EXTRA EXERCISE CHAPTER 3
1. Find a polynomial P(x) of degree 3 with zeros -3, 1 and 2 and satisfying P(-1)=24.
[Ans: 2x3-14x+12]
2. Given P(-1) = P(3) = 0, P(2) = - 9 and P(1) = - 4. Find the polynomial P(x) of degree 3.
[Ans: 2x3-5x2-4x+3]
3. Find a polynomial f (x) of degree three, with leading coefficient 1, and having
x  3, x  1 and x  2 as factors.
[Ans: f ( x)  x 3  4 x 2  x  6 ]
4. Find a polynomial f (x) of degree 4, having  2,  1 as zeros and f (2)  10 .
[Ans:
f ( x) 
5 4
10
x  5x 2 
3
3 ]
5. Find a cubic polynomial Q( x)  ( x  a)( x  b)( x  c) satisfying the following
conditions:
[Ans: x 3  4 x 2  x  6 ]
Q(1)  0, Q(3)  0, Q(4)  10
6. Given that (3i ) is a zero of polynomial P( x)  2 x 5  3x 4  5x 3  15 x 2  207 x  108,
find the remaining zeros using Rational Zeros Theorem and Descartes’ Rule
of Sign.
[Ans: 3i, -3i, -3, ½, 4]
7. Use Descartes’ Rule of Signs to determine how many positive and how many
negative real zeros the polynomial can have. Then, determine the possible total
number of real zeros for P( x)  2 x 6  5x 4  x 3  5x  1 .
[Ans: 1 positive, 1 negative; 2 real]
1
8. For the given polynomial P(x) = 3x4 + 5x3 + 25x2 + 45x – 18 ;
a. Use the Descartes’ Rule of Sign to determine the number of positive and
negative zeros
b. List all possible rational zeros of the polynomial
c. Find all zeros of P
[Ans: -2, -3i, 3i, 1/3]
9. Show that 3x – 2 is a factor of P(x) = 6x3 – 7x2 – x + 2 and find the other two factors.
Solve the equation 6x3 – 7x2 – x + 2 = 0.
[Ans: 2/3, -1/2, 1]
10. Consider polynomial x 3  3x 2  x  3. If x  3 is one of the factors of polynomial, find
the other two factors.
[Ans: x  1 and x  1 ]
11. The polynomial f(x) = 2x3+ax2+bx+3 has factor (x+3). When f(x) is divided by (x-2)
the remainder is 15.
a. Find the values of a and b
b. Find the other two factors of f(x).
[Ans: a: 3, b: -8, (2x-1)(x-1)]
12.
x 1
is a root of multiplicity 2 of the polynomial equation p( x)  0, where
p( x)  x 4  2 x 3  2 x 2  6 x  3. Factorize p(x) completely.
[Ans: p( x)  ( x  1)( x  1)( x  3 )( x  3 )
]
3
x2
13. If the polynomial P( x)  x  px  q is divisible by
and x  1, find the values of p
x3
and q , factorize P(x) completely. Find the remainder when P(x) is divided by
.
2
[Ans: p  3, q  2, P( x)  ( x  2)( x  1) ,16 ]
14. When x3 + px2 + qx + 5 is divided by (x – 1)(x + 2), the remainder is 2x + 7. Find the
values of p and q.
[Ans: p=2 ; q=1]
15. Given ( x  2) is a factor of the polynomial P( x)  x 3  px 2  qx  10. If the remainder
is 5 when P (x ) is divided by ( x  3), find p and q.
5
3
[Ans: p   , q  
17
3
]
2
4
3
2
16. If x  2 and x  3 are factors for expression 3x  px  15 x  qx  12 , find p
and q . Hence, factorize the expression completely.
[Ans: p=8, q=-32, (x-2)(x+3)(3x-1)(x+2)]
17. When a polynomial P(x) is divided by x  1, it leaves a remainder of 5 and when it is
divided by ( x  2), the remainder is 3. Find the remainder when it is divided by
x 2  3x  2 .
[ Ans: 7  2 x ]
18. The polynomial 2x3 – 9x2 + 3x + b has a remainder – 54 when divided by x + 2. Find
the value of b.
[Ans: b = 4]
19. If the polynomial x3 + ax2 + bx + 1 has remainder 5 when divided by x – 1 and
remainder 7 when divided by x – 2, find the values of a and b.
[Ans: a = - 4, b = 7]
20. Show that 3x – 2 is a factor of P(x) = 6x3 – 7x2 – x + 2 and find the other two factors.
Solve the equation 6x3 – 7x2 – x + 2 = 0.
[Ans: 2/3, -1/2, 1]
21. If the polynomial P(x) = x3+px+q is divisible by x – 2 and x + 1, find the values of p
and q, and factorize P(x) completely. Find the remainder when P(x) is divided by
x – 3.
[Ans: p=-3,q=-2 (x-2)(x+1)2,16]
P( x)  x 2  1
Q( x)  ax 4  3x 2  bx  5.
22. Given the polynomials
and
a)
Find all the zeros of P (x ) .
b)
c)
When Q (x) is divided by ( x  2)( x  2), the remainder is 2 x  33 . Find the
values of a and b using the remainder theorem.
Using the values of a and b in (ii) above, show that the remainder is
(6 x  105) when 2 P( x)  3Q( x) is divided by ( x 2  4)
[Ans: i)-1 and 1
ii)a=1, b= -2]
3
23. For the polynomial f ( x)  ( x  2) 3 ( x  3) 2 ( x  4)
i.
Find the x-intercepts and y-intercepts of 𝑓(𝑥)
[ Ans: x=2,3,4 y=288]
ii.
iii.
Determine whether the graph crosses or touches the x-axis at each xintercept.
[Ans: Crosses at x=2 and x=4 touch at x=3]
Determine the maximum number of turning points on the graph.
[Ans : 5]
iv.
Find the intervals on which the graph is above the x-axis and below x-axis.
[Ans: above (-∞,2)U(4, ∞) below (2,3)U(3,4)]
v.
Sketch the graph of f(x).
4
24.
For the given polynomial P(x) = 3x4 + 5x3 + 25x2 + 45x – 18 ;
a. Use the Descartes’ Rule of Sign to determine the number of positive and
negative zeros
b. List all possible rational zeros of the polynomial
c. Find all zeros of P
Ans: -2, -3i, 3i, 1/3
d. Sketch the graph of f(x).
Answer:
5
25. Analyze and graph the following rational function:
f ( x) 
x 2  x  12
x5
26.
6
26. Given the function
(i)
R  x 
 x  1 x  2  x  3
2
x  x  4
(ii)
R( x) 
x 2  3x  2
x 1
Analyze and sketch the graph of R  x  .
Answer:
(i)
(ii)
7
27. Find the horizontal and vertical asymptotes, if any, of the following functions and
sketch the graph of these functions.
a) f ( x) 
2
x2
c) g ( x) 
x 1
x2  4
b)
d)
h( x ) 
x
x3
f ( x)  
8
x 9
2
Answer:
a)
b)
8
c)
d)
9
28. Analyze and graph the following rational function:
f ( x) 
x 2  x  12
x5
10