Exercise 9. Three points in the xy-plane are given, with coordinates(x,y),i = 1,2,3, where 2 p(x) = a + bx + cx y = p(x)y, = p(x), for 2 1 x x x₁ 2 i=1,2,3.V [1 x2 x2 ]det 2 1 (V) = (x² -x₁ )(x¸ −x, )(x, x₁)Va,b,cpV(x, y, ) = (1,4), (₂- and (x)=(4,7)x, We will use linear algebra to show that there is a uniquely definite quadratic polynomial p(x) = a + bx + bx + cx 2 so that the graphy = p(x) passes through the three given points, i.e y = p(x), for i = 1,2,3. i (a) Let 2 1 x x V = [1 x2 1 x 2 2 2 3 3 Show that det(V) = (x2 −x, )(x¸ -x, )(x¸ -x2). (Hint: Row reduction gives the easiest calculation.) (b) Is V invertible? Justify the answer. (c) Show that there exists a uniquely determined set of coefficients a,b,c for the polynomial p, so that (1) is satisfied. (Hint: Set up an equation involving the matrix V.) (d) Find the polynomial in the case where (x₁₁) = (1,4), (2)₂) (2,11) and (x,y) = (4,7). Oppgave 9. Tre punkter i xy-planet er gitt, med koordinater (x,, y,), i = 1,2,3, der x1 < x2 < x3. Vi skal bruke lineær algebra til å vise at det finnes et entydig bestemt annengradspolynom p(x) = a + bx + cx² slik at grafen y = p(x) går gjennom de tre gitte punktene, dvs. (a) La y p(x) for V=1 x2 X3 1,2,3. (1) Vis at det(V) = (x-x1)(x3-x1)(x3-x2). (Hint: Radreduksjon gir enklest regning.) (b) Er V inverterbar? Begrunn svaret. (c) Vis at det eksisterer et entydig bestemt sett med koeffisienter a, b, c for polynomet p, slik at (1) er oppfylt. (Hint: Sett opp en ligning som involverer matrisen V.) (d) Finn polynomet i tilfellet der (x1,y) = (1,4), (x, y) = (2, 11) og (x3, 13) = (4,7).
Exercise 9. Three points in the xy-plane are given, with coordinates(x,y),i = 1,2,3, where 2 p(x) = a + bx + cx y = p(x)y, = p(x), for 2 1 x x x₁ 2 i=1,2,3.V [1 x2 x2 ]det 2 1 (V) = (x² -x₁ )(x¸ −x, )(x, x₁)Va,b,cpV(x, y, ) = (1,4), (₂- and (x)=(4,7)x, We will use linear algebra to show that there is a uniquely definite quadratic polynomial p(x) = a + bx + bx + cx 2 so that the graphy = p(x) passes through the three given points, i.e y = p(x), for i = 1,2,3. i (a) Let 2 1 x x V = [1 x2 1 x 2 2 2 3 3 Show that det(V) = (x2 −x, )(x¸ -x, )(x¸ -x2). (Hint: Row reduction gives the easiest calculation.) (b) Is V invertible? Justify the answer. (c) Show that there exists a uniquely determined set of coefficients a,b,c for the polynomial p, so that (1) is satisfied. (Hint: Set up an equation involving the matrix V.) (d) Find the polynomial in the case where (x₁₁) = (1,4), (2)₂) (2,11) and (x,y) = (4,7). Oppgave 9. Tre punkter i xy-planet er gitt, med koordinater (x,, y,), i = 1,2,3, der x1 < x2 < x3. Vi skal bruke lineær algebra til å vise at det finnes et entydig bestemt annengradspolynom p(x) = a + bx + cx² slik at grafen y = p(x) går gjennom de tre gitte punktene, dvs. (a) La y p(x) for V=1 x2 X3 1,2,3. (1) Vis at det(V) = (x-x1)(x3-x1)(x3-x2). (Hint: Radreduksjon gir enklest regning.) (b) Er V inverterbar? Begrunn svaret. (c) Vis at det eksisterer et entydig bestemt sett med koeffisienter a, b, c for polynomet p, slik at (1) er oppfylt. (Hint: Sett opp en ligning som involverer matrisen V.) (d) Finn polynomet i tilfellet der (x1,y) = (1,4), (x, y) = (2, 11) og (x3, 13) = (4,7).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
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![Exercise 9. Three points in the xy-plane are given, with
coordinates(x,y),i = 1,2,3, where
2
p(x) = a + bx + cx y = p(x)y, = p(x), for
2
1 x
x x₁
2
i=1,2,3.V [1
x2
x2
]det
2
1
(V) = (x² -x₁ )(x¸ −x, )(x, x₁)Va,b,cpV(x, y, ) = (1,4), (₂-
and (x)=(4,7)x,
We will use linear algebra to show that there is a
uniquely definite quadratic polynomial
p(x) = a + bx
+ bx + cx
2
so that the graphy = p(x) passes through the three given
points, i.e
y = p(x), for i = 1,2,3.
