Instead of doing your problem for you, I'll do another one that is
EXACTLY like yours, step by step. Use it as a guide to do yours. The
problem I will do is this one:
Use synthetic division to show that the given x value is a zero of the
polynomial. Then find all other zeros.?
P(x)=x^3-3x^2-10x+24; x=2
We write down the coefficients of P(x) in a row, and the 2 from x=2
to the left
2 | 1 -3 -10 24
|
Bring the 1 down below the bottom line:
2 | 1 -3 -10 24
|
1
Multiply the 1 on the bottom by the 2 at the left, getting 2.
Then write it ABOVE and TO THE RIGHT of the 1 on the bottom,
immediately under the -3, like this:
2 | 1 -3 -10 24
| 2
1
Add the -3 and the 2, getting -1, and write it below the line,
like this:
2 | 1 -3 -10 24
| 2
1 -1
Multiply the -1 on the bottom by the 2 at the left, getting -2.
Then write it ABOVE and TO THE RIGHT of the -1 on the bottom,
immediately under the -10, like this:
2 | 1 -3 -10 24
| 2 -2
1 -1
Add the -10 and the -2 under it, getting -12, and write it below the line,
like this:
2 | 1 -3 -10 24
| 2 -2
1 -1 -12
Multiply the -12 on the bottom by the 2 at the left, getting -24.
Then write that -24 ABOVE and TO THE RIGHT of the -12 on the bottom,
under the 24, like this:
2 | 1 -3 -10 24
| 2 -2 -24
1 -1 -12
Add the 24 and the -24 under it, getting 0, and write the 0 below the line,
under the 24, like this:
2 | 1 -3 -10 24
| 2 -2 -24
1 -1 -12 0
That completes the synthetic division. Now we must interpret what we have.
The right-most number 0 on the bottom is the REMAINDER.
The fact that this REMAINDER number on the bottom right came out to be 0,
shows that the given x value of 2 is a zero of the polynomial.
The numbers on the bottom, all except the remainder 0 on the bottom right, indicate the quotient.
We have factored this polynomial:
P(x) = x3 - 3x2 - 10x + 24
like this:
P(x) = (x - 2)(1x2 - 1x - 12)
We erase the understood 1's
P(x) = (x - 2)(x2 - x - 12)
To find all the zeros, we set the factored polynomial equal to 0
(x - 2)(x2 - x - 12) = 0
We use the zero-factor property, solve the x-2=0 for the zero 2
that we already know we had, and factor the quadratic trinomial
inside the second parentheses:
x - 2 = 0; x2 - x - 12 = 0
x = 2; (x - 4)(x + 3) = 0
x - 4 = 0; x + 3 = 0
x = 4 x = -3
So the other zeros besides the 2 are: 4 and -3.
Now do your problem the exact same way, step by step.
Edwin