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janilaw1
12.07.2019 •
Mathematics
Determine a constant $k$ such that the polynomial $$ p(x, y, z) = x^5 + y^5 + z^5 + k(x^3+y^3+z^3)(x^2+y^2+z^2) $$ is divisible by $x+y+z$.
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Ответ:
k =![\frac{-5}{6}](/tpl/images/0080/8444/ac989.png)
Explanation:
If we assume such a constant exists, then we can solve for it by setting up an equation based on
P(x,y,z)=0
whenever
x+y+z=0.
For example, P(2,-1,-1)=0.
Now we put x = 2, y = -1, z=-1
p(x, y, z) = x⁵ + y⁵ + z⁵ + k(x³+y³+z³)(x²+y²+z²)
p(2,-1,-1) = 2⁵ + (-1)⁵+ (-1)⁵ + k(2³ + (-1)³+ (-1)³)(2² + (-1)²+ (-1)²)
0 = 32 - 1 -1 + k(8-1-1)(4+1+1)
0 = 30 + k(6)(6)
0 = 30 + 36k
solve for k
k =![\frac{-30}{36}](/tpl/images/0080/8444/35236.png)
k =![\frac{-5}{6}](/tpl/images/0080/8444/ac989.png)
Ответ:
X=20
70 x 2 for the other side = 140
every triangle has 180 degrees
180-140=40
There are 40 remaining degrees
there are two x's so we divide by 2
40 = X2
20 = X
Hope this helps
Step-by-step explanation: