Find the zeroes of the polynomial2xsq+7/2x+3/4 by factorisation method and verify the relation between the zeroes and the coefficients of the polynomial
Answers
To find :-
- The zeroes of the given polynomial p(x) and verify the relation between the zeroes and the coefficients of the polynomial.
Solution :-
Given polynomial,
- p(x) = 2x² + 7/2x + 3/4
Here,
⇒ p(x) = 8x² + 14x + 3
⇒ p(x) = 8x² + 2x + 12x + 3
⇒ p(x) = 2x(4x + 1) + 3(4x + 1)
⇒ p(x) = (4x + 1)(2x + 3)
Hence,
x = -1/4 , -3/2
Verification :-
- Sum of zeroes = - ( coefficient of x) / (coefficient of x² )
⇒ -1/4 + (-3/2) = - (14 / 8)
⇒ -(1 + 6)/4 = -7/4
⇒ -7/4 = -7/4
- Product of zeroes = ( constant term ) / (coefficient of x² )
⇒ -1/4 × -3/2 = 3 / 8
⇒ 3/8 = 3/8
Hence, Verified.
_____________________________
Step-by-step explanation:
Let f(x)=2x
2
+
2
7
x+
4
3
.
Comparing it with the standard quadratic polynomial ax
2
+bx+c, we get,
a=2, b=
2
7
, c=
4
3
.
Now, 2x
2
+
2
7
x+
4
3
=2x
2
+
2
6
x+
2
1
x+
4
3
=2x(x+
2
3
)+
2
1
(x+
2
3
)
=(x+
2
3
)(2x+
2
1
).
The zeros of f(x) are given by f(x)=0.
=>(x+
2
3
)(2x+
2
1
)=0
=>(x+
2
3
)=0 or (2x+
2
1
)=0
=>x=−
2
3
or x=−
4
1
.
Hence the zeros of the given quadratic polynomial are −
2
3
, −
4
1
.
Verification of the relationship between the roots and the coefficients:
Sum of the roots =(−
2
3
)+(−
4
1
)
=−
4
7
=−
2×2
7
=
coefficientofx
2
−coefficientofx
.
Product of the roots =(−
2
3
)(−
4
1
)
=−
8
3
=−
4×2
3
=
coefficientofx
2
constantterm
.
Hence, verified.