How to use the abridged multiplication formulas

 

The algorithm to use the abridged multiplication formulae will help understanding on how to use these formulae for simple transformations.

As complex tasks can require additional transformations, then this type algorithm is unsuitable for them. However, in cases when a large number of simple exercises using the abridged multiplication formulae have been solved, the required formulae and transformations will be together.

Using the algorithm it is necessary to keep the task type in mind.

Sometimes it is required to transform an expression so that the factors appear and sometimes the opposite it is required to open the brackets.

Step 1

 

The first thing is to read the question carefully.

In the expression, which is to be transformed, you can try to see one of the abridged multiplication formulas:

  • it can be a left side of the formula (for example, in Task 1):

How to use the abridged multiplication formulas

  • and right side of the formula (for example, in Task 12):

How to use the abridged multiplication formulas

When trying to see a formula in the expression, one should understand that the following elements can take the place of the elements in the formula (i.e. in the place of a and b):

  • Numbers

Numbers

  • Letters

Letters

  • Expressions

Expressions

  • The powers of these elements can be arbitrary

The powers of these elements can be arbitrary

If you find a formula, then you can use it. Having used the formula, you can check the solution by using the formula in reverse.

Step 2

 

If you can’t find a formula straight away, then it is necessary to try to reduce the expression to the one of the formulae.

To this effect, it is necessary to represent each of the elements (in more complicated tasks, a group of elements) of the expression as the square or the cube.

  • if it is not possible to represent several elements as the squares or the cubes, then go to step 3 (for example, in Task 8);
  • if the highest power, which can be indentified with two elements is the second power (squared), then go to step 4 (for example, in Task 4);
  • if the highest power, which can be identified with two elements is the third power (cubed), then go to step 5 (for example, in Task 17);

Note

The expression elements can initially be expressions in a power, for example, a8. In this type of case, it is necessary to try and identify the square. It will be (a4)2. I.e. the highest power identified will be 2. Read the examples examined to get more detailed information.

Number 1 in the tasks can be represented as:

Number 1 in the tasks can be represented as

Or

Number 1 in the tasks can be represented as

Step 3

 

Try to find a common factor in the expression and then put it outside the brackets. Thereafter return to step 1.

If because of the transformations it has not been possible to achieve the required result, then either this expression cannot be solved using the abridged multiplication formulae, or it is necessary to perform additional transformations.

Step 4

 

After it has become possible to mark out the squares of two elements, it is necessary to see if there is the difference of the squares in this expression. If yes, then use the formula (for example, in Task 9):

How to use the abridged multiplication formulas

If there is no difference of two squares, then it is necessary to determine what is twice the product of them; and multiply the elements, which are raised to the power of 2: i.e. if the two elements are (a4)2 and (b3)2, then twice the product of them is as follows:

How to use the abridged multiplication formulas

Now it is necessary to find this twice the product in the expression, which you are solving. Sometimes you will have to carry out additional actions (for example, in Task 6).

If twice the product is present, then it is possible to collect the formula of the square of the sum or of the difference depending upon the sign of the doubled product (for example, in Task 5):

How to use the abridged multiplication formulas

If the expression contains just the product of these elements (rather than twice the product), then go to step 6.

If it has not been possible to find a product of two elements in the expression at all, then either this expression cannot be solved using the abridged multiplication formulae, or it is necessary to perform additional transformations.

Step 5

 

It is necessary to determine what the following will equal (for example, in Task 19):

The algorithm to use the abridged multiplication formulas

note that here it is necessary to multiply together the elements, raised to the power of three: i.e. if the two elements are  (a4)3 and (b3)3, then it is necessary to determine the following:

The algorithm to use the abridged multiplication formulas

Now it is necessary to find the products obtained in the expression, which you are solving. Sometimes you will have to carry out additional actions.

If there is a product, then it is possible to collect the formula of the cube of the sum or of the difference depending upon the signs in the following expression:

The algorithm to use the abridged multiplication formulas

If it has not been possible to mark out the formula of the cube of the sum or of the difference at all, then:

  • or for the sum (difference) of the 2 cubes use the formula of the sum (difference) of the cubes::

The algorithm to use the abridged multiplication formulas

  • or it is necessary to perform additional transformations;
  • or this expression cannot be solved using the abridged multiplication formulae.

 

Note:

If the signs in the expression under consideration alternate. then it is most likely that the formula of the cube of the difference is to be used:

The algorithm to use the abridged multiplication formulas

Step 6

 

If the expression comprises the squares of two elements and their product, then it is necessary to try to find a factor of the sum or difference of these elements in the expression (for example, in Task 22).

If the factor can be found then use the formula of the sum of the cubes or the formula of the difference of the cubes:

The algorithm to use the abridged multiplication formulas

Note:

If the signs in the expression under consideration alternate. then it most likely that the formula of the cube of the difference will be used.