Question

If alpha and beta and are the zeros of the polynomial p(x)=x^2-m(x+1)-n, then find the value of (1+Alpha) (1+beta)​

Answer 1

Answer:

(1 + α) (1 + β) = 1 – n

Step-by-step explanation:

Given :

α and β are the zeroes of the polynomial p(x) = x² - m(x+1) - n

To find :

the value of (1 + α) (1 + β)

Solution :

To solve this question,we must know the relation between zeros and coefficients of the quadratic polynomial.

Sum of zeros = –(x coefficient)/x² coefficient

Product of zeros = constant term/x² coefficient

For the given quadratic polynomial,

First, let's get the given quadratic polynomial to the form ax² + bx + c

p(x) = x² - m(x+1) - n

p(x) = x² - mx - m - n

p(x) = x² - mx - (m + n)

• constant term = –(m + n)
• x coefficient = –m
• x² coefficient = 1

From the relation between zeros and coefficients :

α + β = -(-m)/1 = m

αβ = -(m + n)/1 = -(m + n)

Now, simplify (1 + α) (1 + β)

= (1 + α) (1 + β)

= 1(1 + β) + α(1 + β)

= 1 + β + α + αβ

= 1 + (α + β) + αβ

Substitute,

= 1 + m + [–(m + n) ]

= 1 + m – m – n

= 1 – n

Answer 2

$\huge \fbox \red{Answer}$

x² - p ( x + 1 ) + c

⇒ x² - p x - p + c

⇒ x² - p x + ( c - p )

________________________

Comparing with ax² + bx + c, we get :

a = 1

b = - p

c = c - p .

_______________________

Given :

( α + 1 )( β + 1 ) = 0

⇒ αβ + α + β + 1 = 0

______________________

Note that, sum of roots = - b/a

α + β = - b / a

But b = - p

a = 1

________________________

So α + β = - ( - p ) / 1 = p

Product of roots = αβ = c / a

⇒ αβ = ( c - p )

_______________________

Hence write this as :

αβ + α + β + 1 = 0

⇒ c - p + p + 1 = 0

⇒ c + 1 = 0

⇒ c = -1

______________________

Hence, the value of c is - 1.