Look for the greatest common factor in a polynomial, e.g., 8x + 4, which you can read as "eight x plus four." Four goes into both terms.
Remove the GCF from the expression and then divide both terms by it: 8x ÷ 4 = 2 and 4 ÷ 4 = 1.
Write the remainder in parenthetical notation. 4(2x + 1). Read it as "four times the quantity of two x plus one."
Examine the expression 2x^3 + 3x^2 + 8x + 12, read as "two x cubed plus three x squared plus eight x plus twelve." This polynomial has four terms; you will factor it using a process called grouping.
Divide the expression down the center to factor out one side at a time, e.g., 2x^3 + 3x^2 on one side and 8x + 12 on the other.
Look for the GCF of the first set of terms; x squared divides evenly into both terms. Pull it out and write the remainder in parentheses, x^2(2x + 3), read as "x squared times the quantity of two x plus three."
Repeat the process for the second set of terms. The GCF is 4, so pull it out and write the remainder in parentheses, 4(2x + 3), read as "four times the quantity of two x plus three." Notice that the terms inside the parentheses match. This is the key to factoring by grouping.
Rewrite the terms, multiplying the parenthetical terms by the outside terms, (2x + 3)(x^2 + 4), read as "the quantity of two x plus three times the quantity of x squared plus four."
Examine the expression 9x^ 2 -- 4, "nine x squared minus four." This is a binomial: two terms that are the difference of each other. It is called a difference of squares because although both the first and last terms are square, when you are factoring out, the middle term will disappear.
Find the square root of the first term, 9x^2, which is 3x. Find the square root of the last term, 4, which is 2. Write the square roots in parenthetical notation, using one negative and one positive sign. Read (3x + 2)(3x -- 2) as "the quantity of three x plus two times the quantity of three x minus two."
Distribute the factored solution to check the work. Multiply the first terms, 3x x 3x = 9x^2. Multiply the outside terms, 3x x -2x equals -6x. Multiply the inside terms, 2 x 3x = 6x. Multiply the last terms, 2 x -2 = -4.
Combine like terms, -6x + 6x. The terms cancel each other out and leave the original polynomial.
Examine the expression 49a^2 -- 70a + 25, which you can read as "forty-nine a squared minus seventy a plus twenty-five." This polynomial has three terms, making it a trinomial. Both the first and last terms are perfect squares.
Find the square roots of the first term, which is 7a, and of the last term, which is 5.
Write the terms in parenthetical notation, using all negative signs, (7a -- 5)(7a -- 5), read as "the quantity of seven a minus five times seven a minus five."
Redistribute to check your work, 7a x 7a = 49a^2, 7a x -5 = -35a, -5 x 7a = -35a and -5 x -5 = 25.
Combine like terms, -35a -- 35a = -70a, which leaves the original polynomial.
Examine the expression y^2 -- 16y + 28, read as "y squared minus sixteen y plus twenty-eight." This is another trinomial but the last term is not a square. The inside term must still equal the sum of the products of the inside and outside terms.
Write out the factors of 28, 1 x 28, 2 x 14 and 4 x 7. Because 28 has so many factors, you will have to try each one to see if the middle term equals the sum of the products. An easy way to do this is to look at the factors themselves. Do 4 and 7 equal 16? No, but 2 and 14 do.
Write the square root of y squared, which is y, and the set of factors in parenthetical notation, (y -- 14)(y -- 2), read as "the quantity of y minus fourteen times the quantity of y minus two."
Check your work by distributing. y x y = y^2, y x -2 = -2y, -14 x y = -14y and -14 x -2 = 28.
Combine like terms, -14y -- 2y = -16y, which leaves the original polynomial.