FactoringPerfect Square Trinomial

FactoringPerfect Square Trinomial

A perfect square trinomial is a polynomial or an expression withthree relations that one can factor into a suitable pattern. It is anexpansion of a two-termed polynomial (binomial squared) (López,Robles, & Martínez-Planell, 2015). It is a quadratic that onecan factor into two equal binomials that take the form (ay + b)^{2}or (ay + b) (ay + b). It is essential to note that a factored form ofthese polynomials also has negative forms, for example, ay^{2}– 2aby + b^{2}, which factors to (ay –b) (ay – b). Forexample, the expansion of (y + 4)^{2} = (y + 4) (y + 4) = y^{2}+ 8y + 16, which means that y^{2} + 8y + 16 is the perfectsquare trinomial. For an expansion to fit a perfect square trinomial,the mid-term should be two times the product of the binomial’s lastand first terms while the first and last terms should be perfectsquares (López et al., 2015). For example, in the illustration y^{2}+ 8y + 16, y^{2} and 16 are perfect squares while 8y is twotimes the product of y and 4. y^{2} + 8y + 16 is a perfectsquare trinomial since y^{2 } is the square of y, 16 is thesquare of 4. Moreover, multiplying the two numbers gives 4y, which ifmultiplied by 2 gives 8y, which matches the second terms of theexpansion. On the other hand, an expansion like 4y^{2} –25y + 36 is not a perfect square trinomial since 2y^{2 }and-6 multiplied by 2 gives -24y which is not the same as -25y. It alsomeans that one can factor y^{2} + 8y + 16 into (y + 4)^{2},but cannot factor 4y^{2} – 25y + 36 into (2y – 6)^{2}.

References

López, J., Robles, I., & Martínez-Planell, R. (2015). Students`understanding of quadratic equations. *International Journalof Mathematical Education in Science and Technology*, 1-21.