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Quiz 1, Math 20F  Lecture B (Spring 2007)
1.
In this question we want to determine whether there exists a quadratic polynomial in the
form
p
2
(
x
) =
a
1
+
a
2
x
+
a
3
x
2
that passes through the points (1
,
2) and (

1
,
4) and, if there are more than one such
polynomials, characterize these polynomials.
a) (2.5 points)
Find a system of 2 linear equations in 3 unknowns whose solution set
consists of all triples (
a
1
, a
2
, a
3
) such that
p
2
(
x
) =
a
1
+
a
2
x
+
a
3
x
2
passes through (1
,
2) and
(

1
,
4). Provide also the 2 by 4 augmented matrix associated with this system.
Solution
Using the constraints that the polynomial has to pass through (1
,
2) and (

1
,
4), we have
the linear equations
p
2
(1) =
a
1
+
a
2
(1) +
a
3
(1)
2
= 2
p
2
(

1) =
a
1
+
a
2
(

1) +
a
3
(

1)
2
= 4
with
a
1
,
a
2
and
a
3
unknowns. The augmented matrix for this system is
±
1
1
1 2
1

1 1 4
²
.
b) (2.5 points)
Row reduce the augmented matrix associated with the system of linear
equations in part
a)
into the reduced echelon form. Using the reduced echelon form decide
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 Spring '03
 BUSS
 Math, Linear Algebra, Algebra

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