Given :-

Cubic polynomial : x³ + px² - 37x + q

The given cubic polynomial is exactly divisible by x² - 3x + 2

Required to FIND :-

• Values of p and q

Solution :-

Given information :-

Cubic polynomial : x³ + px² - 37x + q

The given cubic polynomial is exactly divisible by x² - 3x + 2

we need to find the values of p and q .

Let's consider the given polynomial as ;

p ( x ) = x³ + px² - 37x + q

Now,

Let's Factorise x² - 3x + 2

➜ x² - 3x + 2

➜ x² - 2x - 1x + 2

➜ x ( x - 2 ) - 1 ( x - 2 )

➜ ( x - 2 ) ( x - 1 )

From this we can conclude that ;

p ( x ) is exactly divisible by ( x - 2 ) & ( x - 1 )

So,

Let ;

=> x - 1 = 0

=> x = 1

p ( 1 ) =

( 1 )³ + p ( 1 )² - 37 ( 1 ) + q = 0

1 + p ( 1 ) - 37 + q = 0

1 + p - 37 + q = 0

- 36 + p + q = 0

q = - p + 36 ➜ equation - 1

consider this as equation 1

Similarly,

Let ;

=> x - 2 = 0

=> x = 2

p ( 2 ) =

( 2 )³ + p ( 2 )² - 37 ( 2 ) + q = 0

8 + p ( 4 ) - 74 + q = 0

8 + 4P - 74 + q = 0

- 66 + 4p + q = 0

Substitute the value of q from equation 1

- 66 + 4p + ( - p + 36 ) = 0

- 66 + 4p - p + 36 = 0

3p - 30 = 0

3p = 30

p = 30/3

p = 10

Substitute the value of p in equation 1

q = - p + 36

q = - ( 10 ) + 36

q = - 10 + 36

q = 26