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Title: Algebra _ Remainder theorem, factor theorem and synthetic division
Description: Algebra _ Remainder theorem, factor theorem and synthetic division The Remainder Theorem states that the remainder that we end up with when synthetic division is applied actually gives us the functional value. Another use is finding factors and zeros. The Factor Theorem states that if the functional value is 0 at some value c, then x - c is a factor and c is a zero.
Description: Algebra _ Remainder theorem, factor theorem and synthetic division The Remainder Theorem states that the remainder that we end up with when synthetic division is applied actually gives us the functional value. Another use is finding factors and zeros. The Factor Theorem states that if the functional value is 0 at some value c, then x - c is a factor and c is a zero.
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Algebra
Remainder theorem, factor theorem and synthetic division
1
...
Solution:
(i)
f (x)
The remainder is f ( 3)
= 2x2 –7x – 1, when it is divided by x + 3
= 2 (3 )2 – 7 ( 3) –1
= 18 + 21 – 1 = 38
(ii)
f (x)
= 3x3 – 7x2 + 11x + 1,
when it is divided by x + 3
The remainder is f (3) = 3 (3)3 – 7 (3)2 + 11 (3) + 1
= 81 – 63 – 33 + 1
= 176
2
...
Find m if 5x7 – 9x3 + 3x – m leaves a remainder 7 when divided by x + 1
...
when x3 + 3x2 – kx + 4 is divided by x – 2, the remainder is k
...
Solution:
f (x) = x3 + 3x2 – kx + 4
It is divided by x – 2 and the remainder is k
...
3k
= 24
k
= 8
Find the value of p if the division of px3+ 9x2 + 4x – 10 by x + 3 leaves the remainder 5
...
If the polynomials ax3 + 3x2 – 13 and 2x3 – 5x + a leave the same remainder when divided
by x + 2, find the value of a
...
If the polynomials 2x3 + ax2 + 3x – 5 and x3 + x2 – 4x + a leave the same remainder
when divided by x – 2, find the value of a
...
The polynomials ax3 + 3x2 – 3 and 2x3 – 5x + a when divided by x – 4 leave the
remainder R1 and R2 respectively
...
Solution:
(i)
R1
64a + 45
63a
a
= R2
= 108 + a
= 63
= 1
(ii)
R1 + R2
= 0
64a + 45 + 108 + 9
= 0
65a + 153
= 0
a =
2R1 – R2
= 0
2 (64a + 45 ) – (108 + a)
= 0
128a + 90 – 108 – a
= 0
(iii)
127 a
= 18
a
9
...
Solution:
p(x)
Given that p(1)
p(1)
= x3 + lx + m
= 7, p (1) = 7
= (1)3 + l (1) + m = 7
l+m = 6
-----(1)
p (1) = ( 1)3 + l (1) + m = 7
l + m = 8
-----(2)
l+m = 6
l+m = 8
2m
= 14
m
= 7
Substitute m = 7 in equation (1)
l = 1
10
...
Calculate the values of a and b
...
e
...
The remainders when px3 – qx2 – x + 5 and x3 + px2 + 2x – q – 4 are divided by
x + 1 are – 3 and 10 respectively
...
f (x) = px3 – qx2 – x + 5
Solution:
When it is divided by x + 1, the remainder is –3
f (1) = 3
p (1)3 – q (1)2 – (1) + 5 = 3
p – q = 9
p+q
Let g (x)
= 9
-----(1)
= x3 + px2 + 2x – q – 4
When it is divided by ( x + 1) the remainder is 10
g (1)
= 10
(1)3 + p (1)2 + 2 (1) – q – 4 = 10
– 1 + p – 2 – q – 4 = 10
p–q
= 17
p+q
= 9
p–q
= 17
2p
= 26
p
= 13
q
12
...
Solution:
f (x) = px3 + 9x2 + qx + 1
f
p
1 3
1
+ 9
2
2
2
+q –
1
2
1
2
+1 = 4
= 4
p
8
9
q
+1 = 4
4
2
+
p + 18 – 4q = 24
p – 4q = 6
g (x)
-----(1)
3
2
= 9x + qx + px + 1
When it is divided by (3x –1) the remainder is 3
Title: Algebra _ Remainder theorem, factor theorem and synthetic division
Description: Algebra _ Remainder theorem, factor theorem and synthetic division The Remainder Theorem states that the remainder that we end up with when synthetic division is applied actually gives us the functional value. Another use is finding factors and zeros. The Factor Theorem states that if the functional value is 0 at some value c, then x - c is a factor and c is a zero.
Description: Algebra _ Remainder theorem, factor theorem and synthetic division The Remainder Theorem states that the remainder that we end up with when synthetic division is applied actually gives us the functional value. Another use is finding factors and zeros. The Factor Theorem states that if the functional value is 0 at some value c, then x - c is a factor and c is a zero.