Question

The factors of $ {x^4} + {x^2} + 25 $ are
a) $ ({x^2} + 3x + 5),({x^2} - 3x + 5) $
b) $ ({x^2} + 3x + 5),({x^2} + 3x - 5) $
c) $ ({x^2} + x + 5),({x^2} - x + 5) $
d) None of these

Answer 1

Hint: We will check each option one by one to find out which one is correct. As we are given four options out of which if the first three are incorrect then we will choose the last option and if any of the first three options is correct then the last option is of no use and can be discarded. On each of the first three options we are given two factors we will multiply them and find out if it generates the polynomial given to us which is $ {x^4} + {x^2} + 25 $ .

Complete step-by-step answer:
The given polynomial is $ {x^4} + {x^2} + 25 $
We will check the options one by one and find out which option is correct.
In option ‘a’ we are given $ ({x^2} + 3x + 5),({x^2} - 3x + 5) $
Let's multiply these factors,
 $ \Rightarrow ({x^2} + 3x + 5) \times ({x^2} - 3x + 5) $
 $ = {x^2}({x^2} - 3x + 5) + 3x({x^2} - 3x + 5) + 5({x^2} - 3x + 5) $
 $ = ({x^4} - 3{x^3} + 5{x^2}) + (3{x^3} - 9{x^2} + 15x) + (5{x^2} - 15x + 25) $
\[ = {x^4} - 3{x^3} + 5{x^2} + 3{x^3} - 9{x^2} + 15x + 5{x^2} - 15x + 25\]
\[ = {x^4} + {x^2} + 25\]
This is the required polynomial given in the question, we can choose option ‘a’.
Now ,as the factors of a polynomial are always unique so we can guarantee that option ‘b’ and ‘c’ cannot be the factors of $ {x^4} + {x^2} + 25 $ .
So, the correct answer is “Option A”.

Note: We neglected to check if options ‘b’ and ‘c’ are the factors of the given expression because the factors of a given algebraic expression has unique factors or the factors are linearly dependent on each other and the other factors did not satisfy these properties.