The question is "Show that z-2 is a factor of P(z)= z^3+mz^2+(3m-2)z-10m-4 for all values of m. For what values of m is (z-2)^2 a factor of P(z)?"
I've proven the first part, using synthetic division, for all values of m. But the second part, I don't understand why the divisor must be divide again by 2, in order to show the values of m. If it's trying to find the values of P(z), shouldn't be P(z) that should be divided?
after your first division by z-2 using symthetic division you should have had
z^2 + (m+4)z + 5m+2
divide it again by z-2, you should now have a remainder of 7m+10
but if the division is exact that remainder of 7m+10 has to be zero
so
7m+10 = 0
m = -10/7
(dividing by (z-2)^2 is the same as dividing by z-2 and then dividing that result again by z-2.
Just like dividing 20 by 4,
divide 20 by 2 to get 10, then divide by 2 again to get 5)
Thats the results I got, thank you =D
Arithmetic progression
Ah, the joys of polynomial division and finding factors! It can be quite a circus, but fear not, I shall bring some humor to your confusion.
Well, let's dive into this mathematical spectacle, shall we? To show that (z - 2)² is a factor of P(z), we need to divide P(z) by (z - 2)² and see if we end up with a remainder of zero.
Now, you might be wondering why (z - 2)² is being divided instead of P(z), but let me tell you, it's all part of the grand mathematical show! When you divide P(z) by (z - 2)², you're essentially testing if (z - 2)² goes into P(z) evenly without leaving a remainder. If it does, then (z - 2)² is a factor.
But why (z - 2)² specifically? Well, (z - 2)² represents a double root of P(z) at z = 2. In other words, when (z - 2)² is a factor of P(z), it means that the polynomial P(z) has a repeated root at z = 2. It's like a magician pulling the same rabbit out of his hat twice!
So, by dividing P(z) by (z - 2)², you'll end up with some quotient and remainder. If the remainder is zero, then congratulations! This mathemagical trick worked, and (z - 2)² is indeed a factor of P(z).
As for the values of m for which (z - 2)² is a factor, you'll have to examine your quotient and remainder after the division. Again, if the remainder is zero, it means that (z - 2)² is a factor for the given value of m.
I hope this circus act of mathematical explanation has enlightened you. Keep up the good work, and remember, math is all about having fun under the big top!
To show that (z - 2)^2 is a factor of P(z), we need to divide P(z) by (z - 2)^2 and check if the remainder is zero for all values of m.
Let's first understand why we divide P(z) by (z - 2)^2. Remember that (z - 2)^2 represents a quadratic factor, and if it is a factor of P(z), it means that when we divide P(z) by (z - 2)^2, there should be no remainder. In other words, (z - 2)^2 should completely divide P(z).
The process involves performing long division or synthetic division. Since you mentioned you are familiar with synthetic division, let's use that method.
Firstly, we divide P(z) by (z - 2). If (z - 2) is a factor, after division, the remainder should be zero. From the coefficients of P(z), we have:
P(z) = z^3 + mz^2 + (3m - 2)z - (10m + 4)
To divide P(z) by (z - 2), we perform synthetic division as follows:
2 | 1 m (3m - 2) (-10m - 4)
| 2m 10m-4-2m^2 (-20m-4)
| 1 (m+2m) (3m-2-8m)+ (4+20)
| 1
----------------------------------------
| 1 (m+2m) (3m-10m-8m-2) (24+20)
After the division, we obtain a new polynomial whose coefficients are:
1, (m + 2m), (3m - 10m - 8m - 2), (24 + 20)
Simplifying further:
1, 3m, -15m - 8m - 2, 24 + 20
1, 3m, -23m - 2, 44
Now, let's move on to the second part of the question: finding the values of m for which (z - 2)^2 is a factor of P(z). To do this, we need to divide the obtained polynomial above by (z - 2) again.
Dividing by (z - 2) once more:
2 | 1 3m (-23m - 2) 44
| 6m 4m^2 + 2m^2 (-23m - 2 - 6m^2)
| 1 (3m+6m) (-23m - 2 - 6m^2 +48m) + (44 + 101)
| 1 (3m + 6m) (-23m - 6m^2 + 48m - 145)
----------------------------------------
| 1 (3m + 6m) (-23m - 6m^2 + 48m - 145) (-101)
After the second division, we obtain a new polynomial whose coefficients are:
1, (3m + 6m), (-23m - 6m^2 + 48m - 145), (-101)
Simplifying further:
1, 9m, -6m^2 + 25m - 145, -101
Now we need to check if the remainder (-101) is zero for all values of m. If the remainder is zero, it means that (z - 2)^2 is a factor of P(z). Otherwise, if the remainder is non-zero, (z - 2)^2 is not a factor.
By equating the remainder to zero:
-101 = 0
We find that -101 does not equal zero.
Therefore, (z - 2)^2 is NOT a factor of P(z) for any value of m.
In summary, to find the values of m for which (z - 2)^2 is a factor of P(z), we divide P(z) by (z-2) and then divide the resulting polynomial by (z-2) again. If the remainder is zero for all values of m, then (z - 2)^2 is a factor of P(z). In this case, however, we determined that (z - 2)^2 is not a factor of P(z).