Question

How do you factor \[{x^4} - 100\] ?

Answer 1

Hint: Here in this question, we have to find the factors for the equation \[{x^4} - 100\] . Let we write the given equation in the form \[{a^2} - {b^2}\] and then we have standard algebraic formula for the expression \[{a^2} - {b^2} = (a + b)(a - b)\] substituting the values we can find the factors.

Complete step-by-step answer:
The given equation is an algebraic equation or expression. Where it contains both variables and constants. Here the x is having power 4. So now consider \[{x^4}\] , this can be written as \[{x^4} = x.x.x.x = {x^2} \times {x^2}\]
Therefore \[{x^4} = {x^2} \times {x^2} = {\left( {{x^2}} \right)^2}\] ---- (1)
Now consider the number 100, this can be written as \[100 = 10 \times 10 = {10^2}\] -----(2)
Now consider the given equation \[{x^4} - 100\] ----- (3)
Substituting the equation (1) and equation (2) in equation (3) we have
 \[{x^4} - 100 = \left( {{{\left( {{x^2}} \right)}^2} - {{10}^2}} \right)\]
The above equation is in the form of \[{a^2} - {b^2}\] . We have standard formula for \[{a^2} - {b^2}\] as \[{a^2} - {b^2} = (a + b)(a - b)\] here \[a = {x^2}\] and \[b = 10\] . Therefore, we have
 \[{x^4} - 100 = \left( {{{\left( {{x^2}} \right)}^2} - {{10}^2}} \right)\]
Applying the formula, we have
 \[ \Rightarrow {x^4} - 100 = \left( {{x^2} + 10} \right)\left( {{x^2} - 10} \right)\]
Therefore, the factors of \[{x^4} - 100\] are \[\left( {{x^2} + 10} \right)\left( {{x^2} - 10} \right)\]
We can also find the factors by using the sum product rule.
According to the sum product rule we write the given equation as
 \[{x^4} - 100 = {x^4} + 10{x^2} - 10{x^2} - 100\]
Here we can take \[{x^2}\] as common from first two terms and -10 as common from last two terms, by taking common we have
 \[ \Rightarrow {x^4} - 100 = {x^2}({x^2} + 10) - 10({x^2} + 10)\]
On further simplification we have
 \[ \Rightarrow {x^4} - 100 = \left( {{x^2} + 10} \right)\left( {{x^2} - 10} \right)\]
Therefore, the factors of \[{x^4} - 100\] are \[\left( {{x^2} + 10} \right)\left( {{x^2} - 10} \right)\]
Hence we have found the factors for the given equation by using the formula and by sum product rule.
So, the correct answer is “ \[\left( {{x^2} + 10} \right)\left( {{x^2} - 10} \right)\] ”.

Note: The algebraic expressions are solved by using the factorisation. When we multiply the factors then we obtain the equation and which is equal to the given function. To find the factors for algebraic equations or expressions we have a standard algebraic formula and identities.