Why factorise

Step 4 : If we've done this correctly, our two new terms should have a clearly visible common factor. The hardest part is finding two numbers that multiply to give ac , and add to give b. It is partly guesswork, and it helps to list out all the factors. You can practice simple quadratic factoring. Well, one of the big benefits of factoring is that we can find the roots of the quadratic equation where the equation is zero.

We can also try graphing the quadratic equation. Thus, 2 x is a common factor. Since there is no other common factor, 2 x is the highest common factor.

We divide each term by and see what is left. The terms in the expression in the brackets have no common factor except 1 and so this expression cannot be factorised further. The highest common factor of the three terms is 2 ab so. In some instances, there may be no common factor of all the terms in a given expression. It may, however, be useful to factor in pairs. Note that the order in which the brackets are written and the order of the terms within the brackets do not matter.

Students will need some practice with this method, especially with the second step. Note: Different pairing of terms may or may not lead to a useful factorisation. Factoring can give us useful information regarding an expression as the following exercise shows. There are three special expansions and corresponding factorisations that frequently occur in algebra. The first of these is an identity known as the difference of squares. An identity is a statement in algebra that is true for all values of the pronumerals.

Hence the difference between the squares of two numbers equals their sum times their difference. One such application is to mental arithmetic. With practice this can be done mentally, provided the squares of integers up to about 20 are known. The difference of two squares can also be used to solve equations in which we only seek integer solutions.

These identities are harder to use than the difference of two squares and are probably best dealt with as special cases of quadratic factoring, discussed below.

The following example shows how these ideas can be cleverly combined to factor an expression that at first glance does not appear to factor. At first glance this expression does not appear to factor, since there is no identity for the sum of squares. However, by adding and subtracting the term , we arrive at a difference of squares. This expansion produces a simple quadratic.

We would like to find a procedure that reverses this process. We notice that the coefficient x of is the sum of the two numbers 2 and 5 in the brackets and that the constant term 10, is the product of 2 and 5.

This suggests a method of factoring. Hence to reverse the process, we seek two numbers whose sum is the coefficient of and whose produce is the constant term.

Clearly the solutions are 4 and 3 in either order , and no other numbers satisfy these equations. Students should try to mentally expand to check that their answers are correct. Also note that the difference of squares factorisation could also be done using this method. This is, however, not a good method to use. Whoa -- a bit harder to solve, but it's possible. Today let's figure out how factoring works and why it's useful. After the interactions are finished, we should get 6.

Hrm -- this is tricky. So let's fight with a trick of our own: we can make a different system to track the error in our original one this is mind-bending, so hang on. When are we happiest? When there's no difference:. If we have a system and the desired state, we can make a new equation to track the difference -- and try make it zero.

This is deeper than just "subtract 6 from both sides" -- we're trying to describe the error! Factoring the rescue. Imagine taking a pile of sticks our messy, disorganized system and standing them up so they support each other, like a teepee:. If Component A or Component B becomes 0, the structure collapses, and we get 0 as a result.

That is why factoring rocks: we re-arrange our error-system into a fragile teepee, so we can break it. We'll find what obliterates our errors and puts our system in the ideal state.

I've wondered about the real purpose of factoring for a long, long time. In algebra class, equations are conveniently set to zero, and we're not sure why. Here's what happens in the real world:.