How to find the values of a and b?

Question

If the polynomial 6x⁴ + 8x³ - 5x² + ax + b is exactly divisible by the polynomial 2x² - 5, then find the values of a and b.

Answer 1

Hint $\ $ Long divide then equate the remainder to $\,0\,$ to get equations for $\,a,b.$

Answer 2

The roots of $2x^2-5$ must be roots of our polynomial:

$$\left\lbrace \begin{array}{c} 6\cdot\frac{25}4+15\sqrt{\frac52}-\frac{25}2+\sqrt{\frac52}a+b=0\\ \phantom{1}\\ 6\cdot\frac{25}4-15\sqrt{\frac52}-\frac{25}2-\sqrt{\frac52}a+b=0 \end{array} \right.$$

Adding up the equations you get

$$50+2b=0$$

And substracting them:

$$30\sqrt{\frac52}+2\sqrt{\frac52}a=0$$

So $a=-15$, $b=-25$.

Answer 3

You can just write $$6x^4 + 8x^3 - 5x^2 + ax + b=(2x^2-5)(Ax^2+Bx+C)$$ Expand the rhs to get $$6x^4 + 8x^3 - 5x^2 + ax + b=2Ax^4+2Bx^3+(2C-5A)x^2-5Bx-5C$$ Now identify the coefficients of the differents powers of $x$. Using $x^4,x^3,x^2$, this leads to $A=3$, $B=4$,$C=5$.

I am sure that you can take from here.