Answer:
2. a -2b
Step-by-step explanation:
[tex]p(x) = 2 {x}^{3} + a {x}^{2} + bx + 3[/tex]
[tex]f(x) = 2 {x}^{2} + x - 3[/tex]
On factoring f(x) we get
2x^2 - 2x + 3x - 3 = 0
2x(x-1) +3(x-1) = 0
(2x + 3)(x-1) = 0
Thus x = -3/2 and x = 1 are the roots of f(x).
As f(x) divided p(x), -3/2 and 1 are the roots of p(x).
[tex]p( \frac{ - 3}{2} ) = 2( \frac{ - 3}{2} ) {}^{3} + a( \frac{ - 3}{2} ) {}^{2} + b( \frac{ - 3}{2} ) + 3 = 0[/tex]
From this we get: 3a - 2b = 5.
[tex]p(1) = 2(1) {}^{3} + a ({1})^{2} + b(1) + 3 = 0[/tex]
from this we get: a + b = -5
After solving 2 equations we get: a = -1 and b = -4
So p(x) = 2x^3 -x^2 -4x + 3.
So p(2) = 2(2)^3 - 2^2 - 4(2) + 3 = 7 = a -2b
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Verified answer
Answer:
2. a -2b
Step-by-step explanation:
[tex]p(x) = 2 {x}^{3} + a {x}^{2} + bx + 3[/tex]
[tex]f(x) = 2 {x}^{2} + x - 3[/tex]
On factoring f(x) we get
2x^2 - 2x + 3x - 3 = 0
2x(x-1) +3(x-1) = 0
(2x + 3)(x-1) = 0
Thus x = -3/2 and x = 1 are the roots of f(x).
As f(x) divided p(x), -3/2 and 1 are the roots of p(x).
[tex]p( \frac{ - 3}{2} ) = 2( \frac{ - 3}{2} ) {}^{3} + a( \frac{ - 3}{2} ) {}^{2} + b( \frac{ - 3}{2} ) + 3 = 0[/tex]
From this we get: 3a - 2b = 5.
[tex]p(1) = 2(1) {}^{3} + a ({1})^{2} + b(1) + 3 = 0[/tex]
from this we get: a + b = -5
After solving 2 equations we get: a = -1 and b = -4
So p(x) = 2x^3 -x^2 -4x + 3.
So p(2) = 2(2)^3 - 2^2 - 4(2) + 3 = 7 = a -2b