Factor, or divide, the number 9 to its prime factors. Write it as a product of factors: 9 = (3)(3). You can simplify the expression further by writing it as 9 = (3^2).
Factor the number 12. Write it as a product of factors: 12 = (3)(4). Notice, though, that four is not a prime number. Continue the factoring process until the factors are prime: 12 = (3)(2)(2) or 12 = (3)(2^2).
Factor the number 135 into a product of prime factors: 135 = (5)(27) which simplifies to 135 = (5)(3^3).
Factor 12 + 16 to a product of prime factors. The first step to every factoring problem is to look for a greatest common factor (GCF). By pulling out the GCF, you can eliminate much of the factoring process. In this case, 4 divides evenly into both 12 and 16.
Pull the GCF out from the expression and divide the expression by the GCF. Write the remainders inside parentheses next to the GCF: 4(3 + 4). Because the 4 falls into a parenthetical grouping, you do not need to factor further.
Factor the expression n^2 -- 81 as a product of two binomials, or two terms. Find the square root of n^2, which is n and the square root of 81 which is 9. Because 81 is negative, you will use one of each sign. (n -- 9)(n + 9).
Distribute to check your work and to see how the sign arrangements work: n x n = n^2 + 9n - 9n - 81. Combine like terms: +9n - 9n = 0 and simplify: n^2 - 81. If you had used two positive signs, the middle term would have been 18n and the last term would have been positive 81. Two negative signs would have made the middle term -18n and 81 still would have been positive.
Factor the expression g^3 -- 13g^2 -- 90g. This expression has a GCF of g and has an exponent or degree of 3. This is a good indicator that there will be three parts to the factored process. In addition, the last sign is negative, indicating there will one of each sign.
Pull out the GCF and factor the remaining into parentheses: g(g^2 -- 13g -- 90).
Write the factors of 90 down on paper and look for two factors that will combine to equal 13. 90 and 1 do not, 2 and 45 do not, but 5 and 18 do. Because 13g is negative, place the negative sign on the larger of the two factors: g(g + 5)(g -- 18).
Apply the distributive property to check your work. Do the parentheses first according to the order of operations: g x g = g^2 -- 18g + 5g -- 90. Combine like terms: -18g + 5g = -13g. Continue with the distributive property: g(g^2 -- 13g -- 90) = g^3 -- 13g^2 -- 90g. Because you arrived at the original expression, your factoring and sign placement were correct.
Examine the expression: 15ac - 20ad + 3bc - 4bd. It has four terms and will be factored using a process called grouping.
Divide the expression down the center: 15ac -- 20ad and 3bc -- 4bd. In some cases, you may need to reorder the terms to find a GCF for a grouping. Remember the positive sign between the two because it will come into play later.
Factor out 15ac -- 20ad. First pull out the GCF, 5a and factor out the remainder: 5a(3c -- 4d). Factor out 3bc -- 4bd. Pull out the GCF b and factor: b(3c -- 4d). Notice that the parenthetical components match. This is crucial for factoring with grouping.
Write the parenthetical component first and add the outside terms, the GCFs, in parentheses: (3c -- 4d) + (5a + b). Notice the addition sign between the two groups. This is from the original expression.