How to Find a Perfect Square Trinomial


A trinomial is a polynomial, or algebraic expression with exactly three terms. A perfect square trinomial is one that has a squared first term, a squared last term, and the middle term is the sum of the outside base numbers: a^2 + 2ab + b^2. Factoring a perfect square trinomial is the division process, or reverse distribution of the terms to prime polynomials. When factoring a trinomial, both of the signs within the polynomials will be either positive or negative.

  • Examine the expression y^3 + 12y^2 + 36y.

  • Pull out the greatest common factor of all three terms. In this case, all three terms have the variable y in common. Place the greatest common factor before the parenthetical polynomials: y(...)(...)

  • Divide the terms in the trinomial by the GCF: y goes into y^3 twice, goes into 12y^2 once, leaving 12y, and goes into 36 y once, leaving 36. With y factored out, the expression now reads y(y^2 + 12y + 36).

  • Factor the remaining polynomials. Ask yourself what the square root of the first term, y^2, is. The answer is y. Write the root in parenthetical notation: y(y +)(y + ). Ask yourself what the square root of 36 is The answer is 6. Write the answer in the parenthesis y(y + 6)(y + 6). Simplify the polynomial expression: y(y + 6)^2.

  • Multiply the polynomials, (y + 6)(y + 6), using the FOIL (first, outside, inside, last) method to double check your factoring process. Multiply the first two terms, Y x Y = y^2. Multiply the outside terms, y x 6 = 6y. Multiply the inside terms, 6 x y = 6y. Multiply the last terms, 6 x 6 = 36. The expression right now reads y(y^2 + 6y + 6y + 36).

  • Combine like terms, 6y + 6y = 12y: y(y^2 + 12y + 36).

  • Multiply the polynomials in the parenthesis by the GCF, using the distributive process: y x y^2 = y^3, y x 12y = 12y^2 and y x 36 = 36y. Since the terms are the same as the original trinomial, the factoring process proved correct.


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