Question

How do you simplify \[\left( {x + 9} \right)\left( {x - 2} \right)\]?

Answer 1

Hint: Here in this question to simplify the multiplication of polynomials by binomials while using the distributive property i.e., Each term in the second bracket must be multiplied by each term in the first bracket with the help of arithmetic operation we simplify the given polynomials. Hence we obtain the required result for the given question.

Complete step by step solution:
When Multiplying the polynomials the best method is using the distributive property.
Let’s understand that the distributive property is a rule that allows you to remove parentheses.
This is one of the most basic formulas and is frequently used in polynomial calculations. By using the distributive property, we can remove the parentheses and convert it into addition or subtraction equations.
The distribution property says that: \[a \cdot \left( {b + c} \right) = a \cdot b + a \cdot c\]
With more polynomials it gets a bit harder. We will do it the long way:
\[\left( {a + b} \right) \cdot \left( {c + d} \right) = \left( {a + b} \right) \cdot c + \left( {a + b} \right) \cdot d\]
We have distributed the second binomial, and we now distribute the first binomial (twice)
\[\left( {a + b} \right) \cdot c + \left( {a + b} \right) \cdot d = a \cdot c + b \cdot c + a \cdot d + b \cdot d\].
Now consider the given polynomial
\[ \Rightarrow \,\,\left( {x + 9} \right)\left( {x - 2} \right)\]
Remove the parentheses by using a distributive property. i.e., Each term in the second bracket must be multiplied by each term in the first bracket.
\[ \Rightarrow \,\,x\left( {x - 2} \right) + 9\left( {x - 2} \right)\]
Distribute the brackets
\[ \Rightarrow \,\,{x^2} - 2x + 9x - 18\]
Isolate the like terms
\[ \Rightarrow \,\,{x^2} + \left( { - 2 + 9} \right)x - 18\]
On simplification we get
\[ \Rightarrow \,\,{x^2} + 7x - 18\]
For further simplification to solve x, we can use a factorization method.
So, the correct answer is “${x^2} + 7x - 18$”.

Note: To multiply we use operation multiplication, multiplication of numbers is different from the multiplication of algebraic expression. In the algebraic expression it involves the both number that is constant and variables. Variables are also multiplied, if the variable is the same then the result will be in the form of an exponent.