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Assignment 2: The Factor Theorem
1. Find the factors of the following polynomials.
(a) 3x3 − 4x2 − 3x + 4.
(b) 2x3 − 5x2 + 4x − 21.
(c) 2x3 + x2 − 13x + 6.
(d) x4 − 14x3 + 71x2 − 154x + 120.
(e) 2x4 − 9x2 − 4x + 12.
(f) x3 − 3x2 − 4x + 12.
(g) x4 − 4x3 − x2 + 16x − 12.
2. Solve the equations:
(a) x3 − 8x2 − 31x − 22 = 0
(b) 6x3 + x2 − 19x + 6 = 0
(c) x3 − 6x2 + 11x = 6
(d) x4 − 4x3 − x2 + 16x = 12
3. Find the value of p if 4x4 − 12x3 + 13x2 − 8x + p is divisible by 2x − 1.
4. Find what values of p must have in order that x − p may be a factor of
( )
4x3 − (3p + 2)x2 − p2 − 1 x + 3.
5. Find what value of p must have in order that x + 2 may be a factor of 2x3 + 3x2 + px − 6. Find
the other factors.
6. Prove that x − 1 is a factor of 2x3 − 13x2 + 23x − 12 and find the other factors.
Hence solve 2x3 − 13x2 + 23x − 12 = 0.
7. Given that x + 2 is a factor of ax2 + x − 2a, find a. Hence find the other factors.
8. Show that the expression x3 + (k − 2)x2 + (k − 7)x − 4 has a factor x + 1 for all values of k. If
the expression also has a factor x + 2, find the value of k and the third factor.
9. Given that x + 2 is a factor of f (x) = a(x − 1)2 + b(x − 1) + 9. The remainder when f (x) is
divided by x + 1 is −11. Find the value of a and b.
10. Given that 2x2 − x − 1 is a factor of ax4 + x3 − bx2 + 5x + 6, find the values of a and b.
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11. Find the value of p and q if 6x3 + 13x2 + px + q is exactly divisible 2x2 + 7x − 4.
Show also that 3x − 4 is a factor of the given polynomial.
12. The expression x3 + ax2 + bx + 3 is exactly divisible by x + 3 but leaves a remainder of 91 when
divided by x − 4. What is the remainder when it is divided by x + 2 ?
√ √
13. Given that f (x) = x3 + px2 − 2x + 4 3 has a factor x + 2, find the value of p. Show also
√
that x − 2 3 is also a factor and solve the equation f (x) = 0.
14. If x2 + 2x − 3 is a factor of f (x) = x4 + 2x3 − 7x2 + ax + b, find a and b, hence factorize f (x)
completely.
15. The polynomial ax3 + bx2 − 5x + 2a is exactly divisible by x2 − 3x − 4.
Calculate the value of a and b, and factorize the polynomial completely.
16. The expression px3 − 5x2 + qx + 10 has factor 2x − 1 but leaves a remainder of −20 when
divided by x + 2. Find the values of p and q and factorize the expression completely.
17. The polynomials x3 + ax2 − x + b and x3 + bx2 − 5x + 3a have a common factor x + 2. Find a
and b.
18. Given that kx3 + 2x2 + 2x + 3 and kx3 − 2x + 9 have a common factor, what are the possible
values of k ?
19. If the equations ax3 + 4x2 − 5x − 10 = 0 and ax3 − 9x − 2 = 0 have a common root, then show
that a = 2 or 11 .
20. Given that f (x) = 2x3 + ax2 − 7a2 x − 6a3 , determine whether or not x − a and x + a are factors
of f (x). Hence find in terms of a, the roots of f (x) = 0.
( )
21. Given that 4x4 −9a2 x2 +2 a2 − 7 x−18 is exactly divisible by 2x−a, show that a3 −7a−6 = 0
and hence find the possible values of a.
22. Given that 4x4 − 12x3 − b2 x2 − 7bx − 2 is exactly divisible by 2x + b, show that 3b3 + 7b2 − 4 = 0.
Hence find the possible values of b.
23. Factorize completely the expression 4x3 − 13x − 6.
( )
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Hence solve the equation 2 2x − 2
= 13.
x
24. The expression ax2 + bx − 1 leaves a remainder of R when divided by x + 2 and a remainder of
3R + 5 when divided by x − 3. Show that a = 3b − 1. Given also that the expression is exactly
divisible by 2x − 1, evaluate the value of a and b.
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25. Given that f (x) = 8x3 − 12x2 + ax + 3 is exactly divisible by 2x − 3. Find the solution set of
f (x) = 0.
26. Given that x2 + x − 6 is a factor of 2x4 + x3 − ax2 + bx + a + b − 1. Find the value of a and of b.
27. When f (x) = x3 − 3x2 + ax + b is divided by x + 1 the remainder is 12 . If x − 2 is a factor of
f (x), then find a and b. Hence solve f (x) = 0.
28. Given that f (x) = kx3 + (3k − 2)x2 − 4 where k is a constant. Given that x + 2 is a factor
of f (x). Find the value of k, with this value of k. find the remainder when f (x) is divided by
2x − 1.
29. Given that x − p is a common factor of x2 − 5x + k and x2 − 6x + 3k, where k ̸= 0, find the
numerical value of p.
30. Given that the expression 2x3 + px2 − 8x + q is exactly divisible by 2x2 − 7x + 6, evaluate
p and q and factorize the expression completely. Hence find the solution set of the equation
2x3 + px2 − 8x + q = 0.
31. Find the remainder when f (x) = 3x3 + x2 + x − 4 is divided by x + 1. Hence find the value of
k for which g(x) = f (x) + k is divisible by x + 1 and factorise g(x) completely.
32. It is given that x − 1 is a factor of f (x), where f (x) = x3 − 6x2 + ax + b.
(a) Express b in terms of a.
(b) Show that the remainder when f (x) is divided by x − 3 is twice the remainder when f (x)
is divided by x − 2.
33. Given that f (x) = 2x3 + ax2 + 10x + b has a factor of 2x + 1 and it leaves the remainder of 30
when divided by x − 2. Find the value of a and of b. Hence solve the equation f (x) = 21x.
34. The polynomial f (x) is given by f (x) = ax3 + 11x2 + cx − 60, where a and c are constants. If
the roots of f (x) = 0 are 2, −3 and k, find the values of a, c and k.
35. Two cubic polynomials are defined by f (x) = x3 + (a − 3)x + 2b, g(x) = 3x3 + x2 + 5ax + 4b,
where a and b are constants.
(a) Given that f (x) and g(x) have a common factor of (x − 2), show that a = −4 and find
the value of b.
(b) Using these values of a and b, factorise f (x) fully. Hence show that f (x) and g(x) have
two common factors.
