Question

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Answer 1

Question :-

Find the quadratic polynomial in x , when divided by ( x - 1 ) , ( x - 2 ) and ( x - 3 ) leaves remainder 11, 22 and 37 respectively

• a) 2x² + 5x + 4

• b) 3x² - 5x + 5

• c) 2x² - 5x + 4

• d) 3x² + 5x - 5

Answer :-

Given :-

A quadratic polynomial in x , when divided by ( x - 1 ) , ( x - 2 ) & ( x - 3 ) leaves remainder 11 , 22 , 37 respectively

Required to find :-

• The quadratic expression ?

Concept used :-

• Remainder theorem

Solution :-

According to Factor theorem ;

If ( x - a ) divides p ( x ) and leaves remainder . That remainder is equal to p ( a ) .

Similarly,

we know that the standard form of a quadratic polynomial is ;

f ( x ) = ax² + bx + c

In the question it is given that when the quadratic polynomial is divided by ( x - 1 ) , ( x - 2 ) & ( x - 3 ) it leaves remainder 11, 22 ,37 .

So,

This implies ;

➜ x - 1 = 0

➜ x = 1

Substitute the value of x in the standard form of the quadratic polynomial

➜ f ( 1 ) =

a ( 1 )² + b ( 1 ) + c = 11

a ( 1 ) + b + c = 11

a + b + c = 11 ............( 1 )

Consider this as equation 1

Now,

➜ x - 2 = 0

➜ x = 2

➜ f ( 2 ) =

a ( 2 )² + b ( 2 ) + c = 22

a ( 4 ) + 2b + c = 22

4a + 2b + c = 22 ...............( 2 )

Consider this as equation 2

Similarly,

➜ x - 3 = 0

➜ x = 3

➜ f ( 3 ) =

a ( 3 )² + b ( 3 ) + c = 37

a ( 9 ) + 3b + c = 37

9a + 3b + c = 37 ................( 3 )

Consider this as equation 3

Subtract eq 1 from eq 2

$\tt 4a + 2b + c = 22 \\ \: \: \tt \: a + \: \: \: b + c = 11 \\ \underline{ ( - )( - ) \: \: ( - ) \: \: ( - ) \: \: \: } \\ \underline{ \tt 3a + \: \: \: b \: \: \: \: \: \: \: \: \: \: = 11} \dots \dots(4)$

Consider this as equation 4

Subtract eq 2 from eq 3

$\tt 9a + 3b + c = 37 \\ \tt 4 a + 2 b + c = 22 \\ \underline{ ( - )( - ) \: \: ( - ) \: \: ( - ) \: \: \: } \\ \underline{ \tt 5a + \: \: \: b \: \: \: \: \: \: \: \: \: \: = 15} \dots \dots(5)$

Consider this as equation 5

Now,

Subtract equation 4 from equation 5

$\tt 5a + b = 15 \\ \tt 3a + b = 11 \\ \underline{ ( - )( - ) \: \: ( - )} \\ \tt \underline{2a \: \: \: \: \: \: \: \: \: = 4} \\ \sf{This \: implies,} \\ \rm \implies 2a = 4 \\ \rm \implies a = \frac{4}{2} \\ \tt \red{ \implies a = 2}$

Substitute the value of a in equation 5

☞ 5a + b = 15

☞ 5 ( 2 ) + b = 15

☞ 10 + b = 15

☞ b = 15 - 10

☞ b = 5

Similarly,

Substitute the values of a , b in equation 1

➤ a + b + c = 11

➤ 2 + 5 + c = 11

➤ 7 + c = 11

➤ c = 11 - 7

➤ c = 4

The standard form of the quadratic polynomial is ax² + bx + c ;

where ,

a = 2 , b = 5 , c = 4

This implies ;

2x² + 5x + 4

Therefore,

The quadratic polynomial is 2x² + 5x + 4 .

Hence,

Option - a is correct ☑