2
TOPIC: POLYNOMIALS
1. Solve the following quadratic equations:
x 2 + 4x + 3 = 0
(a)
x 2 + 3x − 4 = 0
(d)
x ( x − 3) + x ( x − 4 ) = 20 − x
(g)
(b)
(e)
(h)
x 2 − 2 x − 15 = 0
8 x 2 = −6 x + 9
x+2
x
=
x
2x − 5
2. Divide x 3 − 6 x 2 + 11x − 6 = 0 by:
(a)
(b)
x +1
x −1
3. Divide 12 x 3 + 16 x 2 − 5 x − 3 by
(c)
( 2 x − 1) , stating the quotient
4. Show, without division, that:
(a) ( x − 3) is a factor of x 3 + 2 x 2 − 9 x − 18
(c)
(f)
4x 2 = 9x
6x + 3 = 2x 2
(i)
4 x 2 − 64 = 0
x−2
(d)
x+ 3
and the remainder.
(b) 2 x 3 + 5 x 2 + x − 2 is divisible by ( x + 2 )
(c) (2 x − 1) is a factor of 2 x 3 − 3x 2 − 3x + 2
(d) ( 3 x − 1) is a factor of 9 x3 + 18 x 2 − x − 2
5. Factorise the following completely:
(a) 2 x 3 + 11x 2 + 17 x + 6
(b) x3 - 3x 2 + 4
(c) 2 x 3 + 5 x 2 + x − 2
(d) 2 x 3 − 14x 2 = −24 x
6. Use the Factor Theorem to find the factors of 2 x 3 − 5 x 2 − 4 x + 3 . Hence, solve the equation 2 x 3 − 5 x 2 − 4 x + 3 = 0.
7.
Given that (x + 2) is a factor of x 3 − 4 x − 5 x 2 + p :
(i)
Find the value of p . Hence, factorize x 3 − 4 x − 5 x 2 + p completely.
(ii)
Solve the equation x 3 − 4 x − 5 x 2 + p = 0
8. Solve the following polynomial equations:
(a)
x3 − 2 x 2 − 5 x + 6 = 0
(b)
2 x3 + 3 x 2 − 11x − 6 = 0
(c)
3a 3 − 4a 2 − 17a + 6 = 0
(d)
2 y 3 + y 2 − 25 y + 12 = 0
(e)
3 p3 − 2 p 2 − 37 p − 12 = 0
(f)
2m3 + 7 m 2 − 24m − 45 = 0
9. When the expression 3x 3 + px 2 − x + 1 is divided by (x + 2) the remainder is − 1.
Find the value of p .
10. It is given that ( x - 3) is a factor of the polynomial 2 x 3 + kx 2 − 11x + 60 .
(i)
Show that k = −9 .
(ii)
Factorize 2 x 3 + kx 2 − 11x + 60 completely.
(iii) Solve the equation 2 x 3 + kx 2 − 11x + 60 = 0 .
11. Given that ( x + 2) is a factor of the cubic expression 2 x3 + kx 2 − kx − 2,
(i) find the value of k
(ii)
factorize 2 x3 + kx 2 − kx − 2 completely
Reviewed August 2019