Find all the zeros of the function and write the polynomial as a product of linear factors?

Question

x^3-6x^2+13x-10
Please show details on factoring
(I am supposed to use synthetic division)

Answer 1

Possible roots = +/-1, +/-2, +/-5

Let's test if 2 is a possible zero.

8 - 24 + 26 - 10

-2 + 2 = 0

2 is one of the zeroes.

Use synthetic division and divide the expression by (x-2). Then multiply that by (x-2).

(x^3 - 6x^2 + 13x - 10) / (x-2) = x^2 - 4x + 5

(x-2)(x^2-4x+5)

Use quadratic formula to find the zeros found in x^2-4x+5

x = 2+/-i

Zeroes: 2, 2+/-i

Answer 2

You know that since the coefficiemts are integers, if there is a rational root a/b, then a divides -10 and b divides the leading coefficient 1. This means that all rational roots must occur in the set +-1, +-2, +-5. +-10. Testing these with sybthetic division, you find that 2 is a zero of the function.

Since 2 is a zero, the Factor Theorem tells you that x - 2 is a factor of the given polynomial. Moreover, when you used synthetic division with divisor 2, the bottom row was 1 -4 5 0. This tell you that the complete factorization (although not yet reduced to linear factors) is (x - 2)(x^2 - 4x + 5). You can continue to look fo other rational roots, but there are none.

You can find the roots of x^2 - 4x + 5 = 0 by using the quadratic formula. You find that the two additional zeros of the original polynomial are 2 + i and 2 - i. Now you know that the factorization into linear factors is

(x - 2)[x - (2 + i)][x - (2 - i)].

You should check to see that when you multiply this out, you get the original polynomial.

Answer 3

If there are any integer zeros of a function they must divide the constant term -10. So use synthetic division to check and see if +/- 1,2,5, or 10 are zeros:

After trying +/- we find that 2 is a zero so (x-2) is a root.
Here is the synthetic division:

2/ 1 -6 13 -10
2 -8 10
---------------------
1 -4 5 0

So we can write your equation as

(x-2)(x^2 - 4x +5)=0

but x^2 - 4x +5 will not factor using integers so you must solve x^2 - 4x + 5 using the quadratic formula or completing the square. This will give roots of 2+i and 2-i.

With zeros of 2, 2+i and 2-i you can factor you polynomial
into

(x-2)(x-2-i)(x-2+i)=0

Answer 4

Do it yourself. Don't copy the answer from someone who answers your question and hand in your homework assignment claiming you did it.
Here's a hint: What are the common primes of -6 and -10?
Try these zeros and divide.
Show your work and you can honestly say that YOU did it.

Well, it's been 3 minutes and I noticed someone else decided to answer and do your homework for you.