Factors of xⁿ + aⁿ and xⁿ − aⁿ

Expres­sions such as $x^3 + 8$, $x^4 - 1$ and $x^5 - 32$ appear again and again, and so do num­ber ques­tions like "find the remain­der when $13^{141} + 11^{141}$ is divided by $24$". All of them are about one fam­ily, $x^n + a^n$ and $x^n - a^n$. The fac­tor the­o­rem tells us exactly when $x + a$ and $x - a$ divide these expres­sions, and a pat­tern gives the other fac­tor. This les­son col­lects those results, which you will use for the rest of your alge­bra.

Result 1: x + a divides xⁿ + aⁿ when n is odd#

$x + a$ is a fac­tor of $x^n + a^n$ for every odd pos­i­tive inte­ger $n$.

Proof. Let $f(x) = x^n + a^n$. By the fac­tor the­o­rem we test $x = -a$:

$$f(-a) = (-a)^n + a^n = -a^n + a^n = 0 \quad \text{(since } n \text{ is odd)}$$

So $x + a$ is a fac­tor.

The other fac­tor fol­lows a pat­tern of alter­nat­ing signs, with the power of $x$ falling and the power of $a$ ris­ing:

$$x^n + a^n = (x + a)\left(x^{n-1} - x^{n-2}a + x^{n-3}a^2 - \dots + a^{n-1}\right), \quad n \text{ odd}$$

For exam­ple,

$$x^3 + a^3 = (x + a)(x^2 - xa + a^2)$$

$$x^5 + y^5 = (x + y)(x^4 - x^3y + x^2y^2 - xy^3 + y^4)$$

The first of these is the famil­iar iden­tity $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

What if $n$ is even? Then $f(-a) = (-a)^n + a^n = a^n + a^n = 2a^n$, which is not zero (for $a \neq 0$). So $x + a$ is not a fac­tor of $x^n + a^n$ when $n$ is even. For instance, $x^2 + 1$ has no fac­tor $x + 1$.

Result 2: x − a divides xⁿ − aⁿ for every n#

$x - a$ is a fac­tor of $x^n - a^n$ for every pos­i­tive inte­ger $n$.

Proof. Let $f(x) = x^n - a^n$. Then $f(a) = a^n - a^n = 0$, so $x - a$ is a fac­tor.

This time all the signs in the other fac­tor are plus:

$$x^n - a^n = (x - a)\left(x^{n-1} + x^{n-2}a + x^{n-3}a^2 + \dots + a^{n-1}\right)$$

For exam­ple,

$$x^2 - a^2 = (x - a)(x + a)$$

$$x^3 - a^3 = (x - a)(x^2 + xa + a^2)$$

$$x^4 - a^4 = (x - a)(x^3 + x^2a + xa^2 + a^3)$$

Result 3: both x + a and x − a divide xⁿ − aⁿ when n is even#

$x + a$ and $x - a$ are both fac­tors of $x^n - a^n$ for every even pos­i­tive inte­ger $n$.

We already know $x - a$ is a fac­tor. For $x + a$, test $x = -a$: $f(-a) = (-a)^n - a^n = a^n - a^n = 0$, because an even power of $-a$ is pos­i­tive.

$$x^2 - a^2 = (x + a)(x - a)$$

$$x^4 - a^4 = (x^2 + a^2)(x^2 - a^2) = (x^2 + a^2)(x + a)(x - a)$$

Worked exam­ples#

Exam­ple 1#

Fac­torise $x^4 - 1$ in two ways.

Using Result 2 with $a = 1$:

$$x^4 - 1 = x^4 - 1^4 = (x - 1)(x^3 + x^2 + x + 1)$$

Using the dif­fer­ence of squares twice:

$$x^4 - 1 = (x^2)^2 - 1^2 = (x^2 + 1)(x^2 - 1) = (x^2 + 1)(x + 1)(x - 1)$$

The sec­ond form is com­plete. Com­par­ing the two shows that $x^3 + x^2 + x + 1 = (x + 1)(x^2 + 1)$.

Exam­ple 2#

Fac­torise $x^5 - 1$ and $x^5 + 1$.

$$x^5 - 1 = x^5 - 1^5 = (x - 1)(x^4 + x^3 + x^2 + x + 1)$$

Since $5$ is odd, Result 1 applies to $x^5 + 1$:

$$x^5 + 1 = (x + 1)(x^4 - x^3 + x^2 - x + 1)$$

Exam­ple 3#

Fac­torise $x^3 + 1$ and $x^3 + 8$.

$$x^3 + 1 = x^3 + 1^3 = (x + 1)(x^2 - x + 1)$$

$$x^3 + 8 = x^3 + 2^3 = (x + 2)(x^2 - 2x + 4)$$

In the same way $x + 2$ is a fac­tor of $x^5 + 32$, because $32 = 2^5$ and $5$ is odd.

Remain­ders of large num­bers#

The results hold for num­bers too, because sub­sti­tut­ing whole num­bers for $x$ and $a$ keeps every coef­fi­cient of the other fac­tor a whole num­ber.

Exam­ple 4#

Find the remain­der when $13^{141} + 11^{141}$ is divided by $24$.

Take $x = 13$, $a = 11$ and $n = 141$, which is odd. By Result 1, $x + a = 13 + 11 = 24$ is a fac­tor of $13^{141} + 11^{141}$. So the remain­der is $0$.

Exam­ple 5#

Find the remain­der when $x^{11} - 1$ is divided by $x - 1$.

By Result 2, $x - 1$ is a fac­tor of $x^{11} - 1$, so the remain­der is $0$. (Directly: $f(1) = 1 - 1 = 0$.)

Prac­tice#

1. Fac­torise $x^3 + 27$.
2. Fac­torise $x^5 - 32$.
3. Fac­torise $x^4 - 81$ com­pletely.
4. Fac­torise $x^5 + 243$.
5. Fac­torise $x^6 - 1$ com­pletely.
6. Show that $x - 1$ is a fac­tor of both $x^{10} - 1$ and $x^{11} - 1$.
7. Is $x + 1$ a fac­tor of $x^{10} + 1$? If not, find the remain­der.
8. Find the remain­der when $7^{25} + 5^{25}$ is divided by $12$.
9. Find the remain­der when $17^{20} - 13^{20}$ is divided by $30$.
10. Find the remain­der when $x^{51} + 51$ is divided by $x + 1$.
11. Find $k$ if $x^{140} + 2x^{151} + k$ is divis­i­ble by $x + 1$.

Answers#