Difference of Two Squares
ence of two squares DIFFERENCE OF TWO SQUARESThis formula is used to factorise some algebraic expressions.
Example 5Solution: |
FACTORING THE SUM AND DIFFERENCE OF TWO CUBES
The formula for factoring a sum of two cubes is: | x3+y3=(x+y)(x2−xy+y2) | |
The formula for factoring a difference of two cubes is: | x3−y3=(x−y)(x2+xy+y2) | |
When teaching these factorization methods, it may be a good idea to encourage students to know one method for these factorizations rather than have them memorize two separate formulas.
First of all, the factorization is the product of a binomial and a trinomial. There is amnemonic device for remembering the signs that works when the binomial is put in front of the trinomial as above.
The other two special factoring formulas are two sides of the same coin: the sum and difference of cubes. These are the formulas: a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2)
You'll learn in more advanced classes how they came up with these formulas. For now, just memorize them. First, notice that the terms in each factorization are the same; then notice that each formula has only one "minus" sign. For the difference of cubes, the "minus" sign goes with the linear factor, a – b; for the sum of cubes, the "minus" sign goes in the quadratic factor, a2 – ab + b2. Some people use the mnemonic "SOAP" for the signs; the letters stand for "same" as the sign in the middle of the original expression, "opposite" sign, and "always positive". a3 ± b3 = (a [same sign] b)(a2 [opposite sign] ab [always positive] b2)
Whatever method helps you best keep these formulas straight, do it, because you should not assume that you'll be given these formulas on the test. You really should know them. Note: The quadratic part of each cube formula does not factor, so don't attempt it.
When you have a pair of cubes, carefully apply the appropriate rule. By "carefully", I mean "using parentheses to keep track of everything, especially the
Example 5Solution: |
FACTORING THE SUM AND DIFFERENCE OF TWO CUBES
The formula for factoring a sum of two cubes is: | x3+y3=(x+y)(x2−xy+y2) | |
The formula for factoring a difference of two cubes is: | x3−y3=(x−y)(x2+xy+y2) | |
When teaching these factorization methods, it may be a good idea to encourage students to know one method for these factorizations rather than have them memorize two separate formulas.
First of all, the factorization is the product of a binomial and a trinomial. There is amnemonic device for remembering the signs that works when the binomial is put in front of the trinomial as above.
The other two special factoring formulas are two sides of the same coin: the sum and difference of cubes. These are the formulas: a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2)
You'll learn in more advanced classes how they came up with these formulas. For now, just memorize them. First, notice that the terms in each factorization are the same; then notice that each formula has only one "minus" sign. For the difference of cubes, the "minus" sign goes with the linear factor, a – b; for the sum of cubes, the "minus" sign goes in the quadratic factor, a2 – ab + b2. Some people use the mnemonic "SOAP" for the signs; the letters stand for "same" as the sign in the middle of the original expression, "opposite" sign, and "always positive". a3 ± b3 = (a [same sign] b)(a2 [opposite sign] ab [always positive] b2)
Whatever method helps you best keep these formulas straight, do it, because you should not assume that you'll be given these formulas on the test. You really should know them. Note: The quadratic part of each cube formula does not factor, so don't attempt it.
When you have a pair of cubes, carefully apply the appropriate rule. By "carefully", I mean "using parentheses to keep track of everything, especially the