Question

How do you simplify $3x(6x + 2) - 5x(x - 7y) - 3(2x + 5yy)$?

Answer 1

Hint: In this question, we need to simplify the given polynomial. Firstly, for the first term we will multiply $3x$ to $(6x + 2)$. Then we simplify the second term by multiplying $5x$ to $(x - 7y)$. After that we simplify the term by multiplying $3$ to $(2x + 5yy)$. We do this multiplication using distributive properties of addition and subtraction. After simplifying we combine the like terms and obtain the required algebraic expression.

Complete step by step solution:
Given an algebraic expression of the form,
$3x(6x + 2) - 5x(x - 7y) - 3(2x + 5yy)$ …… (1)
We are asked to simplify the expression given in the equation (1).
We simplify the given expression term by term and then combine all together to obtain the simplified form.
Here we make use of the distributive property of addition and subtraction for each of the terms in the given expression.
According to the distributive property of addition, the sum of two numbers multiplied by the third number is equal to the sum of each number multiplied by the third number.
For instance, $a \cdot (b + c) = a \cdot b + b \cdot c$ ……(2)
According to the distributive property of subtraction, the difference of two numbers multiplied by the third number is equal to the difference of each number multiplied by the third number.
For instance, $a \cdot (b - c) = a \cdot b - b \cdot c$ ……(3)
Now let us consider the first term given as, $3x(6x + 2)$
Here $a = 3x,$ $b = 6x,$ $c = 2$
Using distributive property of addition given in the equation (2), we get,
$3x(6x + 2) = 3x \cdot 6x + 3x \cdot 2$
$ \Rightarrow 3x(6x + 2) = 18{x^2} + 6x$
Now let us consider the second term given as, $5x(x - 7y)$
Here $a = 5x,$ $b = x,$ $c = 7y$
Using distributive property of subtraction given in the equation (3), we get,
$5x(x - 7y) = 5x \cdot x - 5x \cdot 7y$
$ \Rightarrow 5x(x - 7y) = 5{x^2} - 35xy$
Now let us consider the third term given as, $3(2x + 5yy)$
This can also be written as, $3(2x + 5{y^2})$
Here $a = 3,$ $b = 2x,$ $c = 5{y^2}$
Using distributive property of addition given in the equation (2), we get,
$3(2x + 5{y^2}) = 3 \cdot 2x + 3 \cdot 5{y^2}$
$ \Rightarrow 3(2x + 5yy) = 6x + 15{y^2}$
Now substituting all the obtained expressions in the equation (1) and simplifying we will get the desired answer.
Hence from equation (1), we get,
$3x(6x + 2) - 5x(x - 7y) - 3(2x + 5yy)$
$ \Rightarrow 18{x^2} + 6x - (5{x^2} - 35xy) - (6x + 15{y^2})$
Simplifying the above expression we get,
$ \Rightarrow 18{x^2} + 6x - 5{x^2} + 35xy - 6x - 15{y^2}$
Rearranging the above expression we get,
$ \Rightarrow 18{x^2} - 5{x^2} + 35xy + 6x - 6x - 15{y^2}$
Combining the like terms $18{x^2} - 5{x^2} = 13{x^2}$
Combining the like terms $6x - 6x = 0$
Hence we get,
$ \Rightarrow 13{x^2} + 35xy + 0 - 15{y^2}$
$ \Rightarrow 13{x^2} + 35xy - 15{y^2}$

Hence the simplified form of the algebraic expression $3x(6x + 2) - 5x(x - 7y) - 3(2x + 5yy)$
is given by $13{x^2} + 35xy - 15{y^2}$.

Note :
The distributive property applies to the multiplication of a number with the sum or difference of two numbers, i.e. this property holds true for multiplication over addition and subtraction. It simply states that multiplication is distributed over addition or subtraction.
Let a, b, c be any real numbers.
The distributive property of addition is given by,
$a \cdot (b + c) = a \cdot b + b \cdot c$
The distributive property of subtraction is given by,
$a \cdot (b - c) = a \cdot b - b \cdot c$
Also we must know which mathematical expressions have to be used to simplify the equation.