let( x^2)-2x-3 is factor of f(x)=4(x^3)+b(x^2)+9ax+15,then find the value of A and B, and the theros of function

Question

THE QUSTION IS BASED ON ALGEBRA THEOREMS AND ASSOCIATED MATHEMATICAL PRINCIPLES

Answer 1

x²-2x-3=(x+1)(x-3) so -1 and 3 are zeroes and must therefore be solutions to:

4x³+bx²+9ax+15=0.

By substituting the zeroes in the above, we get a system of equations for a and b:

x=-1 gives us -4+b-9a+15=0⇒-9a+b=-11 ①

x=3 gives us 108+9b+27a+15=0⇒27a+9b=-123⇒9a+3b=-41 ②

Add ① and ②:

4b=-52, b=-52/4=-13. So rewriting ① we have 9a=b+11=-2 and a=-2/9.

So now we have the original cubic with values for a and b:

4x³-13x²-2x+15.

We know that this can have no more than three zeroes and we already have two of them. By looking at the coefficient of x³ and the constant we can deduce that the third factor is 4x-5 because 4x³/x²=4 and 15/-3=-5. The x² term and constant come from the quadratic divisor x²-2x-3.

So the zeroes are -1, 3 and 5/4.