SOLUTION: The expression 8x^3 + ax^2 + bx - 9 leaves remainders -95 and 3 when divided by x + 2 and 2x - 3 respectively. Calculate the value of a and of b.

Algebra ->  Test -> SOLUTION: The expression 8x^3 + ax^2 + bx - 9 leaves remainders -95 and 3 when divided by x + 2 and 2x - 3 respectively. Calculate the value of a and of b.      Log On


   

Question 1183387: The expression 8x^3 + ax^2 + bx - 9 leaves remainders -95 and 3 when divided by x + 2 and 2x - 3 respectively. Calculate the value of a and of b.
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!



given:
+8x%5E3+%2B+ax%5E2+%2B+bx+-+9
x+%2B+2 is a factor of given polynomial
x+%2B+2+=+0
x+=+-+2
Remainder = -95
2x+-+3 is a factor of given polynomial
2x+-+3=0
2x+=3
x=3%2F2
Remainder =3

8x%5E3+%2B+ax%5E2+%2B+bx+-+9+..........Put the value of x
8%28-2%29%5E3+%2B+a%28-2%29%5E2+%2B+b%28-2%29+-+9=-95
a%28-2%29%5E2+-+2+b+-+73+=+-95
4a+-2b++=-95%2B73
4a+++=2b-22...........solve for a
a+=2b%2F4+-22%2F4
a+=b%2F2++-11%2F2...........eq.1


8%283%2F2%29%5E3+%2B+a%283%2F2%29%5E2+%2B+b%283%2F2%29+-+9=3
8%2827%2F8%29+%2B+a%289%2F4%29+%2B+b%283%2F2%29+=3%2B9
27+%2B+a%289%2F4%29+%2B+b%283%2F2%29+=12
a%289%2F4%29+%2B+b%283%2F2%29++=12-27
a%289%2F4%29+%2B+b%283%2F2%29++=-15.......both sides multiply by 4
9a+%2B+6b+=-60.......simplify
3a+%2B+2b+=-20.......solve for a
3a=-20-2b
a=-20%2F3-2b%2F3...........eq.2

from eq.1 and eq.2
b%2F2++-11%2F2=-20%2F3-2b%2F3 ......solve for b
b%2F2+%2B+2b%2F3=11%2F2-20%2F3
%287%2F6%29b=-7%2F6
b++=-1

go to
a+=b%2F2++-11%2F2...........eq.1, substitute+b
a+=%28-1%29%2F2++-11%2F2...........eq.1
a+=-6

then your expression is:
8x%5E3+-6x%5E2+-x+-+9


check reminders:
%288x%5E3+-6x%5E2+-x+-+9%29%2F%28x%2B2%29=8x%5E2+-+22+x+%2B+43-95%2F%28x%2B2%29
8x%5E3+-+6+x%5E2+-+x+-+9+=+%28%288+x%5E2+-+22+x+%2B+43%29%29%2A%28x+%2B+2%29++-95


%288x%5E3+-6x%5E2+-x+-+9%29%2F%282x-3%29=4+x%5E2+%2B+3+x+%2B+4%2B3%2F%282x+-+3%29
8+x%5E3+-+6+x%5E2+-+x+-+9+=+%28%284+x%5E2+%2B+3+x+%2B+4%29%29%2A%282+x+-+3%29+%2B+3

Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!

The expression 8x^3 + ax^2 + bx - 9 leaves remainders -95 and 3 when divided by x + 2 and 2x - 3 respectively. Calculate the value of a and of b.
As usual, if you ant to "TORTURE" yourself, then go along with that CONFUSING, TIME-CONSUMING, 
and UNNECESSARY PARTIAL-setup by that woman who responded,

matrix%281%2C3%2C+f%28x%29%2C+%22=%22%2C+8x%5E3+%2B+ax%5E2+%2B+bx+-+9%29
With remainder from factor x + 2 being - 95, we see that x + 2 = 0 ===> x = - 2, which means that: 
                                                                                                                                 
                                                                         +matrix%281%2C6%2C+-+22%2C+%22=%22%2C+4a+-+2b%2C+%22-----%22%2C+eq%2C+%22%28i%29%22%29
With remainder from factor 2x - 3 being 3, we see that 2x - 3 = 0 ===> x = 3%2F2, which means that: 
                                                                        
                                                                         
     - 22 =  4a - 2b --- eq (i)
     - 60 =  9a + 6b --- eq (ii)
     - 66 = 12a - 6b --- Multiplying eq (i) by 3 ------ eq (iii)
    - 126 = 21a -------- Adding eqs (ii) & (iii)
 highlight_green%28matrix%281%2C5%2C+%28-+126%29%2F21%2C+%22=%22%2C+-+6%2C+%22=%22%2C+a%29%29

     - 22 = 4(- 6) - 2b ----- Substituting - 6 for a in eq (i)
     - 22 = - 24 - 2b
- 22 + 24 = - 2b
        2 = - 2b
       highlight_green%28matrix%281%2C5%2C+2%2F%28-+2%29%2C+%22=%22%2C+-+1%2C+%22=%22%2C+b%29%29