12 Higher Engineering Mathematics
ap
3
+ bp
2
+ cp + d, where a = 1,
b = −2, c = −5, d = 6 and p = 3.
Hence the remainder is:
1(3)
3
+ (−2)(3)
2
+ (−5)(3) + 6
= 27 − 18 − 15 + 6 = 0
Hence (x − 3) is a factor.
Thus (x
3
− 2x
2
− 5x + 6)
= (x − 1)(x + 2)(x − 3)
Now try the following exercise
Exercise 7 Further problems on the
remainder theorem
1. Find the remainder when 3x 2 − 4x + 2 is
divided by
(a) (x − 2) (b) (x + 1).
[(a) 6 (b) 9]
2. Determine the remainder when
x 3 − 6x 2 + x − 5 is divided by
(a) (x + 2) (b) (x − 3). [(a) −39 (b) −29]
3. Use the remainder theorem to find the factors
of x 3 − 6x 2 + 11x − 6.
[(x − 1)(x − 2)(x − 3)]
4. Determine the factors of x 3 + 7x 2 + 14x + 8
and hence solve the cubic equation
x
3
+ 7x
2
+ 14x + 8 = 0.
[x = −1, x = −2 and x = −4]
5. Determine the value of ‘a’ if (x + 2) is a
factor of (x 3 − ax 2 + 7x + 10).
[a = −3]
6. Using the remainder theorem, solve the
equation 2x 3 − x 2 − 7x + 6 = 0.
[x = 1, x = −2 and x = 1.5]
ap
3
+ bp
2
+ cp + d, where a = 1,
b = −2, c = −5, d = 6 and p = 3.
Hence the remainder is:
1(3)
3
+ (−2)(3)
2
+ (−5)(3) + 6
= 27 − 18 − 15 + 6 = 0
Hence (x − 3) is a factor.
Thus (x
3
− 2x
2
− 5x + 6)
= (x − 1)(x + 2)(x − 3)
Now try the following exercise
Exercise 7 Further problems on the
remainder theorem
1. Find the remainder when 3x 2 − 4x + 2 is
divided by
(a) (x − 2) (b) (x + 1).
[(a) 6 (b) 9]
2. Determine the remainder when
x 3 − 6x 2 + x − 5 is divided by
(a) (x + 2) (b) (x − 3). [(a) −39 (b) −29]
3. Use the remainder theorem to find the factors
of x 3 − 6x 2 + 11x − 6.
[(x − 1)(x − 2)(x − 3)]
4. Determine the factors of x 3 + 7x 2 + 14x + 8
and hence solve the cubic equation
x
3
+ 7x
2
+ 14x + 8 = 0.
[x = −1, x = −2 and x = −4]
5. Determine the value of ‘a’ if (x + 2) is a
factor of (x 3 − ax 2 + 7x + 10).
[a = −3]
6. Using the remainder theorem, solve the
equation 2x 3 − x 2 − 7x + 6 = 0.
[x = 1, x = −2 and x = 1.5]