Question

If (x+1) and (x-1) are factors of ax3+ x2
-2x+b , find the value of a and b.

Answer 1

Step-by-step explanation:

Given that x

2

+px+1 is a factor of ax

3

+bx+c.

Let ax

3

+bx+c=(x

2

+px+1)(ax+λ), where λ is a constant.

Then equating the coefficients of line powers of x on both sides, we get

0=ap+λ,b=pλ+a,c=λ

⇒p=−

a

λ

=−

a

c

Hence,

b=(−

a

c

)c+a

⇒ab=a

2

−c

2

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Answer 2

S O L U T I O N :

We have cubic polynomial p(x) = ax³ + x² - 2x + b & zero of the polynomial p(x) = 0

Given, two factor are (x+1) & (x-1)

• p(x) = 0

→ x + 1 = 0

→ x = -1

&

→ x - 1 = 0

→ x = 1

Now, putting the value of x = -1 in the given polynomial, we get;

⇒ p(x) = ax³ + x² - 2x + b = 0

⇒ p(-1) = a(-1)³ + (-1)² - 2(-1) + b = 0

⇒ a × (-1) + 1 + 2 + b = 0

⇒ -a + 3 + b = 0

⇒ -a + b = -3.................(1)

Putting the value of x = 1 in the given polynomial, we get;

⇒ p(x) = ax³ + x² - 2x + b = 0

⇒ p(1) = a(1)³ + (1)² - 2(1) + b = 0

⇒ a × (1) + 1 - 2 + b = 0

⇒ a - 1 + b = 0

⇒ a + b = 1..............................(2)

$\underline{\underline{\bf{Using\:by\:substitution\:method\::}}}$

From equation (2),we get;

⇒ a + b = 1

⇒ a = 1 - b..................(3)

∴Putting the value of a in equation (1),we get;

⇒ -(1-b) + b = -3

⇒ -1 + b + b = -3

⇒ -1 + 2b = -3

⇒ 2b = -3 + 1

⇒ 2b = -2

⇒ b = -2/2

⇒ b = -1

∴Putting the value of b in equation (3),we get;

⇒ a = 1 - ( -1)

⇒ a = 1 + 1

⇒ a = 2

Thus,

The value of a & b will be 2 & -1 .