Question

What is/are the factor(s) of $({x^{29}} - {x^{24}} + {x^{13}} - 1)?$
A.$(x - 1)$ only
B.$(x + 1)$ only
C.$(x - 1)$ and $(x + 1)$
D.Neither $(x - 1)$ nor $(x + 1)$

Answer 1

Hint: Here we will take the given expression and place the values from the given multiple choices and then simplify for the given factors. To get the factors, place the value of “x” in the given expression and it should be equal to zero.

Complete step-by-step answer:
Take the given expression: $P(x) = ({x^{29}} - {x^{24}} + {x^{13}} - 1)$
Take the option A:
$x - 1$ to be the factor,
$x - 1 = 0$
Moving constantly on the opposite side, when you move any term from one side to the opposite side then the sign of the term also changes. Positive terms will become negative and vice-versa.
$ \Rightarrow x = 1$
Place the value of “x” in the given expression –
$P(x) = {x^{29}} - {x^{24}} + {x^{13}} - 1$
$P(1) = {1^{29}} - {1^{24}} + {1^{13}} - 1$
Any power to the number one is always one.
$P(1) = 1 - 1 + 1 - 1$
Like terms with the same value and the opposite sign cancels each other.
$P(1) = 0$
So, $(x - 1)$is the factor.
Similarly, take the option B:
$x + 1$to be the factor,
$x + 1 = 0$
Moving constantly on the opposite side, when you move any term from one side to the opposite side then the sign of the term also changes. Positive terms will become negative and vice-versa.
$ \Rightarrow x = - 1$
Place the value of “x” in the given expression –
$P(x) = {x^{29}} - {x^{24}} + {x^{13}} - 1$
$P( - 1) = {( - 1)^{29}} - {( - 1)^{24}} + {( - 1)^{13}} - 1$
Any even power to the number negative one is always positive one whereas the odd power to the negative one gives negative one.
$P( - 1) = - 1 - 1 - 1 - 1$
When you combine negative terms, you have to add the terms and give a negative sign to the resultant value.
$P( - 1) = - 4$
So, $(x + 1)$ is not the factor.
Hence, from the given multiple choices – the option A is the correct answer.
So, the correct answer is “Option A”.

Note: Be careful about the sign convention while simplification and remember the basic fundamental that ${( - 1)^{even{\text{ }}number}} = 1$ and ${( - 1)^{odd\,number}} = - 1$as the product of two negative term gives positive term and the product of one negative and one positive term gives negative term.