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Mathematics
Problem
Consider the cubic polynomial
q
(
x
)
=
x
3
+
p
x
2
+
q
x
+
12
q(x)=x^{3}+px^{2}+qx+12
q
(
x
)
=
x
3
+
p
x
2
+
q
x
+
12
. The remainder is
4
4
4
when
q
(
x
)
q(x)
q
(
x
)
is divided by
(
x
−
2
)
(x-2)
(
x
−
2
)
and
−
23
-23
−
23
when divided by
(
x
+
3
)
(x+3)
(
x
+
3
)
. What is the ordered pair
(
p
,
q
)
(p,q)
(
p
,
q
)
? a)
(
−
56
15
,
−
32
15
)
\left(-\dfrac{56}{15},-\dfrac{32}{15}\right)
(
−
15
56
,
−
15
32
)
b)
(
−
16
15
,
−
88
15
)
\left(-\dfrac{16}{15},-\dfrac{88}{15}\right)
(
−
15
16
,
−
15
88
)
c)
(
−
32
15
,
−
56
15
)
\left(-\dfrac{32}{15},-\dfrac{56}{15}\right)
(
−
15
32
,
−
15
56
)
d)
(
−
32
15
,
56
15
)
\left(-\dfrac{32}{15},\dfrac{56}{15}\right)
(
−
15
32
,
15
56
)
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