Question

Find a and b, if (x+1) and (x+2) are factors of x^3 +3x^2 -2ax + b​

Answer 1

$\mathfrak{\large{\underline{\underline{Answer:-}}}}$

$\boxed{\sf{a = - 1, \: b = 0}}$

$\mathfrak{\large{\underline{\underline{Explanation:-}}}}$

Given :- (x + 1), (x + 2) are factors of x³ +3x² -2ax + b

To find :- Value of a and b

Solution :-

Let f(x) = x³ +3x² -2ax + b

First consider (x + 1) as factor of x³ +3x² -2ax + b

Find zero of x + 1

To find zero of x + 1 equate x + 1 to 0

=> x + 1 = 0

=> x = - 1

So, By Factor theorem f( - 1) = 0

Substitute x = - 1 in x³ +3x² -2ax + b

f(- 1) = 0

(-1)³ + 3(-1)² - 2a(-1) + b = 0

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

- 1 + 3 + 2a + b = 0

2 + 2a + b = 0

Now find b in terms of a

=> b = -2 - 2a----(2)

Consider (x + 2) as factor of x³ +3x² -2ax + b

Find zero of x + 2

To find zero of x + 3 equate x + 2 to 0

=> x + 2 = 0

=> x = - 2

So, By Factor theorem f( - 2) = 0

Substitute x = - 2 in x³ +3x² -2ax + b

f(- 2) = 0

(-2)³ + 3(-2)² - 2a(-2) + b = 0

- 8 + 3(4) + 4a + b = 0

- 8 + 12 + 4a + b = 0

4 + 4a + b = 0

Now find b in terms of a

=> b = - 4 - 4a ---(2)

From (1) and (2)

- 2 - 2a = - 4 - 4a

- 2 - 2a + 4a = - 4

- 2 + 2a = - 4

2a = - 4 + 2

2a = - 2

a = - 2/2

a = - 1

Substitute a = - 3 in (1)

b = - 2 - 2(- 1)

b = - 2 + 2

b = 0

$\Huge{\boxed{\sf{a = - 1, \: b = 0}}}$

Answer 2

Answer:-

Also refer the attachment

x³ +3x² -2ax + b=f(x)

(x + 1)= x³ +3x² -2ax + b

We need to find the zero for( x + 1) and then equate (x+1 to 0)

$(x + 1 = 0) \\ </p><p>(x = - 1)$

By the Therom (Factor theorem)

Using f( - 1) = 0

Let us Substitute the value that is

• (x )= - 1 in x³ +3x² -2ax + b
• f(- 1) = 0
• (-1)³ + 3(-1)² - 2a(-1) + b = 0

$(- 1 + 3(1) + 2a + b = 0) \\ </p><p></p><p>(- 1 + 3 + 2a + b = 0) \\ </p><p></p><p>(2 + 2a + b = 0)$

First term be in a

b = -2 - 2a

(x + 2) =(x³ +3x² -2ax + b)

Let is find zero of x + 2

Then find the zero of x + 3 for x + 2 to 0

$x + 2 = 0 \\ </p><p>x = - 2$

f( - 2) = 0

By Substituting we have

- 2 of x³ +3x² -2ax + b

f(- 2) = 0

(-2)³ + 3(-2)² - 2a(-2) + b = 0

- 8 + 3(4) + 4a + b = 0

- 8 + 12 + 4a + b = 0

4 + 4a + b = 0

Let us find terms b of a

b = - 4 - 4a

From both the equation (1) and (2)

- 2 - 2a = - 4 - 4a

- 2 - 2a + 4a = - 4

- 2 + 2a = - 4

2a = - 4 + 2

2a = - 2

a = - 2/2

a = - 1

Substituting a as - 3 in equation(1)

b = - 2 - 2(- 1)

b = - 2 + 2

b = 0

$\huge\colorbox\Yellow{a=- 1,b = 0}$