Question

By how much is ${a^4} - 4{a^2}{b^2} + {b^4}$ less than ${a^4} + 8{a^2}{b^2} + {b^4}$ ?
A.$12{a^2}{b^2}$
B.-12ab
C.12ab
D.$ - 12{a^2}{b^2}$

Answer 1

Hint: We need to find that by what amount the expression${a^4} + 8{a^2}{b^2} + {b^4}$is greater than ${a^4} - 4{a^2}{b^2} + {b^4}$. We will find the amount required by subtracting the equation ${a^4} - 4{a^2}{b^2} + {b^4}$ from ${a^4} + 8{a^2}{b^2} + {b^4}$. By doing this, we will find the difference in their values and the difference will be the required amount.

Complete step-by-step answer:
We are given two algebraic expressions: ${a^4} - 4{a^2}{b^2} + {b^4}$and${a^4} + 8{a^2}{b^2} + {b^4}$.
We are required to find the amount by which the equation${a^4} - 4{a^2}{b^2} + {b^4}$is less than the equation ${a^4} + 8{a^2}{b^2} + {b^4}$.
For this, we will subtract ${a^4} - 4{a^2}{b^2} + {b^4}$ from ${a^4} + 8{a^2}{b^2} + {b^4}$i.e., we will determine the difference between these two algebraic equations.
$ \Rightarrow $ Difference = $({a^4} + 8{a^2}{b^2} + {b^4}) - ({a^4} - 4{a^2}{b^2} + {b^4})$
We can write this equation as:
$ \Rightarrow $ Difference = ${a^4} + 8{a^2}{b^2} + {b^4} - {a^4} + 4{a^2}{b^2} - {b^4}$
$ \Rightarrow $ Difference = ${a^4} - {a^4} + 8{a^2}{b^2} + 4{a^2}{b^2} + {b^4} - {b^4}$
Upon simplification, we can write this as:
$ \Rightarrow $ Difference = $12{a^2}{b^2}$
Therefore, we can say that the equation ${a^4} - 4{a^2}{b^2} + {b^4}$ is less than ${a^4} + 8{a^2}{b^2} + {b^4}$ by $12{a^2}{b^2}$.
Hence, option (A) is correct.

Note: In this question, you may go wrong with the change of signs while subtracting the equations as we are subtracting the whole equation not only the first term. You may get confused in the subtraction i.e., to decide from which equation you need to subtract the other one.
We have used algebraic expressions here which can be defined as: In mathematics, an algebraic expression is an expression formed with variables, integer constants and algebraic operations (addition, subtraction, etc.).