Question

Simplify \[m\left( {{m^3} - {m^2} + 1} \right) - 2\left( {{m^4} - {m^3} - 1} \right) + {m^2}\left( {{m^2} - m + 2} \right)\]
A.\[2{m^2} + 5m + 2\]
B.\[2{m^2} + m + 2\]
C.\[{m^2} + m + 2\]
D.\[{m^2} + 2m + 2\]

Answer 1

Hint: Here we will open the brackets and multiply the outer terms with the terms inside the brackets using distributive property of multiplication. Then we will simplify the equation and cancel out the common terms with opposite sign. We will then solve it to get the simplified form of the given equation. Distributive property states that when a sum of two or more addends by a number then it gives the same result as multiplying each addend separately by the same number and then adding the products together.

Complete step-by-step answer:
Given expression is \[m\left( {{m^3} - {m^2} + 1} \right) - 2\left( {{m^4} - {m^3} - 1} \right) + {m^2}\left( {{m^2} - m + 2} \right)\].
We will simplify the equation by using the distributive property of multiplication. Therefore, we get
\[ \Rightarrow m\left( {{m^3} - {m^2} + 1} \right) - 2\left( {{m^4} - {m^3} - 1} \right) + {m^2}\left( {{m^2} - m + 2} \right) = {m^4} - {m^3} + m - \left( {2{m^4} - 2{m^3} - 2} \right) + {m^4} - {m^3} + 2{m^2}\]
Simplifying the equation, we get
\[ \Rightarrow m\left( {{m^3} - {m^2} + 1} \right) - 2\left( {{m^4} - {m^3} - 1} \right) + {m^2}\left( {{m^2} - m + 2} \right) = {m^4} - {m^3} + m - 2{m^4} + 2{m^3} + 2 + {m^4} - {m^3} + 2{m^2}\]
Adding the like terms, we get
\[ \Rightarrow m\left( {{m^3} - {m^2} + 1} \right) - 2\left( {{m^4} - {m^3} - 1} \right) + {m^2}\left( {{m^2} - m + 2} \right) = 2{m^4} - 2{m^3} + m - 2{m^4} + 2{m^3} + 2 + 2{m^2}\]
Now we will cancel out the common terms which can be canceled due to the opposite sign. Therefore, we get
\[ \Rightarrow m\left( {{m^3} - {m^2} + 1} \right) - 2\left( {{m^4} - {m^3} - 1} \right) + {m^2}\left( {{m^2} - m + 2} \right) = m + 2 + 2{m^2}\]
Now rearranging the above equation from the higher power of the variable to the constant term, we get
\[ \Rightarrow m\left( {{m^3} - {m^2} + 1} \right) - 2\left( {{m^4} - {m^3} - 1} \right) + {m^2}\left( {{m^2} - m + 2} \right) = 2{m^2} + m + 2\]
Hence the simplified form of \[m\left( {{m^3} - {m^2} + 1} \right) - 2\left( {{m^4} - {m^3} - 1} \right) + {m^2}\left( {{m^2} - m + 2} \right)\] is equal to \[2{m^2} + m + 2\].
So, option B is the correct option.

Note: We should also know that addition is the operation in which two numbers are combined to get the result. Subtraction is the operation which gives us the difference between the two numbers. Multiplication is the operation in which one number is added to itself for some particular number of times. Exponent is defined as the number which represents how many times a number is being multiplied to itself. If the exponent of a number is zero then the value of the number is 1 i.e. \[{a^0} = 1\].