Rewrite each of the following as a product of linear factors
1. x^2+3
My Answer: (x+iroot 3)(x-iroot3)
2. x^2-2x+3
MA: (x-(1+iroot2))(x-(1-iroot2))
3. x^4+x
MA: IM not sure how to do it! can you help?
4. x^5-x^4+5x^3
MA: x^3(x-(1+iroot19/2))(x-(1-iroot19/2))
5. 3x^2+7
MA: NOT sure how to do
6. x^6+3x^5-10x^4-4x^2-12x+40
MA: (x-2)(x+5)(x+iroot2)(x-iroot2)(x+root2) (x-root2)
THANKS SO MUCH!!!
3. I don't see linear factors coming out of this.
5. (xsqrt4+isqrt7)(xsqrt3-isqrt7)
thanks
are all the others correct?
Sure! I can help you with the remaining questions.
3. To factorize x^4 + x into a product of linear factors, we can first try to factorize out the common factor, which in this case is x. So, we have x(x^3 + 1). Now, we need to factorize the expression x^3 + 1 further. We notice that it is a sum of cubes, which can be factored using the formula:
a^3 + b^3 = (a + b)(a^2 - ab + b^2).
So, we can rewrite x^3 + 1 as (x + 1)(x^2 - x + 1). Therefore, the expression x^4 + x can be written as x(x + 1)(x^2 - x + 1).
5. To factorize 3x^2 + 7, we need to see if it can be factored further. However, this expression cannot be factored into linear factors because both terms have different powers of x. So, in this case, 3x^2 + 7 cannot be factored into a product of linear factors.
I hope that helps! Let me know if you have any further questions.