Question

if 6 x^4+ 8 x^3- 5x^2 + a x + b is divisible by 2x^2 - 5 find a and b

Answer 1

Answer:

a = -20,  b = -25

Step-by-step explanation:

Here are a couple of approaches. Use whichever is appropriate for your class work.

Method 1

Doing long division, we get

6x⁴ + 8x³ - 5x² + ax + b = ( 2x² - 5 ) ( 3x² + 4x + 5 )  +  (20+a)x + (b+25).

As the remainder is zero, we have

20 + a = 0  =>  a = -20

and

b + 25 = 0  =>  b = -25

Method 2

Let x be assigned a value such that 2x² - 5 = 0.  Then...

6x⁴ + 8x³ - 5x² + ax + b = 0

=> 3(2x²)(2x²) + 8x(2x²) - 5(2x²) + 2ax + 2b = 0

=> 3(5)(5) + 8x(5) - 5(5) + 2ax + 2b = 0

=> 75 + 40x - 25 + 2ax + 2b = 0

=> (20+a)x + (25+b) = 0.

As this holds for two different values of x (the two solutions of 2x²-5=0), it follows that

20 + a = 0  =>  a = -20

and

25 + b = 0  =>  b = -25