# Notable Products: Explanation And Solved Exercises

The **remarkable products** are algebraic operations, where multiplications of polynomials are expressed, which do not need to be solved traditionally, but with the help of certain rules the results of the same can be found.

Polynomials are multiplied by yes, therefore it is possible that they have a large number of terms and variables. To make the process shorter, the notable product rules are used, which allow multiplication without having to go term by term.

**Notable products and examples**

Each notable product is a formula that results from a factorization, made up of polynomials of several terms, such as binomials or trinomials, called factors.

Factors are the base of a power and have an exponent. When the factors are multiplied, the exponents must be added.

There are several remarkable product formulas, some are more used than others, depending on the polynomials, and they are the following:

**Binomial squared**

It is the multiplication of a binomial by itself, expressed as a power, where the terms are added or subtracted:

**to. Square sum binomial: it** is equal to the square of the first term, plus twice the product of the terms, plus the square of the second term. It is expressed as follows:

(a + b) ^{2} = (a + b) _{*} (a + b).

In the following figure you can see how the product develops according to the aforementioned rule. The result is called the trinomial of a perfect square.

**Example 1**

(x + 5) ² = x² + 2 (x * 5) + 5²

(x + 5) ² = x² + 2 (5x) + 25

(x + 5) ² = x² + 10x + 25.

**Example 2**

(4a + 2b) = (4a) ^{2} + 2 (4a _{*} 2b) + (2b) ^{2}

(4a + 2b) = 8a ^{2} + 2 (8ab) + 4b ^{2}

(4a + 2b) = 8a ^{2} + 16 ab + 4b ^{2} .

**b. Binomial of a squared subtraction:** the same rule of the binomial of a sum applies, only in this case the second term is negative. Its formula is the following:

(a – b) ^{2} = [(a) + (- b)] ^{2}

(a – b) ^{2} = a ^{2} + 2a _{*} (-b) + (-b) ^{2}

(a – b) ^{2} ^{ } = a ^{2} – 2ab + b ^{2} .

**Example 1**

(2x – 6) ^{2} = (2x) ^{2} – 2 (2x _{*} 6) + 6 ^{2}

(2x – 6) ^{2 } = 4x ^{2} – 2 (12x) + 36

(2x – 6) ^{2} = 4x ^{2} – 24x + 36.

**Product of conjugated binomials**

Two binomials are conjugated when the second terms of each have different signs, that is, the first is positive and the second negative or vice versa. It is solved by squaring each monomial and subtracting. Its formula is the following:

(a + b) _{*} (a – b)

In the following figure the product of two conjugated binomials is developed, where it is observed that the result is a difference of squares.

**Example 1**

(2a + 3b) (2a – 3b) = 4a ^{2} + (-6ab) + (6 ab) + (-9b ^{2} )

(2a + 3b) (2a – 3b) = 4a ^{2} – 9b ^{2} .

**Product of two binomials with a common term**

It is one of the most complex and rarely used notable products because it is a multiplication of two binomials that have a common term. The rule states the following:

- The square of the common term.
- Plus the sum the terms that are not common and then multiply them by the common term.
- Plus the sum of the multiplication of the terms that are not common.

It is represented in the formula: (x + a) _{*} (x + b) and is developed as shown in the image. The result is a non-perfect square trinomial.

(x + 6) _{*} (x + 9) = x ^{2} + (6 + 9) _{*} x + (6 _{*} 9)

(x + 6) _{*} (x + 9) = x ^{2} + 15x + 54.

There is a possibility that the second term (the different term) is negative and its formula is as follows: (x + a) _{*} (x – b).

**Example 2**

(7x + 4) _{*} (7x – 2) = (7x _{*} 7x) + (4 – 2) _{*} 7x + (4 _{*} -2)

(7x + 4) _{*} (7x – 2) = 49x ^{2} + (2) _{*} 7x – 8

(7x + 4) _{*} (7x – 2) = 49x ^{2} + 14x – 8.

It can also be the case that both different terms are negative. Its formula will be: (x – a) _{*} (x – b).

**Example 3**

(3b – 6) _{*} (3b – 5) = (3b _{*} 3b) + (-6 – 5) _{*} (3b) + (-6 _{*} -5)

(3b – 6) _{*} (3b – 5) = 9b ^{2} + (-11) _{*} (3b) + (30)

(3b – 6) _{*} (3b – 5) = 9b ^{2} – 33b + 30.

**Squared polynomial**

In this case there are more than two terms and to develop it, each one is squared and added together with twice the multiplication of one term with another; its formula is: (a + b + c) ^{2} and the result of the operation is a squared trinomial.

**Example 1**

(3x + 2y + 4z) ^{2} = (3x) ^{2} + (2y) ^{2} + (4z) ^{2} + 2 (6xy + 12xz + 8yz)

(3x + 2y + 4z) ^{2} = 9x ^{2} + 4y ^{2} + 16z ^{2} + 12xy + 24xz + 16yz.

