factor theorem

**mathaddict** · January 5th 2009, 05:18 AM

Given that (x-3) and (2x+1) are factors of
$f(x)=ax^4+bx^3+13x^2=30x+9$ , find the values of a and b . With these values of a and b , show that $f(x)\geq0$ for all x belongs to real numbers .

My working :

i found that a = 4 and b = -20

$4x^4-20x^3+13x^2+30x+9$
$(x-3)(2x+1)(2x^2-5x-3)$

I am wondering how can i show that $f(x)\geq0$ from here .
Thanks for any help .

**mr fantastic** · January 5th 2009, 05:27 AM

Originally Posted by mathaddict

Given that (x-3) and (2x+1) are factors of
$f(x)=ax^4+bx^3+13x^2=30x+9$ , find the values of a and b . With these values of a and b , show that $f(x)\geq0$ for all x belongs to real numbers .

My working :

i found that a = 4 and b = -20

$4x^4-20x^3+13x^2+30x+9$
$(x-3)(2x+1)(2x^2-5x-3)$

I am wondering how can i show that $f(x)\geq0$ from here .
Thanks for any help .

I suggest you find the coordinates and nature of the turning points of y = f(x) (note that $16x^3 -60x^2 + 26x + 30 = 2(x - 3)(2x + 1)(4x-5)$ .... )

**fanfan1609** · January 5th 2009, 08:15 AM

I think f(x) not >=0 with all real numbers .Maybe the problem is wrong .

**mr fantastic** · January 5th 2009, 11:30 AM

Originally Posted by fanfan1609

I think f(x) not >=0 with all real numbers .Maybe the problem is wrong .

You're wrong. On what grounds do you make this statement?

**Opalg** · January 5th 2009, 12:21 PM

Originally Posted by mathaddict

Given that (x-3) and (2x+1) are factors of
$f(x)=ax^4+bx^3+13x^2=30x+9$ , find the values of a and b . With these values of a and b , show that $f(x)\geq0$ for all x belongs to real numbers .

My working :

i found that a = 4 and b = -20

$4x^4-20x^3+13x^2+30x+9$
$(x-3)(2x+1)(2x^2-5x-3)$

I am wondering how can i show that $f(x)\geq0$ from here .
Thanks for any help .

Try factorising $2x^2-5x-3$ .

**mr fantastic** · January 5th 2009, 02:16 PM

Originally Posted by Opalg

Try factorising $2x^2-5x-3$ .

Well, you only do that if you want to do it the easy way .....

**fanfan1609** · January 5th 2009, 02:30 PM

Originally Posted by mr fantastic

You're wrong. On what grounds do you make this statement?

After factorising . I have $f(x)=(x-3).(2x+1).(2x-3).(x+1)$
If x belongs (3/2,3) ,f(x) <0 .Example x=2 ,I have f(2)=(2-3).(2.2+1).(2.2-3).(2+1)=(-1).5.1.3=-15
I thinks so that .

**mr fantastic** · January 5th 2009, 03:04 PM

Originally Posted by fanfan1609

After factorising . I have $f(x)=(x-3).(2x+1).(2x-3).(x+1)$
If x belongs (3/2,3) ,f(x) <0 .Example x=2 ,I have f(2)=(2-3).(2.2+1).(2.2-3).(2+1)=(-1).5.1.3=-15
I thinks so that .

$2x^2 - 5x - 3 \neq (2x - 3)(x+1)$ .

If you had expanded as a check you would have noticed this.

**fanfan1609** · January 5th 2009, 06:43 PM

Originally Posted by mr fantastic

$2x^2 - 5x - 3 \neq (2x - 3)(x+1)$ .

If you had expanded as a check you would have noticed this.

Oh.I had a stupid mistake .Thanks for your check
$2x^2 -5x -3 = (2x+1)(x-3)$ If that , $f(x)=(2x+1)^2(x-3)^2$ .so that f(x) >= 0 with all x belongs real numbers .
Sorry for my previous comments.Thanks

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