Ballinteer Institute Algebra Cubic Functions

The Cubic equation has three roots such that This information although no longer on the LH course can be use to good effect in certain situations .

Example 1: if X-3 and X - 2 are factors of find k and l and the third factor .

If X - 3 is a factor this means x = 3 is a root , similarly x =2 is also a root . Let us call the 3rd root a ().From above To find l sub x = 2 into the equation or use rule (2) above 8 -4(4)+2l+6 = 0 This gives l = 1 .

Conjugate Roots Theorem : If is a root of Then

p - iq is also a root . also if is a root of the above equation where all the coefficients are rational ( no surds) then is also a root .

Example (Q2P1,1999) If 3 + i is a root of find k , and the other rooots .

From above we know that 3 - i is also a root , call the third root of the equation a . Using (3) above

Example (Q1,P1,1999) If is a factor of one way to do this is to regard as the difference of two cubes this gives this gives

Calculus and the roots of cubic equations ;

(1) Find the y coordinates of the turning points of the curve ( y coordinate of the Max and Min )

The Cubic equation f(x) = 0 has Three real and different roots if .

The Cubic equation f(x) = 0 has Three real roots two of which are equal if (Ymax)(Ymin) = 0 .

The Cubic equation f(x) = 0 has Three roots two of which are complex if (Ymax)(Ymin) > 0 .

The Cubic equation f(x) = 0 has Three roots all equal if no max or min exists .

Example Q7P1LH1999.Find the values of k for which has (1) Three real and different roots .(2)Three real roots two of which are equal .

(1)For three real and different roots

(2)For three roots two of which are equal

Note has no real solutions