1.1 Remainder and Factor Theorems

From Mathematics Digital Library
Jump to navigation Jump to search

Which of the following are polynomials? Write down the order of those which are.

Question 1(a) \[3x^{2}+\frac{7}{x}\]
Question 1(b) \[2x^{3}+5\]
Question 1(c) \[2\sin{x}-4\cos{x}\]
Question 1(d) \[\left(x+3\right)\left(x^{2}-7x^{3}+1\right)\]


Evaluate the following polynomial sums and products:

Question 2(a) \[\left(2x^{3}-3x+1\right)+\left(2x^{3}+4x-4\right)\]
Question 2(b) \[\left(2x^{2}+3x\right)\times\left(x^{3}-1\right)\]
Question 2(c) \[\left(x^{3}+2x\right)-\left(x^{2}-x\right)\]
Question 2(d) \[\left(x^{4}+x^{2}+1\right)\times\left(x^{3}-x\right)\]


Find the quotient and remainder in each of the following division problems.

Question 3(a) \[\left(x^{3}+2x^{2}+x-3\right)\div\left(x-4\right)\]
Question 3(b) \[\left(2x^{2}+3x\right)\div\left(x^{3}-1\right)\]
Question 3(c) \[\left(x^{3}+2x\right)\div\left(x^{2}-x\right)\]
Question 3(d) \[\left(x^{4}+x^{2}+1\right)\div\left(x^{3}-x\right)\]


Find the remainder in each of the following divisions.

Question 4(a) \[\left(x^{2}+3x-2\right)\div\left(x-5\right)\]
Question 4(b) \[\left(3x^{3}+x^{2}+x-1\right)\div\left(x-2\right)\]
Question 4(c) \[\left(x^{3}+7x^{2}-2x-1\right)\div\left(x+1\right)\]
Question 4(d) \[\left(x^{4}+2\right)\div\left(x+2\right)\]


Question 5 The remainder when [math]x^{3}-3x^{2}+ax-1[/math] is divided by [math]x-2[/math] is [math]5[/math]. Find the value of [math]a[/math].
Question 6 The remainder when [math]ax^{3}+3x^{2}+3x+5[/math] is divided by [math]x+3[/math] is [math]8[/math]. Find the value of [math]a[/math].
Question 7 When [math]ax^{3}+3x^{2}+3x+5[/math] is divided by [math]ax^{3}+3x^{2}+3x+5[/math] the remainder is [math]3[/math], and when it is divided by [math]ax^{3}+3x^{2}+3x+5[/math] the remainder is [math]-1[/math]. Find [math]a[/math] and [math]b[/math].
Question 8
Question 9
Question 10
Question 11
Question 12
Question 13


Factorise the following:

Question 14(a) \[x^{3}-2x^{2}-x+2\]
Question 14(b) \[x^{3}+3x^{2}-4x-12\]
Question 14(c) \[x^{3}-2x^{2}-5x+6\]
Question 14(d) \[2x^{3}+11x^{2}+17x+6\]
Question 14(e) \[2x^{3}-9x^{2}+7x+6\]
Question 14(f) \[x^{4}-x^{2}+4x-4\]
Question 14(g) \[x^{3}-x^{2}-x+1\]
Question 14(h) \[x^{3}+5x^{2}+8x+4\]


Solve the following equations:

Question 15(a) \[x^{3}+4x^{2}+x-6=0\]
Question 15(b) \[x^{3}-4x^{2}+x+6=0\]
Question 15(c) \[2x^{3}+5x^{2}+x-2=0\]
Question 15(d) \[6x^{3}-11x^{2}+6x-1=0\]

These questions are from S.C. Solomon's Advanced Level Mathematics.