Show that f(x) = 0 has only one root

[Clever Penguin]

Homework Statement

[/B]
f(x) = 2x³+ax²+bx+10
When f(x)/(2x-1) the remainder is 12
When f(x)/(x+1) there is no remainder
a) Find the value of a and b
b) Show that f(x) = 0 has only one root

Homework Equations

None

The Attempt at a Solution

a) (2x-1)=0
x=1/2

f(1/2) = 12 = 2(1/2)³+b(1/2)²+a(1/2)+10
1/4+a/4+b/2+10=12
1+a+2b+40=48
a+2b=7

(x+1)=0
x=-1

f(-1)= 0 = 2(-1)³+a(-1)²+b(-1)+10
a-b+8=0
b=a+8

b=(7-2b)+8
b=15-2b
3b=15
b=5

a=7-2b
a=7-10
a=-3

a=-3 and b=5

b) 2x³-3x²+5x+10=0
Now I need to factorise this, but I don't know how
I tried using x as a common factor, but its not. 10 does not have a factor of x.

 

[Delta2]

You know x+1 is a factor since the f(-1)=0 or the division by x+1 doesn't leave any reminder.

 

[Ray Vickson]

Homework Statement

[/B]
f(x) = 2x³+ax²=bx+10
When f(x)/(2x-1) the remainder is 12
When f(x)/(x+1) there is no remainder
a) Find the value of a and b
b) Show that f(x) = 0 has only one root

Homework Equations

None

The Attempt at a Solution

a) (2x-1)=0
x=1/2

f(1/2) = 12 = 2(1/2)³+b(1/2)²+a(1/2)+10
1/4+a/4+b/2+10=12
1+a+2b+40=48
a+2b=7

(x+1)=0
x=-1

f(-1)= 0 = 2(-1)³+a(-1)²+b(-1)+10
a-b+8=0
b=a+8

b=(7-2b)+8
b=15-2b
3b=15
b=5

a=7-2b
a=7-10
a=-3

a=-3 and b=5

b) 2x³-3x²+5x+10=0
Now I need to factorise this, but I don't know how
I tried using x as a common factor, but its not. 10 does not have a factor of x.

The formula you ##f(x) = 2x^3+ax^2 = bx + 10## makes no sense. Do you mean ##f(x) = 2x^3+ax^2 + bx + 10## or ##f(x) = 2x^3 + ax^2 - bx -10##, or what?

 

[Clever Penguin]

You know x+1 is a factor since the f(-1)=0 or the division by x+1 doesn't leave any reminder.

So I use long division?

(2x³-3x² + 5x + 10)/(x+1)?

The formula you ##f(x) = 2x^3+ax^2 = bx + 10## makes no sense. Do you mean ##f(x) = 2x^3+ax^2 + bx + 10## or ##f(x) = 2x^3 + ax^2 - bx -10##, or what?

I changed it
I meant f(x) = 2x³+ax² + bx + 10

 

[Delta2]

yes do the polynomial division and you 'll find the quotient polynomial of 2nd degree is such that it doesn't have real roots.

 

[Clever Penguin]

yes do the polynomial division and you 'll find the quotient polynomial of 2nd degree is such that it doesn't have real roots.

I got (2x³-3x² + 5x + 10)/(x+1) = 2x²-5x+10

So 2x³-3x² + 5x + 10 = (x+1)(2x²-5x+10)

 

[Delta2]

Very nice I guess you can see why ##2x^2-5x+10## does not have any real roots... So you original function written as the product of those two polynomials has how many real roots?

 

[Clever Penguin]

Very nice I guess you can see why ##2x^2-5x+10## does not have any real roots... So you original function written as the product of those two polynomials has how many real roots?

One
Thanks