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!
Today we defined the word “inverse”, were taught how to inverse a function and how to figure out if two given functions are the inverse of each other. Well, let’s get started…
Inverse: the set of ordered pairs obtained by interchanging the coordinates of each ordered pair in relation.
Example:
Original set of pairs --> [ ... (-3,3), (1,4), (4,5) ... ]
Inverse of set of pairs --> [ ... (3,-3), (4,1), (5,4) ... ]
* notice that all i did was switch the x and y values around in each pair
When graphing the original and inverse sets of pairs, each point must reflect the other across the y=x line on the graph. So, it will look like this:
Now, the next question is: How do you find the inverse of a function? Well, according to Mrs. Ingram, if you follow these four steps, you'll be fine:
given function: f(x) = 3x - 2 find: f^-1(x)
STEP 1: Replace f(x) with y y = 3x - 2
STEP 2: Interchange the x and y x = 3y -2
STEP 3: Solve for y x + 2 / 3 = y
STEP 4: Replace y with f^-1(x) f^-1(x) = x+2 / 3
And there we have it. The inverse of the function f(x)=3x - 2 is f^-1(x)=x+2 / 3
But, what about if you are given: f(x)=3x-2 & m(x) =x+2 / 3 and it asks if those two functions are inverses of each other? To find out we must do the whole composition of functions thing ... lol. Here we go:
f(m(x)) & m(f(x))
f(x+2 / 3) = 3(x+2 / 3) - 2 ** the threes cancel out = x + 2 - 2 = x
Both compositions equal x which means the two functions are inverses of each other.
And... that's about it. We've come to the conclusion of the blog and let's make this quick. Next blogger shall be ... well remyshire asked for it so yeah.. it's her. Ex. 52 # 1-5,8,9,11 are the homework for tonight. And... well that's all folks. vamoose.
