Question

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) x3 - x2 - (2 + √2)x + √2

Answer 1

To determine if (x + 1) is a factor of each polynomial, we use synthetic division. The polynomials i), ii), iii), and iv) do not have (x + 1) as a factor.

Step-by-step explanation:

To determine which of the given polynomials has (x + 1) as a factor, we can use synthetic division to divide each polynomial by (x + 1). If the remainder is 0, then (x + 1) is a factor.

i) x^3 + x^2 + x + 1:

(x + 1) | 1 1 1 1
| -1 0 -1
--------
1 0 1 0

The remainder is not 0, so (x + 1) is not a factor of x^3 + x^2 + x + 1.

ii) x^4 + x^3 + x^2 + x + 1:

(x + 1) | 1 1 1 1 1
| -1 0 -1 0
--------
1 0 1 0 1

The remainder is not 0, so (x + 1) is not a factor of x^4 + x^3 + x^2 + x + 1.

iii) x^4 + 3x^3 + 3x^2 + x + 1:

(x + 1) | 1 3 3 1 1
| -1 -4 1 -2
--------
1 2 -1 -1 -1

The remainder is not 0, so (x + 1) is not a factor of x^4 + 3x^3 + 3x^2 + x + 1.

iv) x^3 - x^2 - (2 + √2)x + √2:

(x + 1) | 1 -1 -2 -√2
| -1 0 2√2
--------
1 -2 -2√2 -√2

The remainder is not 0, so (x + 1) is not a factor of x^3 - x^2 - (2 + √2)x + √2.