1. Expanding parentheses

Using distributive property, we can expand parentheses in an expression. Consider the expression \mathbf{4(3x - 1)}.

In the expression \mathbf{4(3x - 1)}, we can simplify it by multiplying each term inside the parentheses, \mathbf{3x} and \mathbf{-1}, by the outside factor, which is \mathbf{+4} as follows.

{4(3x - 1) = 4 \times 3x + 4 \times (-1)}

= 12x - 4

Therefore, \mathbf{12x - 4} is the simplified answer for the expression \mathbf{4(3x - 1)}.

• Now let’s see how we can simplify the expression -3(2y - 3).

In the expression -3(2y - 3), we can simplify it by multiplying each term inside the parentheses, 2y and -3, by the outside factor, which is -3 as follows.

            -3(2y - 3) = (-3) \times 2y - 3 \times (-3)

= -6y + 9

Therefore,  \mathbf{-6y + 9}is the simplified answer for the expression -3(2y - 3).

• Consider the following question.

The price of a pen is 2 dollars less than the price of a notebook. If you buy 8 notebooks and 5 pens, find the difference in total cost between the notebooks and pens.

Let x denote the price of a notebook. Since the price of a pen is 2 dollars less than the price of a notebook, we can write an expression for the price of a pen in terms of x and that will be x - 2. Now, we will first find the total cost when buying 8 books by multiplying x and 8.

Total cost of note books = x \times 8 = 8x

Then we can find the total cost when buying 5 pens by multiplying x - 1 and 5.

Total cost of pens = 5 \times (x - 2) = 5(x - 2)

Finally, we can write the difference in total cost between the notebooks and pens by subtracting two expressions as follows.

The difference in total cost between the notebooks and pens = 8x - 5(x - 2)

But, this cannot be the final answer as this expression can be simplified further. With knowledge of distributive law, we can expand -5(x - 2) and then combine like terms to get the final answer. Let’s see how we can simplify the above expression and get the final answer.

8x - 5(x - 2) = 8x - 5 \times x - 5 \times (-2)

= 8x - 5x + 10

= 3x + 10

Therefore, The difference in total cost between the notebooks and pens \mathbf{= 3x + 10}

• Let’s consider a few examples.

Examples

Simplify each of the following

1. (3p + 4) - ( p - 1)

   = 3p + 4 - p + 1

   = 3p - p + 4 + 1

   = 2p + 5

2. 2(x^2 - 1) + 3(x^2 +9)

   = 2x^2 + 2 × (-1) + 3x^2 + 3 × 9

   = 2x^2 - 2 - 3x^2 + 27

   = 2x^2 - 3x^2 - 2 + 27

   = - x^2 + 25

3. (a^2 - 6a +5) - (a^2 - 9)

   = a^2 - 6a +5 - a^2 + 9

   = a^2 - a^2 - 6a + 5 + 9

   = -6a + 14

4. n(3n - 2) + 3(2n + 1)

   = 3n^2 + n × (-2) + 3 × 2n + 3 × 1

   = 3n^2 - 2n + 6n + 3

   = 3n^2 + 4n + 3

2. Factoring

• Consider the expression 6x + 9 and factorize it.

As you can see in the above diagram, we can factor out 3 because both terms in the expression which are 6x and 9 are divisible by 3 without remainder. Therefore, when factoring, the product of factors can be expressed as follows.

6x + 9 = 3 \times 2x + 3 \times 3

= 3 (2x + 3)

• Consider the expression y(y - 3) + 2(y - 3).

In this expression, we have two terms which are y(y - 3) and 2(y - 3). So, these both terms have a common factor which is (y - 3). Therefore, we can factorize the expression y(y - 3) + 2(y - 3) as follows. 

y(y - 3) + 2(y - 3)

= (y - 3) \times y + (y - 3) \times 2

= (y - 3)(y + 2)

Let’s consider a few examples.

1. Factor out each of the following expression 

   1. 2x + 6

      = 2 \times x + 2 \times 3

      = 2(x + 3)

   2. 5a^2 - 10a

      = 5a \times a - 5a \times 2

      = 5a(a - 2)

   3. -3p^3 - 15p^2 - 6p

      = -3p \times p^2 -3p \times 5p -3p \times 2

      = -3p(p^2 + 5p + 2)

   4. m(m + 1) - 3(m + 1)

      = (m + 1) \times m + (m + 1) \times (-3)

      = (m + 1)(m - 3)

   5. (t - 3) + t^2 -3t

      = (t - 3) + t \times t + t \times (-3) 

      = 1(t -3) +t(t - 3)

      = (t - 3)(1 + t)

      = (t - 3)(t + 1)

2. A student has 4x pens. The number of pencils the student has is 2 more than the number of pens. What is the total number of pens and pencils? Put your answer as product of its factors.

Number of pencils = 4x + 2

Total pens and pencils = 4x + (4x + 2)

= 4x + 4x + 2

= 8x + 2

= 2 \times 4x + 2 \times 1

= 2(4x + 1)