Evaluating an Octic
If x = 3 + 5 , what is the value of − x 2 ( x 2 − 8 ) ( x 2 − 1 4 ) ( x 2 − 1 8 ) ?
The answer is 480.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
14 solutions
This is a very nice and clean solution. The text was revised slightly from the original submission, to eliminate a couple of misprints.
Most correct solutions consisted of calculating x^2 and then multiplying out, working with the square root of 15.
You have a misprint when you substitute -x^2 for -(y-8) instead of -(y+8)
Given that x = 3 + 5 , so x 2 = 8 + 2 × 1 5 We then substitute x 2 into the expression which then gives us:
− ( 8 + 2 × 1 5 ) ( 8 + 2 × 1 5 − 8 ) ( 8 + 2 × 1 5 − 1 4 ) ( 8 + 2 × 1 5 − 1 8 ) = − [ ( 2 × 1 5 + 8 ) ( 2 × 1 5 ) ] [ ( 2 × 1 5 − 6 ) ( 2 × 1 5 − 1 0 ) ] = − [ 4 ( 1 5 ) + 1 6 × 1 5 ] [ 4 ( 1 5 ) − 3 2 × 1 5 + 6 0 ] = − [ 4 ( 1 5 ) + 1 6 × 1 5 ] × ( 2 ) × [ 2 ( 1 5 ) − 1 6 × 1 5 + 3 0 ] = − ( 2 ) × [ ( 4 ) 1 5 + 1 6 × 1 5 ] [ 4 ( 1 5 ) − 1 6 × 1 5 ] = − ( 2 ) × [ ( 1 6 ) ( 1 5 ) ( 1 5 ) − ( 1 6 ) ( 1 6 ) ( 1 5 ) ] = − ( 2 ) × [ ( 1 6 ) ( 1 5 ) ( 1 5 − 1 6 ) ] = − ( 2 ) × [ ( 1 6 ) × ( 1 5 ) × ( − 1 ) ] = 4 8 0
x
2
=
(
3
+
5
)
2
=
(
3
+
5
+
2
3
×
5
)
=
8
+
2
1
5
x
2
−
8
=
8
+
2
1
5
−
8
=
2
1
5
x
2
−
1
4
=
8
+
2
1
5
−
1
4
=
2
1
5
−
6
x
2
−
1
8
=
8
+
2
1
5
−
1
8
=
2
1
5
−
1
0
Multiply them all
x
2
(
x
2
−
8
)
(
x
2
−
1
4
)
(
x
2
−
1
8
)
=
(
8
+
2
1
5
)
(
2
1
5
)
(
2
1
5
−
6
)
(
2
1
5
−
1
0
)
=
(
1
6
1
5
+
2
2
×
1
5
)
(
2
2
×
1
5
−
(
1
0
+
6
)
×
2
1
5
+
6
×
1
0
)
x
2
(
x
2
−
8
)
(
x
2
−
1
4
)
(
x
2
−
1
8
)
=
(
1
6
1
5
+
6
0
)
(
1
2
0
−
3
2
1
5
)
x
2
(
x
2
−
8
)
(
x
2
−
1
4
)
(
x
2
−
1
8
)
=
1
2
0
×
6
0
−
3
2
×
1
6
×
1
5
+
(
1
6
×
1
2
0
−
3
2
×
6
0
)
1
5
=
7
2
0
0
−
7
6
8
0
+
0
=
−
4
8
0
−
x
2
(
x
2
−
8
)
(
x
2
−
1
4
)
(
x
2
−
1
8
)
=
4
8
0
And we are done
Let x 2 = 8 + 2 1 5 = y .
Evaluate: f ( y ) = − y ( y − 8 ) ( y − 1 4 ) ( y − 1 8 ) .
f ( y ) = − ( 8 + 2 1 5 ) ( 2 1 5 ) ( − 6 + 2 1 5 ) ( − 1 0 + 2 1 5 ) .
= 2 ( 1 6 1 5 + 6 0 ) ( 1 6 1 5 − 6 0 ) .
= 2 ( 3 8 4 0 − 3 6 0 0 ) = 4 8 0 .
Synthetic division and the remainder polynomial theorem are also useful, but the process is not made any simpler.
( √ 3 + √ 5 ) 2 = 3 + 5 + 2 √ 1 5 = 8 + 2 √ 1 5
− x 2 ( x 2 − 8 ) ( x 2 − 1 4 ) ( x 2 − 1 8 ) = − ( 8 + 2 √ 1 5 ) ( 8 + 2 √ 1 5 − 8 ) ( 8 + 2 √ 1 5 − 1 4 ) ( 8 + 2 √ 1 5 − 1 8 ) = − ( 8 + 2 √ 1 5 ) ( 2 √ 1 5 ) ( 2 √ 1 5 − 6 ) ( 2 √ 1 5 − 1 0 ) = − ( 1 6 √ 1 5 + 6 0 ) ( 6 0 − 2 0 √ 1 5 − 1 2 √ 1 5 + 6 0 ) = − ( 1 6 √ 1 5 + 6 0 ) ( 1 2 0 − 3 2 √ 1 5 ) = − ( 1 9 2 0 √ 1 5 − 7 6 8 0 + 7 2 0 0 − 1 9 2 0 √ 1 5 ) = − ( − 4 8 0 ) = 4 8 0
As x = 3 + 5 , we know that x 2 = 3 + 2 3 ∗ 5 + 5 = 8 + 2 1 5 . Let P = − x 2 ( x 2 − 8 ) ( x 2 − 1 4 ) ( x 2 − 1 8 ) . Then P = − ( 8 + 2 1 5 ) ( 2 1 5 ) ( − 6 + 2 1 5 ) ( − 1 0 + 2 1 5 ) . Then, P = − ( 6 0 + 1 6 1 5 ) ( 1 2 0 − 3 2 1 5 ) = − ( 6 0 ∗ 1 2 0 − 3 2 ∗ 1 6 ∗ 1 5 ) = 4 8 0 .
