Repeated roots: If equation f(x) = 0, where f(x) is a polynomial function, has roots a, a, 3, or a root is repeated root, then f(x)=0 is equivalent to (x - a) (x - 3)..= 0, from which we can conclude that f'(x) = 0 or 2(x - a)(x - 3).+ (- a)*[(x - B3).. = 0 or (a - a) )2{(x-B)..}+ (- a){(x - 8)...}'=0 has root a. Thus, if a root occurs twice in the, equation, then it is common in equations f(x) = 0 and f'(x) = 0. Similarly, if a root occurs thrice in equation, then it is common in the equations f(x)=0, f'(x) = 0, and f"(x) = 0. If a root occurs p times and 3 root occurs q times in polynomial equation f(x)=0 of degree n(1
Repeated roots: If equation f(x) = 0, where f(x) is a polynomial function, has roots a, a, 3, or a root is repeated root, then f(x)=0 is equivalent to (x - a) (x - 3)..= 0, from which we can conclude that f'(x) = 0 or 2(x - a)(x - 3).+ (- a)*[(x - B3).. = 0 or (a - a) )2{(x-B)..}+ (- a){(x - 8)...}'=0 has root a. Thus, if a root occurs twice in the, equation, then it is common in equations f(x) = 0 and f'(x) = 0. Similarly, if a root occurs thrice in equation, then it is common in the equations f(x)=0, f'(x) = 0, and f"(x) = 0. If a root occurs p times and 3 root occurs q times in polynomial equation f(x)=0 of degree n(1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Repeated roots: If equation f(x) = 0, where f(x) is a polynomial function, has roots a, a, B,.. or a root is repeated root, then f(x)=0 is equivalent to (x – a) (x – B)...= 0, from which
we can conclude that f'(x) = 0 or 2(x – a)[(x – B)..]+ (x – a)²[x – B).I'
= 0 or (x – a) 2{(x – B) ..·}+ (x - a){(x – B) ...}'|= 0 has root a. Thus, if a root occurs twice in the,
equation, then it is common in equations f(x) = 0 and f'(x) = 0. Similarly, if a root occurs thrice in equation, then it is common in the equations f(x)=0, f'(x) = 0, and f"(x) = 0.
If a root occurs p times and B root occurs q times in polynomial equation f(x)=0 of degree n(1<p, q<n), then which of the following is not true [where f" (x) represents r th derivative of f(x)
w.r.t. x] ?
(1) If p<q<n, then a and B are two of the roots of the equation fP-1(x) = 0.
(2) If q<p<n, then a and B are two of the roots of the equation f9 (x)= 0.
(3) If p<q<n, then equations f(x)=0 and fP(x) = 0 have exactly one root common.
(4) If q<p<n, then equations f9(x) = 0 and fP(x) = 0 have exactly two roots common.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71d9916a-c176-469e-a26c-d697135b2205%2F1e0a41c9-48d3-497f-a404-a86566a3c934%2Fsghpwxd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Repeated roots: If equation f(x) = 0, where f(x) is a polynomial function, has roots a, a, B,.. or a root is repeated root, then f(x)=0 is equivalent to (x – a) (x – B)...= 0, from which
we can conclude that f'(x) = 0 or 2(x – a)[(x – B)..]+ (x – a)²[x – B).I'
= 0 or (x – a) 2{(x – B) ..·}+ (x - a){(x – B) ...}'|= 0 has root a. Thus, if a root occurs twice in the,
equation, then it is common in equations f(x) = 0 and f'(x) = 0. Similarly, if a root occurs thrice in equation, then it is common in the equations f(x)=0, f'(x) = 0, and f"(x) = 0.
If a root occurs p times and B root occurs q times in polynomial equation f(x)=0 of degree n(1<p, q<n), then which of the following is not true [where f" (x) represents r th derivative of f(x)
w.r.t. x] ?
(1) If p<q<n, then a and B are two of the roots of the equation fP-1(x) = 0.
(2) If q<p<n, then a and B are two of the roots of the equation f9 (x)= 0.
(3) If p<q<n, then equations f(x)=0 and fP(x) = 0 have exactly one root common.
(4) If q<p<n, then equations f9(x) = 0 and fP(x) = 0 have exactly two roots common.
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