Overview
When binomials are squared, the resulting trinomial follows a pattern. The pattern can be derived from the general process of multiplying a binomial by another binomial, except that all the terms are the same. As in the general process, the Distributive Property is used, then like terms are combined.
What Is A Squared Binomial?
A squared binomial is a polynomial with two terms multiplied by itself. For example, 2x + 5 is a binomial, multiplied by itself is (2x + 5)(2x + 5). Similarly (y – 2) is a binomial, multiplied by itself is (y – 2)(y – 2). The concept of squaring a binomial is no different than squaring any number or variable.
Expanding the Expression and Combining Like Terms
The expression (2x + 5)(2x + 5) can be expanded using the Distributive Property, so that solving it results in the expression 2x(2x + 5) + 5(2x + 5) or 4x2 + 10x + 10x + 25. Combining like terms results in the expression 4x2 + 20x + 25. Similarly, expanding (y-2)2 results in y(y-2) – 2(y-2) or y2-2y – 2y + 4. (Multiplying -2 by -2 results in 4.) Combining like terms results in the expression y2– 4y + 4.
What Is the Pattern for Squaring a Binomial Sum?
A binomial sum is in the pattern a + b, so the square is in the form of (a + b)(a + b) algebraically. The general pattern for squaring a sum can be found by multiplying the terms and then combining like terms. Using the Distributive Property, the resulting expansion is a(a + b) + b(a + b). It doesn’t matter if b is another variable or a constant real number. The result is a2 +ab + ba + b2. Combining like terms (because ab and ba are the same because of the Commutative Property), the resulting pattern is a2 + 2ab + b2.
What Is the Pattern for Squaring a Binomial Difference?
A binomial difference is in the pattern a – b, so the square is in the form of (a-b)(a-b). The general pattern is very similar to the pattern for squaring a binomial sum, except that it is important to watch the signs. The expression a(a – b) – b(a – b) results in the expansion a2-ab – ba + b2. Two negative numbers multiplied results in a positive number, just as it did when -2 was multiplied by -2 to result in 4. The resulting pattern is a2– 2ab + b2. Those patterns are a shortcut to factoring polynomials.
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