SOLUTION: Find a possible formula for a polynomial f that has degree 2 or less, f(1)=f(4)=0 and f(2)=6.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find a possible formula for a polynomial f that has degree 2 or less, f(1)=f(4)=0 and f(2)=6.      Log On


   

Question 1188485: Find a possible formula for a polynomial f that has degree 2 or less, f(1)=f(4)=0 and f(2)=6.
Found 3 solutions by MathLover1, greenestamps, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!


formula for a polynomial f that has degree 2 is
f%28x%29=ax%5E2%2Bbx%2Bc
given:
f%281%29=f%284%29=0and+f%282%29=6
these are points
(1,0)
(4,0)
(2,6)
use them to find a,b, and c
0=a%2A1%5E2%2Bb%2A1%2Bc
0=a%2Bb%2Bc............solve for a
a=-b-c..........eq.1
0=a%2A4%5E2%2Bb%2A4%2Bc
0=16a%2B4b%2Bc............solve for a
16a=-4b-c
a=-4b%2F16-c%2F16
a=-b%2F4-c%2F16.........eq.2

6=a%2A2%5E2%2Bb%2A2%2Bc
6=4a%2B2b%2Bc............solve for a
4a=6-2b-c
a=6%2F4-2b%2F4-c%2F4
a=3%2F2-b%2F2-c%2F4.........eq.3
from eq.1 and eq.2 we have
-b-c=-b%2F4-c%2F16........solve for b
c%2F16-c=b-b%2F4
-15c%2F16=3b%2F4
-%2815c%2A4%29%2F3=b
b=-%285c%2F4%29..........eq. 1a
from eq.1 and eq.3 we have
-b-c=3%2F2-b%2F2-c%2F4......solve for b
c%2F4-3%2F2-c=b-b%2F2
-%283c%29%2F4+-+3%2F2=b%2F2
b=-2%283c%29%2F4+-+2%283%2F2%29
b=-%283c%29%2F2+-+3............eq.1b
from eq.1a and eq.1b we have
-5c%2F4=-%283c%29%2F2+-+3......solve for c
3=5c%2F4-%283c%29%2F2+
3=-c%2F4+
c=-12
substitute in eq. 1a
b=-%285%28-12%29%29%2F4
b=15
substitute b and c in eq. 1
a=-b-c..........eq.1
a=-15%2B12
a=-3
so,
a+=+-3, b+=+15, c+=+-12
and your equation is
f%28x%29=-3x%5E2%2B15x-12





Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


The solution from the other tutor is algebraically correct; but it is very inefficient.

Given that there are two roots at x=1 and x=4, and that the degree is to be 2 or less, we know that in fact it is degree 2.

And given those two roots, we know the polynomial is of the form

f%28x%29=a%28x-1%29%28x-4%29

To finish finding the polynomial, us the given information that f(2)=6:

f%282%29=a%282-1%29%282-4%29=a%281%29%28-2%29=-2a=6
a=-3

ANSWER:

One form of the polynomial function is f%28x%29=-3%28x-1%29%28x-4%29

In standard polynomial form that is f%28x%29=-3x%5E2%2B15x-12

Answer by ikleyn(52847) About Me  (Show Source):
You can put this solution on YOUR website!
.

May Lord save you from solving this problem in a way as @MathLover1 does it . . .