Question

How do you multiply $({x^2} + 3x + 1)({x^2} + x + 1)$?

Answer 1

Hint: Certain properties are observed while performing arithmetic operations like addition, subtraction, multiplication and division on the numbers and algebraic expressions, so for making the calculations easier, we have generalized most of them. Commutative, associative and distributive properties are the three most widely used properties. To make the multiplications a lot easier we use the distributive property, so it is also called the distributive law of multiplication. According to this property, the product of a number (a) with a sum or difference of other two numbers is equal to the sum of the product of the number (a) with first number (b) and the product of the number (a) with the second number (c), that is, \[a(b + c) = ab + ac\] .

Complete step by step answer:
We will use the distributive property for multiplying ${x^2} + 3x + 1$ and ${x^2} + x + 1$ as follows –
$ \Rightarrow ({x^2} + 3x + 1)({x^2} + x + 1) = {x^2}({x^2} + x + 1) + 3x({x^2} + x + 1) + 1({x^2} + x + 1)$
Now, we will multiply each term outside the bracket with the whole bracket –
$ \Rightarrow ({x^2} + 3x + 1)({x^2} + x + 1) = {x^4} + {x^3} + {x^2} + 3{x^3} + 3{x^2} + 3x + {x^2} + x + 1$
Now we will group similar terms and perform the given arithmetic operation –
$ \Rightarrow ({x^2} + 3x + 1)({x^2} + x + 1) = {x^4} + 4{x^3} + 6{x^2} + 7x + 2$
Hence, $({x^2} + 3x + 1)({x^2} + x + 1)$ is equal to ${x^4} + 4{x^3} + 6{x^2} + 7x + 2$ .

Note: The equations given in the parenthesis are polynomial equations as the unknown quantity “x” is raised to a positive power. While multiplying ${x^2}$ with ${x^2}$ , we use a law of exponent which states that, when we have to multiply two exponents having same base but different powers, we add the powers keeping the base same, that’s why ${x^2}.{x^2} = {x^4}$ . Any two polynomial equations can be multiplied with each other by distributing each term individually.