if the polynomial 2x^3-ax^2+bx+4 has (x+1) as a factor and leaves remainder 4 when divided by (2x+1), find the value of a and b. Pls answer . I will mark you as the BRILLIANT!
Answers
Solution :
Given, if the polynomial 2x³ - ax² + bx + 4 has (x+1) as a factor and leaves remainder 4 when divided by (2x+1)
We have to find the value of a and b.
Let the polynomial be P(x)
→ P(x) = 2x³ - ax² + bx + 4
If (x + 1) is a factor of P(x) , according to factor theorem, P(-1) = 0
→ P(-1) = 2(-1)³ - a(-1)² + b(-1) + 4
→ 0 = 2(-1) - a(1) - b + 4
→ 0 = -2 - a - b + 4
→ a + b = 2
→ a = 2 - b -(1)
Again, P(x) leaves remainder 4 when divided by (2x + 1) or, 2(x + 1/2) , so according to remainder theorem, P(-1/2) = 4
→ P(-1/2) = 2(-1/2)³ - a(-1/2)² + b(-1/2) + 4
→ 4 = 2(-1/8) - a(1/4) - b/2 + 4
→ 0 = -1/4 - a/4 - b/2
→ 0 = (-1 - a - 2b)/4
→ -a - 2b - 1 = 0
→ a + 2b + 1 = 0
→ 2 - b + 2b + 1 = 0 [From (1)]
→ b + 3 = 0
→ b = -3
Substituting value in (1) :
→ a = 2 - (-3)
→ a = 2 + 3
→ a = 5
Therefore,