factor theorem checking?

Question

Use the factor theorem to show that x-c is a factor of P(x) for the given values of c, where
P(x)=x^3-5x^2+7x-3, c=1

The quotient ?
the remainder ?
please find the quotient and the remainder
i know how to do thid i just need to check my answer with your answer so i could know if i did it right

Answer 1

P(x) = x^3 - 5x^2 + 7x - 3

P(c) = c^3 - 5c^2 + 7c - 3

when c = 1

P(1) = 1 - 5 + 7 - 3 = 8-8 = 0

so 1 is a root the given expression and (x-1) is a factor.

x - 1) x^3- 5x^2+ 7x- 3 ( x^2
-------x^3- x^2
_________________
----------- - 4x^2 + 7x(-4x
------------ -4x^2 + 4x
__________________
---------------------- 3x - 3(3
------------------------3x - 3
____________________
---------------------------0

so the quotient is x^2 - 4x + 3 and remainder is zero

x^2 - 4x +3 = 0

(x - 3)(x-1) = 0

so the three roots of the given expression are

1, 1 and 3

Answer 2

First of all, your excuse is very see through ;). If you use the factor theorem right, you just need to sub it in and get 0 to "check your answer" instead of hopping on the computer. But I'll still do it coz I'm revising for an exam which will probably have the same thing in it somewhere.

P(x)=x^3-5x^2+7x-3 where (x-1) is a factor. So sub in 1.
P(1) = 1 - 5 + 7 - 3
=0

So (x-1) is a factor.

You can find the quotient by long division - but there will be no remainder because (x-1) is a factor. But I wont do the long division on the conputer - thats just nightmarish having to type it out.

But it give me: P(x) = (x-1)(x^2 - 4x + 3)

So the quotient is x^2 - 4x + 3, and the remainder is 0.

Answer 3

If x-c i.e x-1 is a factor of x^3-5x^2+7x-3,then if we substitute x by 1,the value of the expression becomes zero
Let us check
(1)^3-5(1)^2+7(1)-3
=1-5+7-3
=8-8
=0
Therefore x-1 is a factor of the expression
Now to find the quotient divide the expression by x-1
x^2(x-1)-4x(x-1)+3(x-1)
Therefore the quotient is x^2-4x+3
When x-1 is a factor of the expression,there can not be any remainder

Answer 4

If x-c is a factor of P(x) then P(c)=0

P(c) = P(1) = 1³ - 5*1² + 7*1 - 3 = 1 - 5 + 7 - 3 = 0

To get the quotient, Q(x), divide P(x) by x-1:

Q(x) = x² - 4x + 3

Remainder is zero since x-1 is a factor

Answer 5

Basically, you want to know if x-1 is a root of the cubic?

If you divide the cubic by x-1, you get x^2-4x+3.
You should have no remainder if (x-1) is a root.
BTW: the quadratic factors easily.