FACTOR THEOREM and REMAINDER THEOREM
FACTOR THEOREM( x² – 7x + 10 ) => ( x – 5 ) ( x – 2 )
Verify if (x + 1) is a factor ( x² – 9x – 10 )
There are two ways to find it using long division and synthetic division.
using Long Division:
*For ( x + 1 ) to be a factor you must have a zero remainder.
since there is no remainder the (x + 1) is a factor of ( x² – 9x – 10 )
using Synthetic Division:
x + 1 = x – a *we first need to find a
1 = -a
a = -1
using a we do the synthetic division:
Another example:
Divide 3x³ – 2x ² + 3x – 4 by x – 3 using synthetic division
x – 3 = x + a
a = 3
REMAINDER THEOREM
If P(x) is polynomial, then P(a) is equal to the remainder when P(x) is divided by ( x – a )
Find the remainder 6x³ – 5x² + 4x – 17 with (x + 3)
P(3) = 6x³ – 5x²+ 4x – 17
= 6(3)³ – 5(3)² + 4(3) – 17
= 6(27) – 5(9) + 12 – 17
= 6(27) – 5(9) + 12 – 17
= 162 – 45 + 12 – 17
= 112
* (x + 3) is not a factor of 6x³ – 5x² + 4x – 17 because the remainder is not zero
Find the remainder when x^6 + 5x^5 + 5x^4 + 5x³ + 2x² – 10x – 8 with ( x + 1 )
P(1) = x^6 + 5x^5 + 5x^4 + 5x³ + 2x² – 10x – 8
= (1)^6 + 5(1)^5 + 5(1)^4 + 5(1)³ + 2(1)² – 10(1) – 8
= 1 + 5 + 5 + 5 + 2 – 10 – 8
= 0
* ( x + 1 )is a remainder of x^6 + 5x^5 + 5x^4 + 5x³ + 2x² – 10x – 8 because the remainder is zero.
COMPLETELY FACTOR
* using both factor and remainder theorem
f(x) = x³ - 3x² - 13x + 15
steps:
1) find the factors of the constant.
Constant = 15
factors of 15 : ± 1, ± 15, ± 3, ± 5
2) pick one of the factors then evaluate
P(1) = x³ - 3x² - 13x + 15
= (1)³ - 3(1)² - 13(1) + 15
= 1 - 3 - 13 + 15
= 0
*because the remainder is zero ( x - 1 ) is one of the remainders
3) now use synthetic division (because its easier) to reduce and find the other factor.
x³ - 3x² - 13x + 15 = ( x - 1 ) ( x - 5 ) ( x + 3 )
*** can only use if dividing by (x ± a ) coefficient must be 1.
ok.. i think that's all for today.. our homework is exercise 53 #'s 1 - 7.. and the next blogger is Youna.
thats all.. bye!! im going to change the next blogger because i found out that youna isnt gonna be at school tommorow soo here it comes!...... THE NEXT BLOGGER IS ELVIN!
posted by Remyshire @ 11:52 PM
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