10.find all the zeroes of the polynomial function f(x)=x3-5x2+6x-30.if you use synthetic division show all three lines of numbers
i actually figured this out so i dont need help with this problem.
yes x^2(x-5) + 6(x-5) = (x^2+6)(x-5)
To find the zeroes of the polynomial function f(x) = x^3 - 5x^2 + 6x - 30, we can use synthetic division. Synthetic division allows us to test potential zeroes quickly and efficiently.
Step 1: Write down the coefficient of each term of the polynomial in descending order. In our case, it's:
1, -5, 6, -30
Step 2: Identify a potential zero of the function. One way to do this is by using the Rational Root Theorem, which states that if x = p/q is a rational zero of a polynomial, then p must be a factor of the constant term (-30 in our case), and q must be a factor of the leading coefficient (1 in our case). Let's try a few possible rational zeroes: ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30.
Let's assume we start with x = 2 as a potential zero.
Step 3: Perform the synthetic division as follows:
2 | 1 - 5 6 - 30 | ← Coefficients of the polynomial
| 2 - 6 24 |
|________________
1 - 3 0 - 6 | ← Coefficients after one line of synthetic division
| 1 - 3 0 - 6 | ← Coefficients after two lines of synthetic division
Step 4: Read the bottom line of the synthetic division result. The coefficients represent the polynomial after division by x - 2. In our case, after two lines of division, we obtain the polynomial 1x^2 - 3x + 0 - 6, which simplifies to x^2 - 3x - 6.
Step 5: Repeat the process with the new polynomial. In this case, since it is a quadratic polynomial, we can use the quadratic formula or factoring to find the remaining zeros. However, for the purpose of illustrating synthetic division, let's assume we test x = 3 as the new potential zero.
Step 6: Perform the synthetic division again:
3 | 1 - 3 0 - 6 |
| 3 0 0 |
|________________
1 0 0 - 6 |
The result after the second line of division is the polynomial 1x^2 + 0x + 0 - 6, which simply simplifies to x^2 - 6.
Step 7: Now we have a quadratic polynomial. To find its zeros, we can solve the equation x^2 - 6 = 0 using factoring or the quadratic formula:
x^2 - 6 = 0
(x + √6)(x - √6) = 0
x = ±√6
So, the zeroes of the polynomial function f(x) = x^3 - 5x^2 + 6x - 30 are x = 2 and x = ±√6.