Question

10. What number should be added to the polynomial x2 – 5x + 4, so that 3 is the zero of the polynomial?​

Answer 1

let the new polynomial x2 - 5x+(4+k)

a=1,b=-5,c=(4+k)

now 3 is the zero of new polynomial so,

p(X)=(3)^2-5(3)+4+k

p(X)=. 9. -. 15 +4+k___(I)

p(X)=. k-2_____(ii)

but p(X) =0

from (I) and (ii) equation

k-2=0

k=2

2 needs to be added to the polynomial

I hope this answer will help you

Answer 2

Question

What number should be added to the polynomial x² – 5x + 4, so that 3 is the zero of the polynomial

$\colorbox{red}{\orange {Answer}}$

$\rm {\bf 2 }\: \: is \: \: the \: \: required\: number\: to\: be \:added \: in\: \: polynomial$

Solution

Let the required number be ' K ' which should be added to the polynomial i.e.

f ( x ) = x² - 5x + 4 such that when the value of x is taken as 3 as per the remainder & Factor theorem is concerned , it would bear a remainder equals to 0 .

• ➻ f ( x ) = x² - 5x + 4 + k _____ ❶

Now, it is already stated that x = 3 is the factor of polynomial this means f ( 3 ) = 0 ______ ❷

Using x = 3 in eq ❶

• ➻ f ( 3 ) = 3² - 5(3) + 4 + k
• ➻ f ( 3 ) = 9 - 15 + 4 + k
• ➻ f ( 3 ) = 9 - 11 + k ↣ K - 2

From eq ❷ , which is the basic principle of Factor Theorem.

• k - 2 = 0
• k = 2

Conclusion

Therefore, 2 is that required number which when added to the above polynomial whose factor is

( x - 3 ) i.e x = 3 .

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❥︎ Basic Concepts related to the concept :

① Remainder Theorem

This theorem states that if f (x) , a polynomial with x as a function, is divided by ( x - a ) then the remainder of the polynomial would be f(a) .

② Factor Theorem

This Theorem states that if on the application of remainder theorem i.e if on implementing the value of x as a ( say ) in f ( x ) we get the remainder as 0 this means that a is the zero of the polynomial or in other words f (a) is the factor of the given polynomial.

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