Question

The product of two expressions is x⁵+x³+x. If one of them is x²+x+1,find the other.

Answer 1

GIVEN :-

• product of two expression is x⁵ + x³ + x.
• One of the expression is x² + x + 1.

TO FIND :-

• The other expression.

SOLUTION :-

Let the other expression be k.

A.T.Q

=> ( x² + x + 1 ) ( k ) = x⁵ + x³ + x

=> k = ( x⁵ + x³ + x )/( x² + x + 1 )

$\boxed{\begin{array}{l |n|r}{\sf {x}^{2}+x+1 &\sf {x}^{5}+{x}^{3}+x & \sf {x}^{3}-{x}^2}+{x}\\&\sf {x}^{5}+{x}^{3}\:\:\:\: + {x}^{4}\\&(-)\:(-)\:\:\:\:(-)\\&\rule{55}{0.8}\\&\sf\qquad {-x}^{4}+x\\&\sf\qquad {-x}^{4}\:\:\:\:{-x}^{3}-{x}^{2}\\&\qquad (+)\:\:\:\:(+)(+)\\&\rule{55}{0.8}\\&\sf\qquad {x}^{3}+{x}^{2}+x\end{array}}$

=> k = x³ - x² + x

Hence The Other expression is x³ - x² + 2.

VERIFICATION :-

▪︎Product of 2 expression = x⁵ + x³ + x

=> (x² + x + 1) (x³ - x² + 2) = x⁵ + x³ + x

=> x²(x³ - x² + 2) + x(x³ - x² + 2) + 1(x³ - x² + 2) = x⁵ + x³ + x

=> x⁵ - 2x² + 2x² + x⁴ - x³ + 2x + x³ - x² + 2 = x⁵ + x³ + x

=> x⁵ + x³ + x = x⁵ + x³ + x

L.H.S = R.H.S

HENCE VERIFIED