SOLUTION: P(X)=x^3-3x^2-5x+10 A) Evaluate P(x) for integers -3 through 5 B) Find all of the zeros of P(x) C) Prove that five is an upper bound on the zeros of P(X)

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: P(X)=x^3-3x^2-5x+10 A) Evaluate P(x) for integers -3 through 5 B) Find all of the zeros of P(x) C) Prove that five is an upper bound on the zeros of P(X)      Log On


   

Question 918970: P(X)=x^3-3x^2-5x+10
A) Evaluate P(x) for integers -3 through 5
B) Find all of the zeros of P(x)
C) Prove that five is an upper bound on the zeros of P(X)
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

P%28X%29=x%5E3-3x%5E2-5x%2B10

A) Evaluate P%28x%29 for integers -3 through 5:

x=-3
%28-3%29%5E3-3%28-3%29%5E2-5%28-3%29%2B10
-27-27%2B15%2B10
-54%2B25
=-29


x=-2
%28-2%29%5E3-3%28-2%29%5E2-5%28-2%29%2B10
-8-12%2B10%2B10
-20%2B20
=0

x=-1
%28-1%29%5E3-3%28-1%29%5E2-5%28-1%29%2B10
-1-3%2B5%2B10
-4%2B15
=11

x=0
%280%29%5E3-3%280%29%5E2-5%280%29%2B10
0-0%2B0%2B10
=10


x=1
%281%29%5E3-3%281%29%5E2-5%281%29%2B10
1-3-5%2B10
-8%2B11
=3


x=2
%282%29%5E3-3%282%29%5E2-5%282%29%2B10
8-12-10%2B10
-22%2B18
=-4


x=3
%283%29%5E3-3%283%29%5E2-5%283%29%2B10
27-27-15%2B10
0-15%2B10
=-5


x=4
%284%29%5E3-3%284%29%5E2-5%284%29%2B10
64-48-20%2B10
74-68
=6


x=5
%285%29%5E3-3%285%29%5E2-5%285%29%2B10
125-75-25%2B10
135-100
=35

B) Find all of the zeros of P%28x%29
P%28X%29=x%5E3-3x%5E2-5x%2B10
x%5E3-3x%5E2-5x%2B10=0
x%5E3-5x%5E2%2B5x%2B2x%5E2-10x%2B10=0
%28x%5E3%2B2x%5E2%29-%285x%5E2%2B10x%29%2B%285x%2B10%29=0
x%5E2%28x%2B2%29-5x%28x%2B2%29%2B5%28x%2B2%29=0
%28x%2B2%29%28x%5E2-5x%2B5%29+=+0
one zero is:
%28x%2B2%29+=+0=>highlight%28x=-2%29
x%5E2-5x%2B5+=+0 ...use quadratic formula to find other two zeros:
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-%28-5%29+%2B-+sqrt%28+%28-5%29%5E2-4%2A1%2A5+%29%29%2F%282%2A1%29+
x+=+%285+%2B-+sqrt%28+25-20+%29%29%2F2+
x+=+%285+%2B-+sqrt%28+5+%29%29%2F2+
x+=+%285+%2B-+2.236%29%2F2+
solutions:
x+=+%285+%2B+2.236%29%2F2+
x+=+7.236%2F2+
x+=3.6
and
x+=+%285+-+2.236%29%2F2+
x+=+2.764%2F2+
x+=1.4
so, zeros are:highlight%28x=-2%29,highlight%28x=3.6%29, and highlight%28x=1.4%29


C) Prove that five is an upper bound on the zeros of P%28X%29

to find bounds for the real roots of the polynomial means we are looking for a number that is greater than all roots (an upper bound) and a number that is less than all roots (a lower bound)
we are looking if 5 is greater than all roots
zeros are:
5%3Ehighlight%28-2%29,
5%3Ehighlight%283.6%29, and
5%3Ehighlight%281.4%29
so, it's proven that 5 is an upper bound on the zeros of P%28X%29

graph of P%28X%29:

+graph%28+600%2C+600%2C+-20%2C+20%2C+-20%2C+20%2C+x%5E3-3x%5E2-5x%2B10%29+