Determine which of the following polynomials has $(x + 1)$ a factor :
$x^{4} + x^{3} + x^{2} + x + 1$

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Solution

Determining whether polynomial has $(x + 1)$ a factor or not
Let $p (x) = x^{4} + x^{3} + x^{2} + x + 1$ be the polynomial
By factor theorem, if $p (x)$ is a polynomial of degree $n > 1$ and $a$ is any real number, then $x - a$ is a factor of $p (x)$ , if $p (a) = 0$ ,
Therefore, $(x + 1)$ is a factor of $p (x)$ , if $p (- 1) = 0$
$p (x) = x^{4} + x^{3} + x^{2} + x + 1$
$p (- 1) = {(- 1)}^{4} + {(- 1)}^{3} + {(- 1)}^{2} + (- 1) + 1$
$p (- 1) = 1 - 1 + 1 - 1 + 1$
$p (- 1) = - 2 + 3 = 1$
$p (- 1) = 1$
Since, $p (- 1) \neq 0$
Hence, $(x + 1)$ is not a factor of polynomial $x^{4} + x^{3} + x^{2} + x + 1$ .

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Similar questions

Determine which of the following polynomials has (x + 1) a factor:

(i) x³ + x² + x + 1 (ii) x⁴ + x³ + x² + x + 1

(iii) x⁴ + 3x³ + 3x² + x + 1 (iv)

Now in each of the pairs of polynomials given below, check whether the first is a factor of the second:

(i) x + 1, x³ − 1

(ii) x − 1, x³ + 1

(iii) x + 1, x³ + 1

(iv) x² − 1, x⁴ − 1

(v) x − 1, x⁴ − 1

(vi) x + 1, x⁴ − 1

(vii) x − 2, x² − 5x + 1

(viii) x + 2, x² + 5x + 6

(ix)

(x) 1.3x − 2.6, x² − 5x + 6

(x + 1)

x^{4} + x^{3} + x^{2} + x + 1

(x + 1)

x^{3} + x^{2} + x + 1

x^{4} + x^{3} + x^{2} + x + 1

Determine which of the following polynomials has (x+1) as a factor:
(i) $x^{3}$ + $x^{2}$ +x+1 (ii) $x^{4}$ + $x^{3}$ + $x^{2}$ +x+1

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