how do you show that 1/2 is a root of multiplicity 2 of the equation: 4Xcubed 8Xsquared-11X 3=0

Question

show that 1/2 is a root of multiplicity 2 of the equation: 4XCubed+8Xsquared-11X+3=0

Answer 1

If X=½ is a double root then the cubic can be written:

(X-½)²(X-a)=0 where a is the third root.

(X²-X+¼)(X-a)=0. Multiply through by 4:

(4X²-4X+1)(X-a)=4X³-4X²+X-4aX²+4aX-a=4X³+8X²-11X+3.

Equating coefficients:

X²: -4-4a=8, so -4-8=4a, -12=4a, a=-3.

X: 1+4a=-11, so 4a=-12, a=-3. So we have a consistent value for a and the cubic is factorisable:

(X-½)(X-½)(X+3), showing that ½ is a multiple root.