Factorization: Difference of Squares
Factorization using the difference of squares is a mathematical technique that allows for the simplification of expressions involving binomials where each term is a square. Specifically, it applies to expressions of the form \(a^2 - b^2\), which can be factored into the product of two conjugate pairs: \((a + b)(a - b)\). This technique not only streamlines calculations but also serves as a foundational concept in algebra, enabling the resolution of equations and problems involving quadratic expressions.
The difference of squares can be effectively demonstrated through both algebraic manipulation and geometric interpretations. For example, the product of two conjugates can illustrate how certain rectangular arrangements (like a \(4 \times 4\) square) relate to their respective differences and sums. In more advanced contexts, such as with complex numbers, the applicability of this factorization broadens, allowing for situations where real numbers alone would not suffice.
Additionally, the concept finds practical use in rationalizing fractions, especially those containing square roots, by multiplying by the conjugate of the denominator. This approach not only enhances computational ease but also contributes to a deeper understanding of the relationships between numbers. Overall, mastering the difference of squares is essential for anyone looking to build a solid foundation in algebraic concepts and techniques.
Subject Terms
Factorization: Difference of Squares
Let x and y be real variables, and let a and b be any real numbers. Any of the following binomials are called differences of squares:
because the first part of the binomial is a square and the second part, which is subtracted from the first, is also a square. The factorizations are as follows:
Note that in each case the first factor is a sum and the second factor differs from the first only by being a subtraction rather than an addition. Pairs of binomials of the form (ax + by) and (ax – by) are called conjugate pairs or simply conjugates. The difference of two squares can always be factored into a conjugate pair.
Proof
The factorizations above can be easily proven: (ax + by)(ax – by) multiplied out gives ax × ax + ax(−by) + by × ax + by (–by) = a2x2 – axby + axby − b2y2. The desired result is achieved because –axby cancels out axby giving a2x2 – b2y2.
In complex numbers, the binomial a2x2 + b2y2 cannot be factored in real numbers. However, if a and b can be complex numbers (that is, numbers of the form p + qi where i is the square root of −1), then factorization is possible.
Rationalizing Fractions
Conjugates can be used to clear the denominator of a fraction of a binomial, particularly one with square roots. For example, rationalize
Solution: Multiply by the unit fraction consisting of the conjugate of the denominator divided by itself.
Geometric Interpretation
Consider the following three tableaus:
The first tableau is 4 × 4. It is square and has 42 = 16 elements.
The second tableau is 3 × 5 and has 3 × 5 = 15 elements. Note that 3 = 4 – 1 and 5 = 4 + 1. Hence the second tableau has (4 – 1)(4 + 1) = 42 − 1 = 15 elements, that is, 3 × 5 = 15 can be expressed as the difference of two squares.
Similarly, the third tableau is 2 × 6 and has 2 × 6 = 12 elements. Note that 2 = 4 – 2 and 6 = 4 + 2. Hence the third tableau has (4 – 2)(4 + 2) = 42 - 22 = 16 – 4 = 12 elements, that is, 2 × 6 = 12 can be expressed as the difference of two squares.
It is possible to multiply two numbers mentally, if the two numbers can be expressed as conjugates. For example, multiply 15 by 25.
15 = 20 – 5 and 25 = 20 + 5.
Hence 15 × 25 = (20 – 5)(20 + 5) = 202 – 52 = 400 – 25 = 375.
Bibliography
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