x + 1 is a factor of the polynomial
a. x3 + x2 - x + 1
b. x3 + x2 + x + 1
c. x4 + x3 + x2 + 1
d. x4 + 3x3 + 3x2 + x + 1
Solution:
By applying the remainder theorem
x + 1 = 0
x = -1
Let us substitute x = -1 in all the equations
a. x3 + x2 - x + 1 = (-1)3 + (-1)2 - (-1) + 1
= -1 + 1 + 1 + 1
= 2
b. x3 + x2 + x + 1 = (-1)3 + (-1)2 + (-1) + 1
= -1 + 1 - 1 + 1
= 0
c. x4 + x3 + x2 + 1 = (-1)4 + (-1)3 + (-1)2 + 1
= 1 - 1 + 1 + 1
= 2
d. x4 + 3x3 + 3x2 + x + 1 = (-1)4 + 3(-1)3 + 3(-1)2 + (-1) + 1
= 1 - 3 + 3 - 1 + 1
= 1
Therefore, x + 1 is a factor of the polynomial x3 + x2 + x + 1.
✦ Try This: If x + 1 is a factor of the polynomial 5x² + kx, then the value of k is
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 2
NCERT Exemplar Class 9 Maths Exercise 2.1 Problem 13
x + 1 is a factor of the polynomial a. x3 + x2 - x + 1, b. x3 + x2 + x + 1, c. x4 + x3 + x2 + 1, d. x4 + 3x3 + 3x2 + x + 1
Summary:
The standard form of a polynomial refers to writing a polynomial in the descending power of the variable. x + 1 is a factor of the polynomial x3 + x2 + x + 1
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