Question

If x +1 and x -1 are factors of f(x) = x³ +2 ax +b, calculate the values of a and b.

Answer 1

Step-by-step explanation:

$\sf \underline \red{Given:-}$

• A polynomial f(x) = x³+2ax+b
• (x+1) and (x-1) are factor of f(x)

$\sf \underline \red{To \: \: Find:-}$

• The value of a and b

$\sf \underline \red{Answer:-}$

In first case,

f(x) = x³+2ax+b

g(x) = x+1

g(x) = 0

x+1=0

x=-1

$\sf \therefore \: f( - 1) = 0$

$\sf \implies \purple{ {( - 1)}^{3} + 2 \times a \times ( - 1) + b = 0}$

$\sf \implies \purple{ - 1 - 2a + b = 0}$

$\sf \implies \purple{b - 2a = 1}......(1)$

In second case,

f(x) = x³+2ax+b

g(x) = x-1

g(x) = 0

x-1=0

x=1

$\sf \implies \purple{ {(1)}^{3} + 2 \times a(1) + b = 0}$

$\sf \implies \purple{1 + 2a + b = 0}$

$\sf \implies \purple{2a + b = - 1}........(2)$

From (1) and (2)

$\huge \bf b \: \: - \: \: 2a \: \: = \: \: 1$

$\huge \bf \underline{ b \: \: + \: \: 2a \: \: = \: \: - 1}$

$\huge \bf \underline{2b \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 0 \: \: \: \: \: }$

now ,

b= 0

Sub b in eq (1)

$\sf \implies \purple{b - 2a = 1}$

$\sf \implies \purple{0 - 2a = -1}$

$\sf \implies \purple{a = - \frac{1}{2} }$

$\therefore \tt \large \green { \: the \: \: value \: \: of \: \: a \: \: and \: \: b \: \: is \: \: - \frac{1}{2} \: \: and \: \: 0 \: \: respectively \: }$