Polynomials and Their Factors
24. Subtract the second polynomial from the first in the follo
(i) x3 + x +1; 1- x - x2
(ii) 6x3 + 5x
(iii) x3 + x2 + x + 1; X3 - x2 + x - 1 (iv) x4 - 3x
(v) 3x7 - 2x2 + 3; x6 - 3x4 + x2 + x
25. Find the product of the following polynomials :
(i) x2 - 4x + 4; x-2
(ii) x2 + 3x
(iii) 3x3 + x2 + x; x + 2
(iv) 3x + 2;
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Answer:
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Apply remainder theorem
x+1=0
x=−1
Put the value of x=−1 in all equations.
(i) x
3
+x
2
+x+1=(−1)
3
+(−1)
2
+(−1)+1=−1+1−1+1=0
Then x+1 is the factor of equation
(ii) x
4
+x
3
+x
2
+x+1=(−1)
4
+(−1)
3
+(−1)
2
+(−1)+1=1−1+1−1+1=1
This is not zero.Then x+1 is not the factor of equation
(iii) x
4
+3x
3
+3x
2
+x+1=(−1)
4
+3(−1)
3
+3(−1)
2
+(−1)+1=1
This is not zero.Then x+1 is not the factor of equation
(iv)x
3
−x
2
−(2+
2
)x+
2
=(−1)
3
−(−1)
2
−(2+
2
)(−1)+
2
=−1−1+2−
2
+
2
=0
This is zero. Then x+1 is the factor of equation
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