i
(a) Let
2
1 x
x
V = [1
x2
1 x
2
2
2
3
3
Show that det(V) = (x2 −x, )(x¸ -x, )(x¸ -x2). (Hint: Row
reduction gives the easiest calculation.)
(b) Is V invertible? Justify the answer.
(c) Show that there exists a uniquely determined set of
coefficients a,b,c for the polynomial
p, so that (1) is satisfied. (Hint: Set up an equation
involving the matrix V.)
(d) Find the polynomial in the case where
(x₁₁) = (1,4), (2)₂) (2,11) and (x,y) = (4,7).
Oppgave 9. Tre punkter i xy-planet er gitt, med koordinater (x,, y,), i = 1,2,3, der
x1 < x2 < x3.
Vi skal bruke lineær algebra til å vise at det finnes et entydig bestemt annengradspolynom
p(x) = a + bx + cx²
slik at grafen y = p(x) går gjennom de tre gitte punktene, dvs.
(a) La
y p(x) for
V=1
x2
X3
1,2,3.
(1)
Vis at det(V) = (x-x1)(x3-x1)(x3-x2). (Hint: Radreduksjon gir enklest regning.)
(b) Er V inverterbar? Begrunn svaret.
(c) Vis at det eksisterer et entydig bestemt sett med koeffisienter a, b, c for polynomet
p, slik at (1) er oppfylt. (Hint: Sett opp en ligning som involverer matrisen V.)
(d) Finn polynomet i tilfellet der (x1,y) = (1,4), (x, y) = (2, 11) og (x3, 13) = (4,7).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F771475ed-14ea-4d4d-bf15-67681461306e%2F271b8bf8-8150-4dcf-b8f9-a11358703d94%2Fcl6lrv4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 9. Three points in the xy-plane are given, with
coordinates(x,y),i = 1,2,3, where
2
p(x) = a + bx + cx y = p(x)y, = p(x), for
2
1 x
x x₁
2
i=1,2,3.V [1
x2
x2
]det
2
1
(V) = (x² -x₁ )(x¸ −x, )(x, x₁)Va,b,cpV(x, y, ) = (1,4), (₂-
and (x)=(4,7)x,
We will use linear algebra to show that there is a
uniquely definite quadratic polynomial
p(x) = a + bx
+ bx + cx
2
so that the graphy = p(x) passes through the three given
points, i.e
y = p(x), for i = 1,2,3.
i
(a) Let
2
1 x
x
V = [1
x2
1 x
2
2
2
3
3
Show that det(V) = (x2 −x, )(x¸ -x, )(x¸ -x2). (Hint: Row
reduction gives the easiest calculation.)
(b) Is V invertible? Justify the answer.
(c) Show that there exists a uniquely determined set of
coefficients a,b,c for the polynomial
p, so that (1) is satisfied. (Hint: Set up an equation
involving the matrix V.)
(d) Find the polynomial in the case where
(x₁₁) = (1,4), (2)₂) (2,11) and (x,y) = (4,7).
Oppgave 9. Tre punkter i xy-planet er gitt, med koordinater (x,, y,), i = 1,2,3, der
x1 < x2 < x3.
Vi skal bruke lineær algebra til å vise at det finnes et entydig bestemt annengradspolynom
p(x) = a + bx + cx²
slik at grafen y = p(x) går gjennom de tre gitte punktene, dvs.
(a) La
y p(x) for
V=1
x2
X3
1,2,3.
(1)
Vis at det(V) = (x-x1)(x3-x1)(x3-x2). (Hint: Radreduksjon gir enklest regning.)
(b) Er V inverterbar? Begrunn svaret.
(c) Vis at det eksisterer et entydig bestemt sett med koeffisienter a, b, c for polynomet
p, slik at (1) er oppfylt. (Hint: Sett opp en ligning som involverer matrisen V.)
(d) Finn polynomet i tilfellet der (x1,y) = (1,4), (x, y) = (2, 11) og (x3, 13) = (4,7).
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