a)find p(1) if p(x) =x²+2x+5
b)if x-1 is a factor x²+2x+k.what is the value of K
c) express the polynomial 2x²+x-3
d)(x-2)is a factor of x³-2x²+Kx+10.find the value of K
★Answer this question plz
Answers
Step-by-step explanation:
1. Given: P(x)=x
2
−2x+5
Forx=−1, P(−1)=(−1)
2
−2(−1)+5=1+2+5=8
For x=5, P(5)=(5)
2
−2(5)+5=25−10+5=20
Hence P(−1)=8 and P(5)=20
2.Therefore the value of k is 1.
3. TO DETERMINE
The polynomial 2x² + x + 3 in terms of Legendre polynomials
EVALUATION
We know that in case of Legendre polynomials :
\displaystyle \sf{P_0(x) = 1}P
0
(x)=1
\displaystyle \sf{P_1(x) = x}P
1
(x)=x
\displaystyle \sf{P_2(x) = \frac{1}{2} (3 {x}^{2} - 1) }P
2
(x)=
2
1
(3x
2
−1)
From above we get
\displaystyle \sf{ {x}^{2} = \frac{2}{3} P_2(x) + \frac{1}{3} }x
2
=
3
2
P
2
(x)+
3
1
Now the given polynomial
\displaystyle \sf{ = 2 {x}^{2} + x + 3 }=2x
2
+x+3
\displaystyle \sf{ = 2 \bigg( \frac{2}{3} P_2(x) + \frac{1}{3} \bigg) +P_1(x) + 3 }=2(
3
2
P
2
(x)+
3
1
)+P
1
(x)+3
\displaystyle \sf{ = \frac{4}{3} P_2(x) + \frac{2}{3} +P_1(x) + 3 }=
3
4
P
2
(x)+
3
2
+P
1
(x)+3
\displaystyle \sf{ = \frac{4}{3} P_2(x) + P_1(x) + 3 + \frac{2}{3} }=
3
4
P
2
(x)+P
1
(x)+3+
3
2
\displaystyle \sf{ = \frac{4}{3} P_2(x) + P_1(x) + \frac{9 + 2}{3} }=
3
4
P
2
(x)+P
1
(x)+
3
9+2
\displaystyle \sf{ = \frac{4}{3} P_2(x) + P_1(x) + \frac{11}{3} }=
3
4
P
2
(x)+P
1
(x)+
3
11
\displaystyle \sf{ = \frac{4}{3} P_2(x) + P_1(x) + \frac{11}{3} P_0(x) }=
3
4
P
2
(x)+P
1
(x)+
3
11
P
0
(x)
Which is the required expression in terms of Legendre polynomials