognizing the difference of squares pattern is crucial for efficiently multiplying and factoring polynomials.
Solving polynomial equations often involves identifying and factoring the difference of squares.
The difference of squares is a special case of the more general formula for the difference of two perfect powers.
Review Questions
How can the difference of squares formula be used to multiply polynomials?
The difference of squares formula, $a^2 - b^2 = (a + b)(a - b)$, can be used to efficiently multiply binomials where one term is the sum of two numbers and the other term is the difference of the same two numbers. For example, $(x + 3)(x - 3) = x^2 - 9$. This pattern can be extended to multiply more complex polynomials by identifying the difference of squares within the expression.
Explain how the difference of squares can be used to factor trinomials.
When factoring a trinomial of the form $x^2 + bx + c$, if $b^2 - 4c = d^2$ for some integer $d$, then the trinomial can be factored as the difference of squares: $x^2 + bx + c = (x + \frac{b}{2})^2 - (\frac{d}{2})^2$. This allows the trinomial to be expressed as the product of two binomials, $(x + \frac{b}{2} + \frac{d}{2})(x + \frac{b}{2} - \frac{d}{2})$.
Describe how the difference of squares can be used to solve polynomial equations.
When solving polynomial equations, recognizing the difference of squares pattern can simplify the factorization process. For example, to solve the equation $x^2 - 16 = 0$, we can factor the left-hand side as $(x + 4)(x - 4)$, leading to the solutions $x = 4$ and $x = -4$. Similarly, more complex polynomial equations can be solved by identifying and factoring the difference of squares within the expression.
Related terms
Perfect Square: A number or expression that can be written as the product of two equal factors, such as $x^2$ or $9$.
Binomial: An algebraic expression with two terms, such as $x + 3$ or $2y - 5$.
Factoring: The process of expressing a polynomial as a product of simpler polynomials.