Question

x + 1 is a factor of the polynomial ______.

Options

• x³ + x² – x + 1 

• x³ + x² + x + 1

• x⁴ + x³ + x² + 1  

• x⁴ + 3x³ + 3x² + x + 1

Answer 1

x + 1 is a factor of the polynomial x³ + x² + x + 1.

Explanation:

We know that if x + a is a factor of f(x) then, f(–a) = 0.

(A) Let f(x) = x³ + x² – x + 1

Now, f(–1) = (–1)³ + (–1)² – (–1) + 1

= –1 + 1 + 1 + 1

= 2 ≠ 0

So, f(x) is not a factor of x + 1.

(B) Let f(x) = x³ + x² + x + 1

Now, f(–1) = (–1)³ + (–1)² + (–1) + 1

= –1 + 1 – 1 + 1

= 0

So, f(x) is a factor of x + 1.

(C) Let f(x) = x⁴ + x³ + x² + 1

Now, f(–1) = (–1)⁴ + (–1)³ + (–1)² + 1

= 1 – 1 + 1 + 1

= 2 ≠ 0

So, f(x) is not a factor of x + 1.

(D) Let f(x) = x⁴ + 3x³ + 3x² + x + 1

Now, f(–1) = (–1)⁴ + 3 × (–1)³ + 3 × (–1)² + (–1) + 1

= 1 – 3 + 3 – 1 + 1

= 1 ≠ 0

So, f(x) is not a factor of x + 1.