Answer:
(i) (x + 3) is a factor of p(x)= x⁴ + 4x³ + 4x² + 4x + 3 and the remaining factors are (x + 1) and ( x² + 1).
(ii) (x + 1) is not a factor of p(x)= 2x⁴ - 5x² - 2 and the remainder is -5.
Step-by-step explanation:
(i) Given;
Polynomial, p(x)= x⁴ + 4x³ + 4x² + 4x + 3
Factor, = x + 3
x = -3
if (x + 3) is a factor of the polynomial, then p(-3) = 0
p(-3) = (-3)⁴ + 4(-3)³ + 4(-3)² + 4(-3) + 3
= 81 -108 + 36 - 12 + 3
= 0
(x + 3) is a factor of the given polynomial and the remainder is 0.
The remaining factors can be obtained as follows;
x³ + x² + x + 1
--------------------------------
x + 3 √x⁴ + 4x³ + 4x² + 4x + 3
- ( x⁴ + 3x³)
--------------------------------
x³ + 4x² + 4x + 3
-(x³ + 3x²)
----------------------------------
x² + 4x + 3
-(x² + 3x)
---------------------------------------
x + 3
-(x + 3)
---------------------------------------
0
Further divide x³ + x² + x + 1 by x + 1
x² + 1
---------------------
x + 1 √x³ + x² + x + 1
-(x³ + x²)
------------------------
x + 1
-( x + 1)
-----------------
0
Thus, the factors of x⁴ + 4x³ + 4x² + 4x + 3 = (x + 3)(x + 1)( x² + 1)
(ii) Given;
Polynomial, p(x)= 2x⁴ - 5x² - 2
Factor, = x + 1
x = -1
if (x + 1) is a factor of the polynomial, then p(-1) = 0
p(-1) = 2(-1)⁴ - 5(-1)² - 2
= 2 - 5 - 2
= -5
Thus, (x + 1) is not a factor of the given polynomial and the remainder is -5.
