Find all the zeroes of the polynomial function f(x)=x^3-5x^2+6x-30. If you use synthetic division, show all three lines of numbers.
Please help, thank you.
Look for the low-hanging fruits first
Descartes' Rule of Signs says there are no negative roots, so that narrows things down a bit.
x^3-5x^2+6x-30 = (x-5)(x^2+6)
There is a nice synthetic division calculator at
https://www.mathportal.org/calculators/polynomials-solvers/synthetic-division-calculator.php
Thank you oobleck :)
To find the zeroes of the polynomial function f(x) = x^3 - 5x^2 + 6x - 30, you can use synthetic division to test possible values that could potentially be zeroes. Here's how you can do it:
Step 1: Write down the coefficients of the polynomial in descending order:
1 -5 6 -30
Step 2: Start with a possible value for a zero of the polynomial. You can try factors of the constant term (-30) divided by factors of the leading coefficient (1). Possible values could be ±1, ±2, ±3, ±5, ±6, ±10, ±15, or ±30. Let's start with x = 1.
Step 3: Set up the synthetic division. Write the possible zero (1) on the left side and line up the coefficients on the right side:
1 | 1 -5 6 -30
Step 4: Bring down the first coefficient (1) directly below:
1 | 1 -5 6 -30
1
Step 5: Multiply the possible zero (1) by the first coefficient (1) and write the result below the next coefficient:
1 | 1 -5 6 -30
1
-------------
-4
Step 6: Add the result (-4) to the next coefficient (-5) and write the sum below the next coefficient:
1 | 1 -5 6 -30
1
-------------
-4 -9
Step 7: Repeat steps 5 and 6 until you reach the end of the coefficients:
1 | 1 -5 6 -30
1 -4
-------------
-4 -9 -3
Step 8: The last number in the bottom row (-3) represents the remainder. If the remainder is zero, then the possible value you chosen (1) is a zero of the function. Otherwise, you need to try a different possible zero.
In this case, since the remainder is -3 and not zero, 1 is not a zero of the function. You can try other possible zeros using the same process until you find one that gives a remainder of zero.
Hope this helps! Let me know if you have any more questions.