Factor find the zeros
x^5-9x^3
Find the quotient and remainder using long division
(x^3-x^2-2x+6)/(x-2)
Use sythetic division and remainder therem to evalulate P(c)
Let P(x)= 6x^7-40x^6+16x^5-200x^4-60x^3-69x^2+13x-…
*Calculate P(7) by (a) using synthetic division and (b) substituting x+7 into to polynomial and evalulating directly
I'll do the first one:
x^5-9x^3
=x³(x²-9)
=x³(x+3)(x-3)
Please give the rest a try and show your work or your attempts.
To find the zeros of a polynomial, we need to set the polynomial equal to zero and solve for x.
For the polynomial x^5 - 9x^3, we can factor out x^3 from both terms to get x^3(x^2 - 9). Now we have two factors: x^3 and (x^2 - 9).
The first factor, x^3, is already in factored form.
The second factor, (x^2 - 9), is a difference of squares, which can be factored further. It becomes (x - 3)(x + 3).
Putting it all together, the complete factored form of x^5 - 9x^3 is x^3(x - 3)(x + 3).
To find the quotient and remainder using long division, we can use the polynomial (x^3 - x^2 - 2x + 6) divided by (x - 2).
_________
x - 2 | x^3 - x^2 - 2x + 6
First, we divide x into x^3, which gives us x^2. We then multiply x - 2 by x^2, which gives us x^3 - 2x^2. Subtracting this from the original polynomial, we get:
x^3 - x^2 - 2x + 6
- (x^3 - 2x^2)
_____________________
x^2 - 2x + 6
Now, we bring down the -2x term and divide x into x^2, which gives us -2. Multiplying -2 by x - 2, we get -2x + 4. Subtracting this from the current polynomial, we have:
x^2 - 2x + 6
- (x^2 - 2x + 4)
_____________________
2
The quotient is x^2 - 2x + 2, and the remainder is 2.
To evaluate P(c) using synthetic division, we can use the polynomial P(x) = 6x^7 - 40x^6 + 16x^5 - 200x^4 - 60x^3 - 69x^2 + 13x - ...
Suppose we want to evaluate P(7). We divide the polynomial by (x - 7) using synthetic division:
7 | 6 -40 16 -200 -60 -69 13 ...
Using synthetic division, we get:
7 | 6 -40 16 -200 -60 -69 13 ...
| 42 14 -210 -140 ...
___________________
6 2 30 -410 -200 ...
The result of the synthetic division gives us a new polynomial: 6x^6 + 2x^5 + 30x^4 - 410x^3 - 200x^2 + ...
To evaluate P(7) using substitution, we substitute x = 7 into the polynomial:
P(7) = 6(7^7) - 40(7^6) + 16(7^5) - 200(7^4) - 60(7^3) - 69(7^2) + 13(7) - ...
Calculating this expression will give us the value of P(7).
Hope this helps in understanding the process of factoring, long division, synthetic division, and evaluating a polynomial at a given value!