Question

Check whether x + 1 is the factor of the polynomial x

4

+ 3 x3

+ 3 x2

+ x + 1​

Answer 1

Answer:

Let we consider the polynomial is f(x)

So,

f(x) = \bold{x^{4} \: + \: x^{3} \: +\: x^{2} \:-\: 5x\: + \:1}

Now we have (x+1)

Now we consider (x+1) is a factor of polynomial and x = -1

Using factor theorem [f(x) = 1]

\bold{f(x)\: = \:x^{4} \: + \: x^{3} \: + \: x^{2} \: -\: 5x\: +\: 1}

\bold{f(-1)\: = \:(-1)^{4} \: + \: (-1)^{3} \: + \: (-1)^{2} \: -\: 5(-1)\: +\: 1}

\bold{f(-1) \: = \: 1 \: -\: 1 \: + \: 1 \: +\: 5 \: +1}

\bold{f(-1) \: = 8\: -\:1}

\bold{f(-1) \: = 7}

The factor theorem say if

f(x) = 0

Then the (x+1) is a factor of polynomial f(x) but here it's equal to \bold{7}

From this statement \bold{x+1} is not a factor of polynomial \bold{x^{4} + x^{3} + x^{2} -5x +1}