29 is great
( r 1 + 2 9 ) ( r 2 + 2 9 ) ( r 3 + 2 9 ) ( r 4 + 2 9 )
Let r 1 , r 2 , … , r 4 be the roots of x 4 + 7 x 3 − 2 x 2 + 3 x + 1 7
Find the value of the expression above.
The answer is 534806.
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2 solutions
Let y=x+29 and substitute (y-29) for x in the equation. We get
(
y
−
2
9
)
4
+
7
∗
(
y
−
2
9
)
3
−
2
∗
(
y
−
2
9
)
2
+
3
∗
(
y
−
2
9
)
+
1
7
=
0
.
∴
a
4
∗
y
4
+
a
3
∗
y
3
+
a
2
∗
y
2
+
a
1
∗
y
+
{
(
−
2
9
)
4
+
7
∗
(
−
2
9
)
3
−
2
∗
(
−
2
9
)
2
+
3
∗
(
−
2
9
)
+
1
7
}
.
⟹
a
4
∗
y
4
+
.
.
.
+
a
1
∗
y
+
5
3
4
8
0
6
.
By Vieta the product of each root of y is the constant term 534806.
But y=x+29. So roots of y are roots of x plus 29. So 534806 is the required product.
Great job on that!
x 4 + 7 x 3 − 2 x 2 + 3 x + 1 7 = ( x − r 1 ) ( x − r 2 ) ( x − r 3 ) ( x − r 4 ) ⇒ ( 2 9 + r 1 ) ( 2 9 + r 2 ) ( 2 9 + r 3 ) ( 2 9 + r 4 ) = ( − 1 ) 4 ⋅ ( − 2 9 − r 1 ) ( − 2 9 − r 2 ) ( − 2 9 − r 3 ) ( − 2 9 − r 4 ) = = ( − 2 9 ) 4 + 7 ( − 2 9 ) 3 − 2 ( − 2 9 ) 2 + 3 ( − 2 9 ) + 1 7 = 5 3 4 8 0 6 .