verify that 2, 1, 1 are zeros of the polynomial p(x) = x? - 4x² + 5x – 2. Also verify the
relationship between the zeros and the coefficients. jaldi bato aukat se
Answers
Answer:
(i) 2x
3
+x
2
−5x+2;
2
1
,1,−2
p(x)=2x
3
+x
2
−5x+2 .... (1)
Zeroes for this polynomial are
2
1
,1,−2
Substitute the x=
2
1
in equation (1)
p(
2
1
)=2(
2
1
)
3
+(
2
1
)
2
−5(
2
1
)+2
=
4
1
+
4
1
+
2
5
+2
=0
Substitute the x=1 in equation (1)
p(1)=2×1
3
+1
2
−5×1+2
=2+1−5+2=0
Substitute the x=−2 in equation (1)
p(−2)=2(−2)
3
+(−2)
2
−5(−2)+2
=−16+4+10+2=0
Therefore,
2
1
,1,−2 are the zeroes of the given polynomial.
Comparing the given polynomial with ax
3
+bx
2
+cx+d we obtain,
a=2,b=1,c=−5,d=2
Let us assume α=
2
1
, β=1, γ=−2
Sum of the roots = α+β+γ=
2
1
+1=2=
2
−1
=
a
−b
αβ+βγ+αγ=
2
1
+1(−2)+
2
1
(−2)=
2
−5
=
a
c
Product of the roots = αβγ=
2
1
×x×(−2)=
2
−2
=
a
d
Therefore, the relationship between the zeroes and coefficient are verified.
(ii) x
3
−4x
2
+5x−2;2,1,1
p(x)=x
3
−4x
2
+5x−2 .... (1)
Zeroes for this polynomial are 2,1,1
Substitute x=2 in equation (1)
p(2)=2
3
−4×2
2
+5×2−2
=8−16+10−2=0
Substitute x=1 in equation (1)
p(1)=x
3
−4x
2
+5x−2
=1
3
−4(1)
2
+5(1)−2
=1−4+5−2=0
Therefore, 2,1,1 are the zeroes of the given polynomial.
Comparing the given polynomial with ax
3
+bx
2
+cx+d we obtain,
a=1,b=−4,c=5,d=−2
Let us assume α=2, β=1, γ=1
Sum of the roots = α+β+γ=2+1+1=4=−
1
−4
a
−b
Multiplication of two zeroes taking two at a time=αβ+βγ+αγ=(2)(1)+(1)(1)+(2)(1)=5=
1
5
=
a
c
Product of the roots = αβγ=2×1×1=2=−
1
−2
=
a
d
Therefore, the relationship between the zeroes and coefficient are verified.