Question 1. Determine which of the following polynomials has (x + 1) a factor :
(i) x3 + x3 + x + 1
(ii) x4 + x3 + x3 + x + 1
(iii) x4 + 3x3 + 3x3 + x + 1
(iv) x3 – x3 – (2+√2)x + √2
Apply remainder theorem
=>x + 1 =0
=> x = - 1
Replace x by – 1 we get
=>x3 + x2 + x + 1
=>(-1)3 + (-1)2 + (-1) + 1
=> -1 + 1 - 1 + 1
=> 0
Remainder is 0 so that x+1 is a factor of x3 + x3 + x + 1
Apply remainder theorem
=>x + 1 =0
=> x = - 1
Replace x by – 1 we get
=>x4 + x3 + x2 + x + 1
=> (-1)4+ (-1)3 + (-1)2 + (-1) + 1
=> 1 -1 + 1 - 1 + 1
=> 1
Remainder is not equal to 0 so that x+1 is not a factor of x4 + x3 + x3 + x + 1
(iii)x4 + 3x3 + 3x3 + x + 1
Apply remainder theorem
=>x + 1 =0
=> x = - 1
Replace x by – 1 we get
=>x4 + 3x3 + 3x3 + x + 1
=> (-1)4+ 3(-1)3 + 3(-1)2 + (-1) + 1
=> 1 -3 + 3 - 1 + 1
=> 1
Remainder is not equal to 0 so that x+1 is not a factor of x4 + 3x3 + 3x3 + x + 1
Apply remainder theorem
=>x + 1 =0
=> x = - 1
Replace x by – 1 we get
=>x3 – x3 – (2+√2)x + √2
=> (-1)3 – (-1)2 – (2 + √2)(-1) + √2
=> 1 - 1 + 2 + √2 + √2
=> 2 + 2√2
Remainder is not equal to 0 so that x+1 is not a factor of x3 – x3 – (2+√2)x + √2
(i) x3 + x3 + x + 1
(ii) x4 + x3 + x3 + x + 1
(iii) x4 + 3x3 + 3x3 + x + 1
(iv) x3 – x3 – (2+√2)x + √2
Determine x3 + x3 + x + 1 polynomials has (x + 1) a factor
Solution: (i) x3 + x3 + x + 1 Apply remainder theorem
=>x + 1 =0
=> x = - 1
Replace x by – 1 we get
=>x3 + x2 + x + 1
=>(-1)3 + (-1)2 + (-1) + 1
=> -1 + 1 - 1 + 1
=> 0
Remainder is 0 so that x+1 is a factor of x3 + x3 + x + 1
Determine x4 + x3 + x3 + x + 1 polynomials has (x + 1) a factor
(ii)x4 + x3 + x3 + x + 1Apply remainder theorem
=>x + 1 =0
=> x = - 1
Replace x by – 1 we get
=>x4 + x3 + x2 + x + 1
=> (-1)4+ (-1)3 + (-1)2 + (-1) + 1
=> 1 -1 + 1 - 1 + 1
=> 1
Remainder is not equal to 0 so that x+1 is not a factor of x4 + x3 + x3 + x + 1
Determine x4 + 3x3 + 3x3 + x + 1 polynomials has (x + 1) a factor
(iii)x4 + 3x3 + 3x3 + x + 1
Apply remainder theorem
=>x + 1 =0
=> x = - 1
Replace x by – 1 we get
=>x4 + 3x3 + 3x3 + x + 1
=> (-1)4+ 3(-1)3 + 3(-1)2 + (-1) + 1
=> 1 -3 + 3 - 1 + 1
=> 1
Remainder is not equal to 0 so that x+1 is not a factor of x4 + 3x3 + 3x3 + x + 1
Determine x3 – x3 – (2+√2)x + √2 polynomials has (x + 1) a factor
(iv) x3 – x3 – (2+√2)x + √2Apply remainder theorem
=>x + 1 =0
=> x = - 1
Replace x by – 1 we get
=>x3 – x3 – (2+√2)x + √2
=> (-1)3 – (-1)2 – (2 + √2)(-1) + √2
=> 1 - 1 + 2 + √2 + √2
=> 2 + 2√2
Remainder is not equal to 0 so that x+1 is not a factor of x3 – x3 – (2+√2)x + √2
Total NCRT 9th class 1.1 Exercise
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