We get x 2 = ( 3 + 5 ) 2 = 8 + 2 1 5
So, ( x 2 − 8 ) ( x 2 − 1 8 ) = 2 1 5 ( 2 1 5 − 1 0 ) = 6 0 − 2 0 1 5 = 2 0 ( 3 − 1 5 )
And, x 2 ( x 2 − 1 4 ) = ( 8 + 2 1 5 ) ( 2 1 5 − 6 ) = 6 0 − 4 8 + ( 1 6 − 1 2 ) 1 5 = 1 2 + 4 1 5 = 4 ( 3 + 1 5 )
So, − x 2 ( x 2 − 8 ) ( x 2 − 1 4 ) ( x 2 − 1 8 ) = − 4 ⋅ 2 0 ( 3 + 1 5 ) ( 3 − 1 5 ) = − 8 0 ⋅ ( 9 − 1 5 ) = − 8 0 ⋅ − 6 = 4 8 0
Firstly, I calculated the value of x^2. Then I used it in the given equation and multiplied 2 brackets at a time using algebra, which finally got me the answer.
Because the question asked you to compute an equation in x^2, it would be easier to compute x^2 first, rather than dealing with three terms multiplication.
So, you get x^2 = 8 +2 \sqrt{15}
Then, by inputting this value into x^2, you just do the math slowly and carefully, and you will end up with an answer of 480
x 2 = 3 + 5 + 2 ( 3 ) ( 5 ) = 8 + 2 1 5 x 2 ( x 2 − 1 4 ) = ( 2 1 5 + 8 ) ( 2 1 5 − 6 ) = 6 0 + 1 6 1 5 − 1 2 1 5 − 4 8 = 1 2 + 4 1 5 x 2 ( x 2 − 1 4 ) ( x 2 − 1 8 ) = ( 4 1 5 + 1 2 ) ( 2 1 5 − 1 0 ) = 1 2 0 + 2 4 1 5 − 4 0 1 5 − 1 2 0 = − 1 6 1 5 − x 2 ( x 2 − 8 ) ( x 2 − 1 4 ) ( x 2 − 1 8 ) = − ( 2 1 5 ) ( − 1 6 1 5 ) = 3 2 × 1 5 = 4 8 0
x= \sqrt{3} + \sqrt{5}
x^2=[\sqrt{3} + \sqrt{5}]^2
x^2=[\sqrt{3}]^2+[\sqrt{5}]^2+2 \sqrt{3} \sqrt{5}
x^2=8+2(\sqrt{15}
So, On the substitution of these values in the equation, we get the required solution as 480.
The value of x^2 is 8+2 \sqrt {15} and it become to 2 \sqrt {15} (-8- \sqrt {15} ) (2 \sqrt {15} -6)(2 \sqrt {15} -10). By do it one by one, you can find the value soon.
X 2 = 8 − 2 1 5 . So, the required product is just − ( 2 1 5 + 8 ) ( 2 1 5 ) ( 2 1 5 − 6 ) ( 2 1 5 − 1 0 ) = ( 4 1 5 + 2 0 ) ( 6 0 − 1 2 1 5 ) = 4 8 0 .
We have that x 2 = 8 + 2 1 5 . Now, x 3 = ( 8 + 2 1 5 ) ( 3 + 1 5 ) = 1 8 3 + 1 4 5 . So, we have x ( x 2 − 1 8 ) = x 3 − 1 8 x = − 4 5 and x ( x 2 − 1 4 ) = x 3 − 1 4 x = 4 3 . Hence − x 2 ( x 2 − 8 ) ( x 2 − 1 4 ) ( x 2 − 1 8 ) = − ( x 2 − 8 ) ( x 3 − 1 4 x ) ( x 3 − 1 8 x ) = − ( 2 1 5 ) × ( 4 3 ) × ( − 4 5 ) = 4 8 0
Noticed that,
x 2 = 8 + 2 1 5
Let y be x 2 − 8 = 2 1 5 for the equation
− ( y − 8 ) ( y ) ( y − 6 ) ( y − 1 0 )
Then, we expand and arrange the equation
− y ( y − 8 ) ( y 2 − 1 6 y + 6 0 )
Noticed that y 2 = 6 0 and substitute to the equation in a special way
= − y ( y − 8 ) ( y 2 − 1 6 y + y 2 )
= − y ( y − 8 ) ( 2 y 2 − 1 6 y )
= − 2 y 2 ( y − 8 ) ( y + 8 )
= − 2 y 2 ( y 2 − 6 4 )
Hence, we substitute y 2 = 6 0 to the equation
= − 2 ( 6 0 ) ( 6 0 − 6 4 )
= 4 8 0