Formulas of abbreviated multiplication

When calculating algebraic polynomials, simplified multiplication formulas are used to simplify calculations. There are seven such formulas. They all need to know by heart.

It should also be remembered that instead of a and b in the formulas there can be both numbers and any other algebraic polynomials.

Difference of squares

The difference of squares of two numbers is equal to the product of the difference of these numbers and their sum.

a 2 - b 2 = (a - b) (a + b)

Examples:

• 15 2 - 2 2 = (15 - 2) (15 + 2) = 13 x 17 = 221
• 9a 2 - 4b 2 with 2 = (3a - 2bc) (3a + 2bc)

Square amount

The square of the sum of two numbers is equal to the square of the first number plus the doubled product of the first by the second plus the square of the second number.

(a + b) 2 = a 2 + 2ab + b 2

Note that using this short multiplication formula makes it easy to find the squares of large numbers without using a calculator or column multiplication. Let us explain by example:

Find 112 2 .

• We decompose 112 into a sum of numbers whose squares we remember well.2
  112 = 100 + 1
• We write the sum of the numbers in brackets and put a square above the brackets.
  112 2 = (100 + 12) 2
• We use the sum square formula:
  112 2 = (100 + 12) 2 = 100 2 + 2 x 100 x 12 + 12 2 = 10 000 + 2 400 + 144 = 12 544

Remember that the square sum formula is also valid for any algebraic polynomials.

• (8a + s) 2 = 64a 2 + 16ac + c 2

Warning!

(a + b) 2 is not equal to a 2 + b 2

Difference square

The square of the difference of two numbers is equal to the square of the first number minus twice the product of the first by the second plus the square of the second number.

(a - b) 2 = a 2 - 2ab + b 2

Also worth remembering is a very useful conversion:

(a - b) 2 = (b - a) 2

The formula above is proved by simply opening the parentheses:

(a - b) 2 = a 2 - 2ab + b 2 = b 2 - 2ab + a 2 = (b - a) 2

Cube amount

The cube of the sum of two numbers is equal to the cube of the first number plus the tripled product of the square of the first number by the second plus the tripled product of the first by the square of the second plus the cube of the second.

(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3

Remembering this scary-looking formula is pretty simple.

• Learn that in the beginning comes a 3 .
• Two polynomials in the middle have coefficients 3.
• Recall that any number in the zero degree is 1. (a 0 = 1, b 0 = 1). It is easy to see that in the formula there is a decrease in the degree of a and an increase in the degree of b. This can be seen:
  (a + b) 3 = a 3 b 0 + 3a 2 b 1 + 3a 1 b 2 + b 3 a 0 = a 3 + 3a 2 b + 3ab 2 + b 3

Warning!

(a + b) 3 is not equal to a 3 + b 3

Difference cube

The cube of the difference of two numbers is equal to the cube of the first number minus the threefold product of the square of the first number by the second plus the threefold product of the first number by the square of the second minus the cube of the second.

(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3

This formula is remembered as the previous one, but only taking into account the alternation of the signs "+" and "-". Before the first member of a 3 is "+" (according to the rules of mathematics, we do not write it). So, the next member will be “-”, then again “+”, etc.

(a - b) 3 = + a 3 - 3a 2 b + 3ab 2 - b 3 = a 3 - 3a 2 b + 3ab 2 - b 3

Sum of cubes

Not to be confused with the cube amount!

The sum of cubes is equal to the product of the sum of two numbers by the incomplete squared difference.

a 3 + b 3 = (a + b) (a 2 - ab + b 2 )

The sum of cubes is the product of two brackets.

• The first bracket is the sum of two numbers.
• The second bracket is the incomplete square of the difference of numbers. The incomplete squared difference is the expression:
  
  a 2 - ab + b 2
  
  This square is incomplete, since in the middle, instead of the doubled product, it is the usual product of numbers.

Cube difference

Not to be confused with the difference cube!

The difference of cubes is equal to the product of the difference of two numbers by an incomplete squared sum.

a 3 - b 3 = (a - b) (a 2 + ab + b 2 )

Be careful when writing characters.

Application of reduced multiplication formulas

It should be remembered that all the formulas above are also used from right to left.

Many examples in the textbooks are designed for the fact that you use the formulas to collect the polynomial back.

Examples:

• a 2 + 2a + 1 = (a + 1) 2
• (ac - 4b) (ac + 4b) = a 2 c 2 - 16b 2