**Binomial cubed**

It is a remarkably complex product. To develop it, the binomial is multiplied by its square, as follows:

**to. For the binomial cubed of a sum:**

- The cube of the first term, plus triple the square of the first term times the second.
- Plus the triple of the first term, times the second squared.
- Plus the cube of the second term.

(a + b) ^{3} = (a + b) _{*} (a + b) ^{2}

(a + b) ^{3} = (a + b) _{*} (a ^{2} + 2ab + b ^{2} )

(a + b) ^{3} = a ^{3} + 2a ^{2} b + ab ^{2} + ba ^{2} + 2ab ^{2} + b ^{3}

(a + b) ^{3} = a ^{3} + 3a ^{2} b + 3ab ^{2} + b ^{3} .

**Example 1**

(a + 3) ^{3} = a ^{3} + 3 (a) ^{2 }_{*} (3) + 3 (a) _{*} (3) ^{2} + (3) ^{3}

(a + 3) ^{3} = a ^{3} + 3 (a) ^{2 }_{*} (3) + 3 (a) _{*} (9) + 27

(a + 3) ^{3} = a ^{3} + 9 a ^{2} + 27a + 27.

**b. For the binomial cubed of a subtraction:**

- The cube of the first term, minus three times the square of the first term times the second.
- Plus the triple of the first term, times the second squared.
- Minus the cube of the second term.

(a – b) ^{3} = (a – b) _{*} (a – b) ^{2}

(a – b) ^{3} = (a – b) _{*} (a ^{2} – 2ab + b ^{2} )

(a – b) ^{3} = a ^{3} – 2a ^{2} b + ab ^{2} – ba ^{2} + 2ab ^{2} – b ^{3}

(a – b) ^{3} = **a ^{3} – 3a ^{2} b + 3ab ^{2}**

**– b**

^{3}.**Example 2**

(b – 5) ^{3} = b ^{3} + 3 (b) ^{2 }_{*} (-5) + 3 (b) _{*} (-5) ^{2} + (-5) ^{3}

(b – 5) ^{3} = b ^{3} + 3 (b) ^{2 }_{*} (-5) + 3 (b) _{*} (25) -125

(b – 5) ^{3} = b ^{3} – 15b ^{2} + 75b – 125.

**Cube of a trinomial**

It is developed by multiplying it by its square. It is a very extensive remarkable product because you have 3 terms cubed, plus three times each term squared, multiplied by each of the terms, plus six times the product of the three terms. Seen in a better way:

(a + b + c) ^{3} = (a + b + c) _{*} (a + b + c) ^{2}

(a + b + c) ^{3} = (a + b + c) _{*} (a ^{2} + b ^{2} + c ^{2} + 2ab + 2ac + 2bc)

(a + b + c) ^{3} = a ^{3} + b ^{3} + c ^{3} + 3a ^{2} b + 3ab ^{2} + 3a ^{2} c + 3ac ^{2} + 3b ^{2} c + 3bc ^{2} + 6abc.

**Example 1**

**Solved exercises of notable products**

**Exercise 1**

Expand the following binomial cubed: (4x – 6) ^{3} .

**Solution**

Remembering that a binomial cubed is equal to the first term cubed, minus three times the square of the first term times the second; plus the triple of the first term, times the second squared, minus the cube of the second term.

(4x – 6) ^{3} = (4x) ^{3} – 3 (4x) ^{2} (6) + 3 (4x) _{*} (6) ^{2} – (6) ^{2}

(4x – 6) ^{3} = 64x ^{3} – 3 (16x ^{2} ) (6) + 3 (4x) _{*} (36) – 36

(4x – 6) ^{3} = 64x ^{3} – 288x ^{2} + 432x – 36.

**Exercise 2**

Develop the following binomial: (x + 3) (x + 8).

**Solution**

There is a binomial where there is a common term, which is x and the second term is positive. To develop it, you only have to square the common term, plus the sum of the terms that are not common (3 and 8) and then multiply them by the common term, plus the sum of the multiplication of the terms that are not common.

(x + 3) (x + 8) = x ^{2} + (3 + 8) x + (3 _{*} 8)

(x + 3) (x + 8) = x ^{2} + 11x + 24.

**References**

- Angel, AR (2007).
*Elementary Algebra.*Pearson Education ,. - Arthur Goodman, LH (1996).
*Algebra and trigonometry with analytical geometry.*Pearson Education. - Das, S. (nd).
*Maths Plus 8.*United Kingdom: Ratna Sagar. - Jerome E. Kaufmann, KL (2011).
*Elementary and Intermediate Algebra: A Combined Approach.*Florida: Cengage Learning. - Pérez, CD (2010). Pearson Education.