Difficult polynomial question involving factor and remainder theorems

In summary, the conversation discusses how to prove that (a-b) is a factor of a^5-b^5 and how to find the other factor. The suggested methods include using the remainder theorem and the factor theorem, as well as long division with polynomials or synthetic division. An example of polynomial division is given and it is suggested to practice with simpler exercises first.
  • #1
stfz
35
0

Homework Statement



Prove that ##(a-b)## is a factor of ##a^5-b^5##, and find the other factor.

Homework Equations



Remainder theorem : remainder polynomial ##p(x)## divided by ##(x-a)## is equal to ##p(a)##
Factor theorem : if remainder = 0, then divisor was a factor of dividend.

The Attempt at a Solution



I think am able to prove that it is a factor:

##P(x) = x^5 - b^5##; we replace a with x
##P(x) = (x-b)Q(x) + 0## ; we assume that (x-b) is a factor
##P(b) = (b-b)Q(x) + 0 = 0 ##; proves that .. um.. I think I'm going the wrong way anyway. This doesn't really prove anything? er.. eh.. ?

And I can't really find the other factor :grumpy:
I could do most of the other questions in the exercise, but not this one (and other related ones!)

Any help would be appreciated
 
Last edited:
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  • #2
stfz said:

Homework Statement



Prove that ##(a-b)## is a factor of ##a^5-b^5##, and find the other factor.

Homework Equations



Remainder theorem : remainder polynomial ##p(x)## divided by ##(x-a)## is equal to ##p(a)##
Factor theorem : if remainder = 0, then divisor was a factor of dividend.

The Attempt at a Solution



I think am able to prove that it is a factor:

##P(x) = x^5 - b^5##; we replace a with x
##P(x) = (x-b)Q(x) + 0## ; we assume that (x-b) is a factor
##P(b) = (b-b)Q(x) + 0 = 0 ##; proves that .. um.. I think I'm going the wrong way anyway. This doesn't really prove anything? er.. eh.. ?

And I can't really find the other factor :grumpy:
I could do most of the other questions in the exercise, but not this one (and other related ones!)

Any help would be appreciated
Do you know long division with polynomials, or synthetic division ?
 
Last edited:
  • #3
Yes. However, attempting polynomial division on (x^5+b^5) didn't quite work for me. If that's how to solve the question, could someone give me an example?
 
  • #5
stfz said:
Yes. However, attempting polynomial division on (x^5+b^5) didn't quite work for me. If that's how to solve the question, could someone give me an example?

Maybe that's because you want to divide (a[itex]^{5}[/itex]-b[itex]^{5}[/itex]) by (a - b).

Start off with a few warm-up exercises:

(a[itex]^{2}[/itex]-b[itex]^{2}[/itex])/(a-b)

(a[itex]^{3}[/itex]-b[itex]^{3}[/itex])/(a-b)

You should know the answer to the first exercise by inspection.
 
  • #6
Woops, got what I missed. Thanks!
 

1. What is the purpose of the factor theorem in solving a difficult polynomial question?

The factor theorem is used to factorize a polynomial into simpler expressions, making it easier to solve and find its roots. It helps to identify possible factors of a polynomial and determine if a given value is a root or not.

2. How does the remainder theorem help in solving a difficult polynomial question?

The remainder theorem states that if a polynomial f(x) is divided by (x-a), then the remainder is equal to f(a). This helps in finding the remainder of a polynomial division, which can be used to solve difficult polynomial questions involving long division or synthetic division.

3. Can the factor and remainder theorems be used interchangeably?

No, the factor and remainder theorems are used for different purposes. The factor theorem is used to factorize a polynomial, while the remainder theorem is used to find the remainder of a polynomial division. However, they both involve finding roots of a polynomial, and can be used together in some cases.

4. Are the factor and remainder theorems applicable to all polynomials?

Yes, the factor and remainder theorems are applicable to all polynomials, including difficult ones. However, the process of applying these theorems may become more complex for higher degree polynomials.

5. Can the factor and remainder theorems be used to solve all polynomial equations?

No, the factor and remainder theorems can only be used to solve polynomial equations with rational roots. They cannot be used to solve equations with irrational or complex roots. Other methods, such as the quadratic formula or graphing, may be needed to solve these types of polynomial equations.

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