Chapter 2 Polynomials Exercise 2.3
Question 1: Find the remainder when x³ + 3x² + 3x + 1 is divided by
i) x + 1
ii) x - 1/2
iii) x
iv) x + π
v) 5 + 2x
Answer:
i)
Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).
The zero of x + 1 is -1.
p(-1) = (-1)³ + 3(-1)² + 3(-1) + 1
= -1 + 3 - 3 + 1
= 0
Using long division method
![Chapter 2 Polynomials Ex 2.3 Q1 i)](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYRRRnyn7O1wTrZXWWCml8YUSuYF_mDkzy6-DV7-tzrVTW0OqIKPWjUjNMmAd1pal3vkB3i2grCkwvVt95KCbULmcmdACW-yQGjLc2kGwn1K_sJu6QX3Vozqo9G1GGmuqVYicGL36267wCYK2va1ezgLtnKhpnWpWDT8AZjOWUC-WOCN-wqcACQTVONQ/s16000-rw/1.png)
ii)
Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).
The zero of x - 1/2 is 1/2.
p(1/2) = (1/2)³ + 3(1/2)² + 3(1/2) + 1
= 1/8 + 3/4 + 3/2 + 1
= 1 + 6 + 12 + 8/8
= 27/8
Using long division
![Chapter 2 Polynomials Ex 2.3 Q1 ii)](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6R0Y93Cxdf01xdPdHLWg2qoBVLo4H75MGQA-4I-3LPwJKf2xR_UOqBVzjd2DviDDBLBXyRBAyqtPgQ8RTQvz1GOtqYqyTkblILxUbPnBE_TQ1ijNLfbQn_zogc-LMEkS96t9892isUbbjKz-8Ot8b-YmUEALVRIjBmBUbsAhZhdDnFS1lrWA77vz2Ig/s16000-rw/2.png)
iii)
Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).
The zero of x is 0.
p(0) = (0)³ + 3(0)² + 3(0) + 1
= 0 + 0 + 0 + 1
= 1
Using long division
![Chapter 2 Polynomials Ex 2.3 Q1 iii)](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_LfGnOmlh3leq-Vu1iFE6G-CXOcF7DfMZ-d4Ldmq2Rf855jP-lyDLsvnD_9vTaBG2cVSkMvOAfF1RmbOjzXABnnVbO4SPzBbru44IHvJDO81tL5DfnX7PTO9ugyLWCA9GGk402xB3GDstwdLLs91QZV1Df4GbxmfZAYZlEQpKcL06UuxpuOrM22wEtw/s16000-rw/3.png)
iv)
Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).
The zero of x + π is -π.
p(-π) = (-π)³ + 3(-π)² + 3(-π) + 1
= -π³ + 3π² - 3π + 1
Using long division
![Chapter 2 Polynomials Ex 2.3 Q1 iv)](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlmgBKEytFi3FmhXqLZy9rJDBBWJzUU__CbXok4TGjHsoovlY4hXsHID7FKx_W8SkxyIyDdxbPr7OCTjhCMyaOYmbNyVJUmT_oXIKRuifP-afoqZkR9mt31Q3Hj_eQ4t6ki_diTf0axLBmdQL7u-vHWqCCkLY0EiuEkt037f2xoV2m8yjhmJcisdyZ6w/s16000-rw/4.png)
v)
Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).
The zero of 5 + 2x is -5/2.
p(-5/2) = (-5/2)³ + 3(-5/2)² + 3(-5/2) + 1
= -125/8 + 75/4 + -15/2 + 1
= -27/8
Using long division
![Chapter 2 Polynomials Ex 2.3 Q1 v)](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHBMt8g19voXLTNkhN-TVAeAJRipnjrNwjE9AM-gMwEjRKqBoKdmw3Taz3y8KW4msAcRP2DFA15MjLbLNhwxog-b_nuSi2s2djCl5AfM_ASrtRjsZgtPzYFWga3n3-55mbaAEdwlChmo2TsGIsPHDZPU5r_G0BcPIFNEChh78bK6woHnRBTJInjAwoog/s16000-rw/5.png)
Question 2: Find the remainder when x³ - ax² + 6x - a is divided by x - a.
Answer: Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).
The zero of x - a is a.
p(a) = x³ - ax² + 6x - a
= (a)³ - a(a)² + 6(a) - a
= a³ - a³ + 6a - a
= 5a
Question 3: Check whether 7 + 3x is a factor of 3x³ + 7x.
Answer: Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).
The zero of 3x + 7 is -7/3.
p(-7/3) = 3(-7/3)³ + 7(-7/3)
= 3(-343/27) - 49/3
= -343 - 147/9
= -490/9
Therefore, 7 + 3x is not a factor of 3x³ + 7x.
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