Question

The polynomial x^2+3x-1 is a factor of x^4+3x^3-2x^2-3x+1 true or false?

Answer 1

Yes, it is true that
`x^2+3x-1` is a factor of
`x^4+3x^3-2x^2-3x+1`.

Explanation:

Let us try to factorize
`x^4+3x^3-2x^2-3x+1`

`x^4+3x^3-2x^2-3x+1\\\Rightarrow x^4-2x^2+1-3x+3x^3`

Let us try to make a whole square of the given terms:

`\Rightarrow (x^2)^2-2* x^2 * 1+1^2+3x^3-3x\\\Rightarrow (x^2-1)^2+3x^3-3x\\`

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Formula used above:

`a^(2) -2 * a * b +b^(2) = (a-b)^2`

In the above equation, we had
`a = x, b = 1`.

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Further solving the above equation, taking
`3x` common out of
`3x^3-3x`

`\Rightarrow (x^2-1)^2+3x(x^2-1)\\`

Taking
`(x^(2) -1)` common out of the above term:

`\Rightarrow (x^2-1)((x^2-1)+3x)\\\Rightarrow (x^2-1)(x^2+3x-1)`

So, the two factors are
`(x^2-1)\ and\ (x^2+3x-1)`.

`\therefore` The statement that
`x^2+3x-1` is a factor of
`x^4+3x^3-2x^2-3x+1` is True.