Question Video: Factorisation by Grouping Mathematics

Factorize fully 4π₯π + π₯π + 4π¦π + π¦π.

05:46

Video Transcript

Factorize fully the expression four π₯π plus π₯π plus four π¦π plus π¦π.

This expression has four terms. Weβre going to answer this question using the method of factorizing by grouping. Iβll begin by separating the expression into two pairs of terms and then looking at them individually. Letβs start with the first half of the expression.

Both terms here have a common factor of π₯, and so they can be factorized by π₯. We can therefore write the first two terms as π₯ multiplied by a bracket. Inside the bracket, we need four π for the first term and then plus π for the second. You could check by expanding the bracket. This does indeed give four π₯π plus π₯π. So we factorized the first pair of terms. Now letβs consider the second pair.

Both terms are positive and they have a common factor of π¦. So we can take π¦ out as a common factor, writing the second half of the expression as plus π¦ multiplied by a bracket. Inside the bracket, we need to have four π for the first term so that it gives four π¦π when multiplied by π¦, and we need to have plus π for the second so that it gives plus π¦π when multiplied by π¦. So now we factorized the two halves of the expression.

And what youβll notice is that they have a common bracket of four π plus π. Now this isnβt a coincidence. This is what I hoped would happen when I decided to factorize by grouping. As the two halves of the expression have a common bracket of four π plus π, we can bring this bracket out as a common factor. The whole expression is therefore four π plus π multiplied by a second bracket. The terms in the second bracket are what four π plus π has been multiplied by in the two halves of the expression.

So thatβs π₯ for the first half and plus π¦ for the second. This gives the fully factorized form of the original four term expression. Itβs equal to four π plus π multiplied by π₯ plus π¦. Now you may be wondering well what would happen if the terms in the original expression have been written in a different order. So Iβve chosen to swap the middle two terms around, giving four π₯π plus four π¦π plus π₯π plus π¦π.

Iβll now attempt to use factorizing by grouping on this expression. So Iβve divided it up into two pairs of terms as before. Looking at the first pair of terms, four π₯π plus four π¦π, we can see that we have a common factor of four π. Their remaining factors, which need to be included inside the bracket, are π₯ and plus π¦. Now letβs consider the second half of this expression. Both terms are positive, and they have a common factor of π.

So we can write the second half as plus π multiplied by a bracket. The leftover factors to be included in the bracket are π₯ and plus π¦. Now letβs look at this expression. We can see that the two halves do again have a common bracket. But this time, the common bracket is π₯ plus π¦. We can bring this bracket out as a common factor, leading to π₯ plus π¦ multiplied by a second bracket, which we now need to fill in.

The terms that go in the second bracket are what π₯ plus π¦ has been multiplied by each time. So thatβs four π for the first half of the expression and plus π for the second. Now compare this with the answer that we already found. Youβll notice that we have the same two brackets each time, but theyβre just in a different order.

This means that we have the same answer, because of course multiplication is commutative. It doesnβt matter which order you multiply two numbers or two expressions together in. Therefore, we could write the brackets either way round. What weβve seen then is that if the terms of the original expression are a different order, we still get the same result for the factorized form.

Letβs consider one final possible reordering of the terms. So Iβve brought the final term π¦π and moved it to become the second term of the expression. So we now have four π₯π plus π¦π plus π₯π plus four π¦π. Suppose we attempt to factorize by grouping as weβve just done. Looking at the first two terms, four π₯π and plus π¦π, we see that they have no common factors. The same is true of the second pair of terms. They also have no common factors.

Does this mean that the expression doesnβt factorize? Well, no. As weβve already seen twice, it can be factorized into two brackets. It just means that this particular ordering of the terms doesnβt work for factorizing by grouping. If youβre answering a question like this and factorizing by grouping doesnβt appear to be working, try reordering the terms. The answer to this problem was four π plus π multiplied by π₯ plus π¦.