Question

Show that x^(2)+1 is a factor of f(x)=x^(3)-3x^(2)+x-3. Find the quotient of dividing f(x) by x^(2)+1.

Answer 1

By performing polynomial division, it is shown that x^2 + 1 is a factor of the polynomial f(x) = x^3 - 3x^2 + x - 3, giving a quotient of x - 3 and a remainder of 0, confirming that x^2 + 1 is indeed a factor.

Step-by-step explanation:

To show that x^2 + 1 is a factor of f(x) = x^3 - 3x^2 + x - 3, we can use polynomial division or apply the remainder theorem. For the sake of clarity, polynomial division will be demonstrated here.

Divide f(x) by x^2 + 1:

1. Dividend: x^3 - 3x^2 + x - 3
2. Divisor: x^2 + 1
3. Divide the first term of the dividend by the first term of the divisor, which is x. Multiply the entire divisor by x and subtract the result from the dividend.
4. This process is repeated until the terms of the dividend have been fully divided or a remainder is left that is of a lower degree than the divisor.

The division yields the following:

• x(x^2) = x^3
• x(1) = x
• -3x^2 - x + (x + 3) = -3x^2 + 3

Bringing down the next term from the dividend:

• -3(x^2) = -3x^2
• -3(1) = -3
• (x - 3) - (-3x^2 + 3) = 0

The quotient from this division is x - 3, and the remainder is 0, which confirms that x^2 + 1 is a factor of f(x). Thus the quotient when dividing f(x) by x^2 + 1 is x - 3.