Grade 12 Calculus Problem - Differentiation and Division

[G-S]

Homework Statement

Let p be a polynomial function with p(a)= 0 =p'(a) for some real a. Which of the following must be true?

A) p(x) is divisible by x+a
B) p(x) is divisible by x^2+a^2
C) p(x) is divisible by x^2-a^2
D) p(x) is divisible by x^2+2ax+a^2
E) p(x) is divisible by x^2-2ax+a^2

Homework Equations

The Attempt at a Solution

I'm honestly beyond stumped for this one. Having a hard time finding out where to start on this problem.

 

[SammyS]

What does it mean for a polynomial (or any function) if p(x) = 0 at x=a ?

What does it mean for a polynomial, if p'(x) = 0 at x=a ?

Put the answers together.

 

[vela]

Hint: p(x) is a polynomial and p(a)=0 means a is a root of the polynomial. If you were to factor p(x), what factor do you know has to be there if a is a root?

 

[G-S]

Thanks for the replies.
So this is the conclusion I've come to so far,

p(x)=0 at x=a
p'(x)=0 at x=a
a is a root of the polynomial

So,
x²-a²=0
x²=a²
x=a

Therefore, C) must be true.

Am I in the right direction?

 

[SammyS]

No, unless you know that -a is also a root.

What is true about a function if it's first derivative is zero at some value of x ?

 

[G-S]

Sorry SammyS but the only thing that comes to mind is the function is a constant. Either that or the first step to find critical points.

After looking at the problem again I've come to the conclusion that in order to fulfill the condition p(a)=0 only C) and E) can be true.

Considering that we only know that a is a root (and not -a) E) seems like the only one that fulfills the conditions.

0=x²-2ax+a²
0=(x-a)²
0=x-a
x=a

Taking stabs in the dark here.

 

[e(ho0n3]

After looking at the problem again I've come to the conclusion that in order to fulfill the condition p(a)=0 only C) and E) can be true.

Considering that we only know that a is a root (and not -a) E) seems like the only one that fulfills the conditions.

You have narrowed down the list of choices to just E, so E must be the answer. Is there anything more you need to know?

 

[G-S]

You have narrowed down the list of choices to just E, so E must be the answer. Is there anything more you need to know?

I'm trying to get confirmation as to whether the answer and my thought process is correct and if not, how one would approach this question.

 

[SammyS]

Sorry SammyS but the only thing that comes to mind is the function is a constant. Either that or the first step to find critical points.

...

And, in general, why is it that you look for critical points?

Also, what is the slope of a constant function? (I'm assuming that this is not a constant polynomial.)

 

[G-S]

And, in general, why is it that you look for critical points?

Also, what is the slope of a constant function? (I'm assuming that this is not a constant polynomial.)

To find max/min values. And the slope of a constant function is 0.

 

[SammyS]

More to the point, what is the slope of the line tangent to the polynomial at x = a, if p'(a) = 0 ?

This is the question I should have asked in my previous post.

 

[G-S]

The slope of the tangent line is 0 if p'(a)=0.

 

[SammyS]

The behavior of a polynomial in the neighborhood of one of its zeros is due mainly to the factor which 'causes' the zero, and the multiplicity of that zero.

So, in the neighborhood of x=a, the polynomial, p(x), behaves like ±(x - a)ⁿ, where n ≥ 2 .

How do we know it's not like ±(x - a)¹ ?

 

[G-S]

How do we know it's not like ±(x - a)¹ ?

Because it's not one of the options? Haha, I haven't got a clue.

 

[SammyS]

The derivative of (x - a) ≠ 0 at x = 1 .

 

[HallsofIvy]

The derivative of (x - a) ≠ 0 at x = 1 .

I presume you mean at x= a.

 

[SammyS]

Yes. Thanks for the correction.