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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

Solution:

When a polynomial p(x) is divided by (x - a) and if p(a) = 0 then (x - a) is a factor of p(x).

The root of x + 1 = 0 is -1.

(i) Let p(x) = x3 + x2 + x + 1

Therefore, p(-1) = (-1)3 + (-1)2 + (-1) + 1

⇒ -1 + 1 - 1 + 1 = 0

Since the remainder of p(-1) is 0, we conclude that x + 1 is a factor of x3 + x2 + x + 1.

(ii) Let p(x) = x4 + x3 + x2 + x + 1

Therefore, p(-1) = (-1)4 + (-1)3 + (-1)2 + (-1) + 1

= 1 - 1 + 1 - 1 + 1

= 1 ≠ 0

Since the remainder of p(-1) ≠ 0, we conclude that x + 1 is not a factor of x4 + x3 + x2 + x + 1.

(iii) Let p(x) = x4 + 3x3 + 3x2 + x + 1

Therefore, p(-1) = (-1)4 + 3(-1)3 + 3(-1)2 + (-1) + 1

= 1 - 3 + 3 - 1 + 1

= 1 ≠ 0

Since the remainder of p(-1) ≠ 0, x + 1 is not a factor of x4 + 3x3 + 3x2 + x + 1.

(iv) Let p(x) = x3 - x2 - (2 + √2)x + √2

∴ p(-1) = (-1)3 - (-1)2 - (2 + √2)(-1) + √2

= -1 - 1 + 2 + √2 + √2

= 2√2 ≠ 0

Since the remainder of p(-1) ≠ 0, x + 1 is not a factor of x3 - x2 - (2 + √2)x + √2.

☛ Check: Class 9 Maths NCERT Solutions Chapter 2

Video Solution:

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

NCERT Solutions Class 9 Maths Chapter 2 Exercise 2.4 Question 1

Summary:

(x + 1) is a factor of x3 + x2 + x + 1 whereas, (x + 1) is not a factor of the polynomials x4 + x3 + x2 + x + 1, x4 + 3x3 + 3x2 + x + 1, and x3 - x2 - (2 + √2)x + √2.

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