find zeroes of quadratic polynomial and find the relation hip between the zeroes and the coefficients 1.x2-2x-8
Answers
S O L U T I O N :
⠀ ⠀
We're given with a polynomial that is x² – 2x – 8. And, we're asked to find the zeroes of this given polynomial and verify the relationship b/w the zeroes and the coefficients of the polynomial. Firstly, we'll find out the zeroes of the polynomial by using the method given below.
_____________________________________
• S P L I T T I N G⠀T H E⠀M I D D L E⠀T E R M :
↠ x² – 2x – 8 = 0
↠ x² – 4x + 2x – 8 = 0
↠ x(x – 4) + 2(x – 4) = 0
↠ (x – 4) (x + 2) = 0
↠ x = 4 & – 2
∴ Therefore, the zeroes of the polynomial are α = 4 and β = – 2 respectively.
_____________________________________
★ On Comparing with the Quadratic polynomial (ax² + bx + c = 0) —
- a = 1
- b = – 2
- c = – 8
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
★ V E R I F I C A T I O N
For any Quadratic polynomial the sum and product of the roots are Given by :
- (α + β) = –b/a
- (αβ) = c/a
_____________________________________
S U M⠀O F⠀Z E R O E S :
➟ α + β = –b/a
➟ 4 + (– 2) = – (– 2)/1
➟ 2 = 2
_____________________________________
P R O D U C T⠀O F⠀Z E R O E S :
➟ αβ = c/a
➟ (4) (– 2) = – 8/1
➟ – 8 = – 8
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
⠀ ⠀ ⠀⠀⠀⠀ ∴ Hence, Verified!
x^2 - 2x - 8
X^2 -4x + 2x - 8
x(x - 4) + 2 ( x - 4 )
( x + 2 ) ( x - 4 )
X = -2 or 4
Hence zeroes are -2 and 4
Verification:
A = 1
B= -2
C = -8
Alpha + Beta = -b/a
-2 + 4 = -(-2)/1
2 = 2
Alpha Beta = c /a
-2 (4) = -8/1
-8 = - 8
Hence Verified