A Factor Theorem Question

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In summary, the conversation is about how to show that a given expression is divisible by a certain polynomial. The hint given is to notice the factors of the polynomial and use them to simplify the expression. The final suggestion is to continue writing out the terms until the desired result is achieved.
  • #1
Hyperreality
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My maths teacher says this problem is not as impossible as it seems, but I just can't solve it.

Show that (x + a + b)^7 - x^7 - a^7 - b^7 is divisble by
x^2 + (a + b)x +ab.
 
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  • #2
Looking for the easy way out huh ?
You can always solve the entire excercise...

Live long and prosper.
 
  • #3
Show that (x + a + b)^7 - x^7 - a^7 - b^7 is divisble by x^2 + (a + b)x +ab.

Hint:
Notice that (x+a) and (x+b) are the 2 factors of x^2 + (a + b)x +ab.
So it is equivalent to show that (x + a + b)^7 - x^7 - a^7 - b^7 is divisible by both (x+a) and (x+b).

Let f(x) = (x + a + b)^7 - x^7 - a^7 - b^7
...
...
...
...


Can you continue from here?

Hope this help. :smile:
 
  • #4
just write everything out
eg. (x+a)^2=x^2+2xa+a^2

maybe rewrite some terms then and you will see that it is divisible by x^2 + (a + b)x +ab
 
  • #5
KL has the easy way!

Writing it out however... *shudder* I wouldn't wish writing out a trinomial to the 7th power to anyone!

Hurkyl
 

1. What is the Factor Theorem?

The Factor Theorem is a fundamental theorem in algebra that states that if a polynomial function f(x) has a root x = a, then (x-a) is a factor of the polynomial.

2. How do you use the Factor Theorem to factor a polynomial?

To use the Factor Theorem, you first need to find a root of the polynomial by setting the polynomial equal to zero and solving for the variable. Then, divide the polynomial by (x-a) to find the other factor of the polynomial.

3. What is the relationship between the Factor Theorem and the Remainder Theorem?

The Factor Theorem and the Remainder Theorem are closely related. The Factor Theorem states that if (x-a) is a factor of a polynomial, then x = a is a root of the polynomial. The Remainder Theorem states that if you divide a polynomial f(x) by (x-a), the remainder is equal to f(a).

4. Can the Factor Theorem be used to solve all polynomial equations?

No, the Factor Theorem can only be used to solve polynomial equations that have a root. If a polynomial does not have a root, the Factor Theorem cannot be applied.

5. What are some real-world applications of the Factor Theorem?

The Factor Theorem has many real-world applications in fields such as engineering, physics, and economics. It can be used to solve problems involving optimization, finding roots of equations, and predicting the behavior of systems.

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