Chapter 4: Q. 91 (page 210)

Use the Factor Theorem to prove that $x - c$ is a factor of
$x^{n} - c^{n}$ for any positive integer n.

Short Answer

Expert verified

If $f (x) = x^{n} - c^{n}$ then $f (c) = 0$ so $x - c$ is a factor of $x^{n} - c^{n}$

Step by step solution

Step 1. given data

The given expression is

$x^{n} - c^{n}$

Factor is $x - c$

Step 2. proof

Consider function $f (x) = x^{n} - c^{n}$

According to Factor Theorem, If $x - c$ is a factor of function then $f (c) = 0$

substitute c for x in the function

$f (x) = x^{n} - c^{n} f (c) = c^{n} - c^{n} f (c) = 0$

the function value is zero for $x = c$

so $x - c$ is a factor of $x^{n} - c^{n}$

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Use the Factor Theorem to prove that $x - c$ is a factor of
$x^{n} - c^{n}$ for any positive integer n.

Short Answer

Step by step solution

Step 1. given data

Step 2. proof

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Most popular questions from this chapter

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Use the Factor Theorem to prove that $x - c$ is a factor of
$x^{n} - c^{n}$ for any positive integer n.