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.
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{{{matrix(1,3, f(x), "=", 8x^3 + ax^2 + bx - 9)}}}
With remainder from factor x + 2 being - 95, we see that x + 2 = 0 ===> x = - 2, which means that: 
                                                                     {{{matrix(4,3, f(- 2), "=", 8(- 2)^3 + a(- 2)^2 + b(- 2) - 9, 
- 95, "=", 8(- 8) + 4a - 2b - 9, - 95, "=", - 64 + 4a - 2b - 9, - 95 + 73, "=", 4a - 2b)}}}                                                            
                                                                         {{{ matrix(1,6, - 22, "=", 4a - 2b, "-----", eq, "(i)")}}}</pre><pre>With remainder from factor 2x - 3 being 3, we see that 2x - 3 = 0 ===> x = {{{3/2}}}, which means that: 
                                                                        {{{ matrix(5,3, f(3/2), "=", 8(3/2)^3 + a(3/2)^2 + b(3/2) - 9, 
3, "=", 8(27/8) + 9a/4 + 3b/2 - 9, 3, "=", 27 + 9a/4 + 3b/2 - 9, 3 - 18, "=", 9a/4 + 3b/2, - 15, "=", 9a/4 + 3b/2)}}}
                                                                         {{{ matrix(1,7, - 60, "=", 9a + 6b, "=", "-----", eq, "(ii)")}}}
     - 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(matrix(1,5, (- 126)/21, "=", - 6, "=", a))}}}

     - 22 = 4(- 6) - 2b ----- Substituting - 6 for a in eq (i)
     - 22 = - 24 - 2b
- 22 + 24 = - 2b
        2 = - 2b
       {{{highlight_green(matrix(1,5, 2/(- 2), "=", - 1, "=", b))}}}</pre>