HELP show me how to do this so i could learn it please, thank you?

Question

Given the polynomial

p(x)=2x^3+17x^2+23x-42 and one of its factors x+6

a. how many zeros will there be? how do u know that?

b. find the remaining factors of the polynomial

c. please explain in words how you found the others.

Answer 1

Hi,

There can be as many roots as the largest exponent of the polynomial. In this case, there could be as many as 3 zeros. There could also be a double or triple roots, so there could also be 2 or just 1 solution.

The zeros of this polynomial are x = -6, -3.5, and 1. They can be found by putting the polynomial into a graphing calculator and using the zero command to find these zeros.

They can also be found by doing synthetic division if you divide by the known zero, x = -6.
....._____________
-6.|.2...17...23..-42
...........-12.-30...42
.....----------------------
.......2.....5.....-7....0 This zero is the remainder. It shows that -6 is a zero. The polynomial remaining in front of it is

2x² + 5x - 7. This factors into:
(2x + 7)(x - 1) = 0 this solves to:
x = -7/2 and x = 1

I hope this helps!! :-)

Answer 2

If you divide the cubic by (x+6), you will get a quadratic. The quadradic can be either factored or solved by quadratic formula to get other two factors. There is a change of signs rules about polynomials and roots but I forgot what it is.

Answer 3

Since it is a third-order polynomial, there should be 3 zeros (even if two are imaginary).

b. Use long division (or whatever method you're comfortable with) to divide polynomial by x+6

............2x^2 + 5x - 7
..........---------------------------------
x+6..|..2x^3 + 17x^2 + 23x - 42
...........2x^3 + 12x^2
...........------------------
..............0.+.....5x^2 + 23x
........... ............ 5x^2 + 30x
............. ..........----------------
....... .......... .......0 - 7x - 42
........... ...... ....... ...-7x - 42
........ ................. ...-----------
.......... ................. .........0

So 2x^2 + 5x - 7 is your resulting quadratic. Factor normally to get (2x +7)(x -1).
So your factors are x = -6, x = -7/2, x= 1