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Title: Algebra _ Factor Theorem
Description: Algebra _ Factor Theorem Factor Theorem Factor theorem is mainly used to factor the polynomials and to find the n roots of the polynomials. Factor theorem is very helpful for analyzing polynomial equations. In real life, factoring can be useful while exchanging money, dividing any quantity into equal pieces, understanding time, and comparing prices.
Description: Algebra _ Factor Theorem Factor Theorem Factor theorem is mainly used to factor the polynomials and to find the n roots of the polynomials. Factor theorem is very helpful for analyzing polynomial equations. In real life, factoring can be useful while exchanging money, dividing any quantity into equal pieces, understanding time, and comparing prices.
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Algebra
Factor Theorem
1
...
Solution:
p(x) = 2x2 – 11x + 5
p(5) = 2 (5)2 – 11 (5) + 5
= 50 – 55 + 5 = 0
(x – 5) is a factor
p 1
2
1
= 2
2
=
2
1
2
11
+ 5
1
11
+5 = 0
2
2
(2x – 1) is a factor
2
...
Solution:
Let p(x)
7
p
2
=
2x3 + 7x2 – 4x – 14
=
7
2
2
=
343
4
3
+
2
7
7
+ 7 4
14
2
2
343
4
+ 14 – 14 = 0
2x + 7 is a factor
3
...
Solution:
Let p(x) = 3x2 + kx + 6
p(3) = 3 (3)2 + k (3) + 6 = 0
27 – 3k + 6 = 0
3k = 33
k = 11
4
...
Find the value of k, if ( x + 2) is a factor of (x + 1)7 + 2x + k
...
= 5
Determine the value of a for which the polynomial 2x4 – ax3 + 4x2 + 2x + 1 is
divisible by 1 – 2x
...
Find the value of a, if x + a is a factor of x3 + ax2 – 2x + a + 4
...
= 4
3
Using factor theorem, show that x + a is a factor of xn + an when n is any
odd positive integer
...
If f (x) = x4 – 5x2 + 4 and g(x) = x2 – 3x + 2, show that g (x) is a factor of f (x)
...
If ( x – 1) and ( x + 1) are factors of x3 + 3x2 + ax + b find a and b
...
If x2 –5x + 6 is a factor of 3x3 + ax2 + bx + 24, find a and b
...
Given that x2 – 9 is a factor of x3 + px2 + qx – 45 find p and q
...
If x3 + ax2 + bx + 6 has x – 2 as a factor and leaves a remainder 3 when
divided by x – 3 find a and b
Solution:
= x3 + ax2 + bx + 6
Let p(x)
Since (x – 2) is a factor
p (2) = 0
(2)3 + a(2)2 + b(2) + 6 = 0
4a + 2b = 14
2a + b = 7
-----(1)
Since the remainder is 3 when divided by (x – 3)
p (3)
= 3
(3)3 + a (3)2 + b (3) + 6 = 3
9a + 3b = 30
3a + b
= 10
(1)
2a + b
= 7
(2)
3a + b
= 10
a
= 3
----(2)
or
a = 3
Substitute a = -3 in equation (2)
3a + b = 10
3(3) + b = 10
9 + b = 10
b = 10 + 9
a = 3,
14
...
Solution:
Let f ( x, y, z) = x2 ( y – z) + y2 ( z – x ) + z2 ( x – y )
Put x = y in the above
Then RHS
= y2 ( y – z ) + y2 ( z – y) + z2 (y – y)
= y3 – y2 z + y2 z – y3 + 0 = 0
( x – y ) is a factor
Similarly (y – z), (y – x) are also factors
...
Using factor theorem show that a – b, b – c and c – a are the factors of
a (b2 – c2) + b (c2 – a2) + c (a2 – b2)
...
similarly (b – c) (c – a) are also a factor
...
If ( x – 2) is a factor of x2 + ax + b and a + b = 1, find the values of a and
b
Title: Algebra _ Factor Theorem
Description: Algebra _ Factor Theorem Factor Theorem Factor theorem is mainly used to factor the polynomials and to find the n roots of the polynomials. Factor theorem is very helpful for analyzing polynomial equations. In real life, factoring can be useful while exchanging money, dividing any quantity into equal pieces, understanding time, and comparing prices.
Description: Algebra _ Factor Theorem Factor Theorem Factor theorem is mainly used to factor the polynomials and to find the n roots of the polynomials. Factor theorem is very helpful for analyzing polynomial equations. In real life, factoring can be useful while exchanging money, dividing any quantity into equal pieces, understanding time, and comparing prices.