Question

How do you simplify \[{\left( { - 3x + 1} \right)^3}\]?

Answer 1

Hint: In order to determine the multiplication of the given expression, first rewrite the expression in the product form of the three binomials. Now combine the first two binomials obtained to convert them into one polynomial by combining like terms using distributive property of multiplication Now repeat the same steps to multiply the remaining product of polynomial and binomial.

Complete step-by-step solution:
The binomial concept will come under the topic of algebraic expressions. The algebraic expression is a combination of variables and constant. The alphabets are known as variables and the numerals are known as constants. In algebraic expression or equation, we have 3 types namely, monomial, binomial and polynomial.
A polynomial equation with two terms joined by the arithmetic operation + or – is called a binomial equation.
Now let us consider the given expression \[{\left( { - 3x + 1} \right)^3}\]
It can be written as
\[ \Rightarrow \,\left( { - 3x + 1} \right)\left( { - 3x + 1} \right)\left( { - 3x + 1} \right)\]
From above we can clearly see that we have obtain the three binomials as \[\,\left( { - 3x + 1} \right),\left( { - 3x + 1} \right)\,and\,\left( { - 3x + 1} \right)\]which are same.
Now first we will be multiplying first two binomials in order to combine them into one factor
\[ \Rightarrow \,\left( { - 3x\left( { - 3x + 1} \right) + 1\left( { - 3x + 1} \right)} \right)\left( { - 3x + 1} \right)\]
On multiplication we get
\[ \Rightarrow \,\left( {9{x^2} - 3x - 3x + 1} \right)\left( { - 3x + 1} \right)\]
Now combining the like terms having same variable and same power
\[ \Rightarrow \,\left( {9{x^2} - 6x + 1} \right)\left( { - 3x + 1} \right)\]
We can see, now we have obtained the product of a polynomial and a binomial.
Multiplying both the remaining factors, we get
\[ \Rightarrow \,9{x^2}\left( { - 3x + 1} \right) - 6x\left( { - 3x + 1} \right) + 1\left( { - 3x + 1} \right)\]
Applying distributive property of multiplication for every term as $A\left( {B + C} \right) = AB + AC$
\[ \Rightarrow \, - 27{x^3} + 9{x^2} + 18{x^2} - 6x - 3x + 1\]
Combining all the like terms , we get
\[ \Rightarrow \, - 27{x^3} + 27{x^2} - 9x + 1\]
Hence, we have multiplied our given polynomial.

Therefore, we have \[{\left( { - 3x + 1} \right)^3} = \, - 27{x^3} + 27{x^2} - 9x + 1\]

Note: We can also solve the given question by using the standard algebraic formula \[{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)\]. Here \[a = - 3x\]and \[\;b = 1\]. On substituting the values in the formula, we get
\[
   \Rightarrow {\left( { - 3x + 1} \right)^3} = {\left( { - 3x} \right)^3} + {\left( 1 \right)^3} + 3\left( { - 3x} \right)\left( 1 \right)\left( { - 3x + 1} \right) \\
   \Rightarrow {\left( { - 3x + 1} \right)^3} = \, - 27{x^3} + 27{x^2} - 9x + 1 \\
 \]

And few points to remember
1.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 exponent.
2. Like terms are the terms which are having the same variable and same power.
3. Make sure all the terms are all combined properly.
4. To cross-check your answer , use the alternate method to verify your result.