Examine the trinomial 6x^2 + 48x + 72. The GCF is six. Pull the GCF out of the trinomial and set it before parentheses,6( ). Remember, factoring is writing the trinomial as a product of factors.
Divide the GCF out of each term, (6x^2 ÷ 6 = x^2) + (48x ÷ 6 = 8x) + (72 ÷ 6 = 12). Simplify the expression, 6(x^2 + 8x + 12).
Break down the first parenthetic term to its prime factor, x^2 = (x)(x), or x times x. Simplify the expression to a product of binomials, 6(x )(x ).
Write out the factors of 12, 1 x 12, 2 x 6 and 3 x 4. The middle term, 8x, will be the sum of the two factors that equal 12. One and 12 do not equal eight, but two and six will. Write the factors into the parentheses, 6(x 6)(x 2).
Look at the expression from before, 6(x^2 + 8x + 12). Both 8x and 12 are positive. This means that the signs within the parentheses will be positive; therefore, the factored solution is 6x^2 + 48x + 72 = 6(x + 6)(x + 2).
Examine the trinomial a^3 -- 13a^2 -- 90a. The GCF is a, so pull it out and write it in front of the parentheses, a( ).
Divide the terms by the GCF, and write the remainders in the parentheses, a(a^2 -- 13a -- 90).
Divide the first parenthetic term down to its prime factor and simplify the expression, a(a )(a ).
Write the factors for 90, 1 x 90, 2 x 45, 3 x 30, 5 x 18. Look for the two numbers that when combined equal -13. Write the factors into the parentheses, a(a 18)(a 5).
Look at the expression from before, a(a^2 -- 13a -- 90). The middle and last terms are negative, meaning that there will be one of each sign, a positive and a negative. Because the middle term is negative, put the negative number in front of the larger of the two factors. The factored solution is a^3 -- 13a^2 -- 90a = a(a -- 18)(a + 5).
Examine the trinomial 4x^2y + 8xy -- 12y. The greatest common factors are 4 and y. Pull the two out and simplify, 4y( ).
Divide the terms by the GCFs and write the remainders in the parentheses, 4y(x^2 + 2x -- 3).
Break the first term in parentheses down and simplify, 4y(x )(x ). The last term is already prime, so write the factors of three into the parentheses, 4y(x 1)(x 3).
Look at the expression 4y(x^2 + 2x -- 3). The second term is positive and the last term is negative, so this means there will be one of each sign. Because the second term is positive, place the positive sign before the largest factor, 4y(x -- 1)(x + 3), which is the factored solution.
Examine the trinomial -100 -- y^2 -- 20y. All three terms are negative; no sign arrangement, (+/-) or (-/-), will result in two negative terms. Instead, you can remove the negative sign by factoring out the GCF, which is the understood -1. Simplify the trinomial, -1( ).
Divide the terms by the GCF and write the remainders in parentheses, -1(100 + y^2 + 20y). Remember, a negative divided by a negative will leave a positive answer.
Reorder the terms in the parentheses so that the variables are in descending order, -1(y^2 + 20y + 100).
Break the first parenthetical term to its prime and simplify the expression, -1(y + )(y + ). Because both terms are positive, you will have two positive signs in the parenthetical binomials.
Factor out 100, 1 x 100, . . ., 10 x 10. The sum of 10 and 10 is 20, so write the factors into the parentheses, -(y + 10)(y + 10). Although the one is removed here, it is automatically understood as present.