Use factor theorem- p(x)=x^2-5x+6,q(x)=x-2?

Question

use factortheorem to verify in the question that q(x) is a factor of p(x).
Factor Theorem
x-a is a factor of polynomial p(x) if p(a) = 0.

Answer 1

You can try 2,3,4........... to know the value of x for which p(x) vanishes.

When x = 2
p(x) = x^2-5x+6 = 2^2-5*2+6 = 0

p(x) vanishes for x = 2
Therefore, x-2 is is a factor of p(x)

But x-2 = q(x)
Therefore, q(x) is a factor of p(x)

Answer 2

In order for you to know if q(x) is a factor of p(x), p(a) must be equal to 0 where a is a root of q(x).

So that: for q(x) = x - 2
To solve for the root, let q(x) = 0, x - 2 = 0; x = 2
Therefore, a = 2 where a is the root of q(x)

Then we solve for p(a):
p(a) = p(2) = (2)^2 - 5*(2) + 6 = 4 - 10 + 6 = 0

Since p(a) = 0, then we can conclude that x-2 is a factor of x^2-5x+6

To simplify:
To solve for the REMAINDER of a polynomial ratio: P(x)/Q(x)
We should solve for P(a), where a is the root of Q(x)
So if the remainder is 0, then P(x) is divisible by Q(x) making Q(x) a factor of P(x).

Hope it helps!

Answer 3

You know the factor theorem;
"x-a is a factor of polynomial p(x) if p(a) = 0."

So if for q(x) to be a factor of p(x) and q(x) = x-2 then "a" must be 2.

To prove that q(x) is a factor of p(x) you simply use the theorem and carry out p(2)

i.e.
p(2) = (2)^2 - 5(2) + 6
= 4 - 10 + 6 = 0

Therefore q(x) must be a factor of p(x)

Hope this helps,
Cheers,
T

Answer 4

Start with your factor theorem:
x - a is a factor of polynomial p(x) if p(a) = 0

since q(x) = x -2, then a = 2

so,
p(x) = x^2 - 5x + 6
p(a) = a^2 -5a + 6
p(2) = 2^2 - 5 * 2 + 6 = 4 - 10 + 6 = -6 + 6 = 0

therefore, yes, q(x) is a factor of p(x)

Answer 5

q(x=x-2) is a factor of P(x=x^2-5x+6),if
P(2)=0
Now we calculate P(x) at x=2
2^2-5x2+6=4-10+6=0.
so x-2 is a factor of x^2-5x+6
x^2-5x+6=x(x-2)-3(x-2)=(x-2)(x-3). ANS.

Answer 6

x^2-5x+6=x^2-2x-3x+6
=x(x-2)-3(x-2)
=(x-2)(x-3)
since x-a is a factor of polynomial p(x)ifp(a)=0
x-2
p(2)=2^2-5*2+6
=4-10+6
=0
thus q(x) is a factor of p(x)



is it clear
bye bye

Answer 7

p(x) = x² - 5x + 6
p(2) = 2² - (5 x 2) + 6
p(2) = 4 - 10 + 6
p(2) = 0
Thus (x - 2) is a factor of x² - 5x + 6
Further:-
x² - 5x + 6 = (x - 3) (x - 2)

Answer 8

x^2 -5x+6 = (x-2)(x-3)
x-2 = 0, x-3 = 0

Answer 9

Hey Buddy,
Well I don't remember much of the mathematics but still giving it a shot, I hope it'll solve your query.

=X^ - 5X + 6
= X2-3X-2X + 6
=X (X-3)- 2(X-3)
=(X-3) (X-2)

Hence it clearly proves that qx=(X-2) is a factor of p(x).

Answer 10

To factor p(x) go ahead and plug in q(x) as a factor so you get

p(x)=(x-_)(x-2)

Now go ahead and figure out what the missing value is. we know that it multiplied by -2 has to be +6 so it's going to be -3 and when you add -3 and -2 you get -5

so p(x)=(x-3)(x-2)