Question

How do you rewrite the expression \[\left( 6x+15 \right)\div 3\] in standard form using the distributive property ?

Answer 1

Hint:In the given question, we have been asked to write an expression using distributive property. In order to solve the question, first we need to change the divide sign into multiplication sign by putting multiplication of reciprocal in place of divide sign, later we need to use the distributive property of multiplication i.e. \[a\times \left( b+c \right)=\left( a\times b \right)+\left( a\times c \right)\] to simplify the given expression.

Formula used:
Distributive property of multiplication over subtraction states that the product of a number say ‘a’ with product of two numbers, say ‘b’ and ‘c’ is equal to the subtraction of products of the number ‘a’ multiplied separately with ‘b’ and ‘c’:
\[a\times \left( b-c \right)=ab-ac\]

Complete step by step answer:
We have given that, \[\left( 6x+15 \right)\div 3\]
To change the divide sign into the multiplication sign;
To divide: multiply the reciprocal.
Applying this in the above expression, we get
\[\left( 6x+15 \right)\times \dfrac{1}{3}\]
Rewrite the above expression as,
\[\dfrac{1}{3}\times \left( 6x+15 \right)\]
Using the distributive property of multiplication, i.e.
\[a\times \left( b+c \right)=\left( a\times b \right)+\left( a\times c \right)\]
Applying the distributive property in the above expression, we get
\[\dfrac{1}{3}\times \left( 6x+15 \right)=\left( \dfrac{1}{3}\times 6x \right)+\left( \dfrac{1}{3}\times 15 \right)\]
Simplifying the above expression, we get
\[\dfrac{6x}{3}+\dfrac{15}{3}\]
Converting each fraction into simplest form, we get
\[2x+5\]
\[\therefore\left( 6x+15 \right)\div 3=2x+5\]

Hence, the standard form of the expression \[\left( 6x+15 \right)\div 3\] is equal to \[2x+5\].

Note:We note that the polynomial with degree 1 is called the linear polynomial. To solve these types of questions, we need to know there are 4 properties for the arithmetic operation called closure, commutative, associative and distributive. The distributive property of multiplication requires one more operation either addition or subtraction. We know that the Distributive property of multiplication over subtraction states that the product of a number say ‘a’ with product of two numbers, say ‘b’ and ‘c’ is equal to the subtraction of products of the number ‘a’ multiplied separately with ‘b’ and ‘c’:\[a\times \left( b-c \right)=ab-ac\].