find the lowest positive zero of :

Question

f(x)= x⁴+3x³-21x²-83x-60

Answer 1

Graphing is a way of seeing the zeros but if you'd like to use "old school" methods, let's start with the rational zeros (roots) theorem which says, if there are any zeros that are rational, they will be the factors of the constant term (in this case "60") divided by the factors of the leading coefficient (in this case "1"). The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 30, and 60 so the POSSIBLE rational zeros are ± all those.

Using Descartes' rules of signs, since f(x) = x⁴ + 3x³ - 21x² - 83x - 60 has only 1 sign change, then there will be only 1 positive real root. and since f(-x) = x⁴ - 3x³ - 21x² + 83x - 60 has 3 sign changes, then there may be either 3 real negative zeros or 1 real negative zero.

Since there is a better chance of finding a negative zero, I will start doing synthetic division there. Let's try -1:

-1 | 1 3 -21 -83 -60

-1 -2 23 60

------------------------------------

1 2 -23 -60 0

So we got lucky and -1 is a zero so (x + 1)(x³ + 2x² - 23x - 60) = x⁴ + 3x³ - 21x² - 83x - 60

Now, let's try -2 (but we use the reduced polynomial:

-2 | 1 2 -23 -60

-2 0 46

--------------------------

1 0 -23 -14

So, since that didn't work, let's try -3:

-3 | 1 2 -23 -60

-3 3 60

--------------------------

1 -1 -20 0

So that worked and now we have another factor, (x + 3) meaning:

f(x) = x⁴ + 3x³ - 21x² - 83x - 60 = (x + 1)(x + 3)(x² - x - 20)

For x² - x - 20, we can factor it easily using quadratic factoring techniques to (x - 5)(x + 4). So now we have:

f(x) = x⁴ + 3x³ - 21x² - 83x - 60 = (x + 1)(x + 3)(x - 5)(x + 4) meaning the zeros are x = -1, -3, 5, and -4.

The lowest positive zero is x = 5

Answer 2

Looking at all the ± factors of 60, use synthetic division to help factor the polynomial.

Try 1:

1 3 -21 -83 -60 |-1

-1 -2 23 60

-------------------------

1 2 -23 -60 0

Try 3:

1 2 -23 -60 |-3

-2 3 60

--------------------

1 -1 -20 0

x²-x-20 = 0

(x+4)(x-5) = 0

So zeroes are -1,-3,-4, 5

Only 5 is positive.

Answer 3

Let look for rational zeros, by Rational Zeros Theorem this is factors of - 60: ±1, ±2,±3 ,±4, ±5, ±6,±10, ±12, ±15, ±30, ±60. We have to check only positive and we see that only positive zero x = 5.