Factoring Nonquadratic Rational Expressions

• 1). Rewrite the expression so the terms appear in descending order. This means you should write the term with the largest exponent first, followed by the term with the second-largest exponent and so on, writing any constants farthest to the right. For example, rewrite -9p^2 + 6p^3 as 6p^3 -- 9p^2 because the exponent 3 is higher than the exponent 2.
  
  
• 2). Determine the greatest common factor, or GCF, which is the largest factor of all terms. Find the GCF of the coefficients, which in 6p^3 -- 9p^2 is 3. Find the GCF of the variables, which in this example is p^2. Combine the numeric and variable GCFs. In this case, you get 3p^2.
  
  
• 3). Factor out the GCF. Draw a set of parentheses and write the GCF to their left. Inside the parentheses, write the terms that, when multiplied by the GCF, produce the original equation. Basically, this is the distributive property reversed. The factored expression in the example reads 3p^2 (2p -- 3).
  
  
• 4). Check your answer by applying the distributive property. Multiply the term outside the parentheses by every term in the parentheses. The original expression -- 6p^3 -- 9p^2 in the example -- should result.
  
  

Factoring Quadratic Rational Expressions

• 1). Rewrite the expression with the terms in descending order. For example, in -3 + 4x^2 + 4x, write 4x^2 + 4x -- 3.
  
  
• 2). Multiply the coefficient of the first term by the third term, which is a constant. Here, multiply 4 and -3 to get -12.
  
  
• 3). List all factor pairs of this number, in this case -12, until you've found a set that adds up to the coefficient of the middle term, in this case 4. In this example, you want to find 2 numbers that when multiplied equal -12 and when added equal 4. The numbers are -2 and 6.
  
  
• 4). Rewrite the original equation so that the middle term is replaced by the factor pair found in the previous step. The example becomes 4x^2 -- 2x + 6x -- 3. Order doesn't matter: writing 4x^2 + 6x -- 2x -- 3 produces the same answer.
  
  
• 5). Calculate the GCF of the first two terms and factor it out as outlined in the previous section. In the example, factor out the GCF of 4x^2 -- 2x to obtain 2x (2x -- 1). Repeat this process with the second two terms, factoring 6x -- 3 into 3 (2x -- 1). The terms inside both sets of parentheses should be identical. Combine these factored expressions, rendering 2x (2x -- 1) + 3 (2x -- 1).
  
  
• 6). Draw two sets of parentheses side by side. In one, write the terms from the previous step that appeared outside the parentheses. In the example, you have (2x + 3). In the other, write the terms from the previous step that were already inside parentheses, (2x -- 1) in this case. In total, your answer is (2x + 3)(2x -- 1).
  
  
• 7). Check your solution by multiplying the two terms in the first set of parentheses by both terms in the second set of parentheses and then combining like terms. The original equation, 4x^2 + 4x -- 3, should result.