What is this answer?

Question

given that the polynomial function has the given​ zero, find the other zeros.

​f(x)=x3−11x2+49x−75​; 3

The other​ zero(s) is/are ?

​(Do not factor. Use a comma to separate answers as needed. Express complex numbers in terms of

i.

Type each solution only​ once.)

Answer 1

Use synthetic division to divide x³ - 11x² + 49x - 75 by "x - 3" Note that "x - 3" is equivalent to a zero of x = 3

To use synthetic division:

Set up the division by putting the "zero" to the left and the coefficients of the polynomial to the right. Like this:

Bring down the "1" and multiply 3 times 1 and put the result (3) below the "-11" then add them. Put the answer (-8) below the line. Now, multiply 3 times -8 and put the result (-24) under the "49" and add them. Put the answer (25) below the line. Now multiply 3 times 25 and put the result (75) under the "-75" and add them. Put the answer (zero) below the line. Like this:

If you get zero as the last result (like you did here) then the remainder is zero and "x - 3" divides evenly into x³ - 11x² + 49x - 75. The result (quotient) is coded in the numbers below the line which are 1, -8, and 25 (ignore the zero). Starting with an "x" that is one degree less than the highest degree in the original polynomial), add "x" values in decreasing order to the numbers. So, since you started with x³ in your original polynomial, the quotient will start with x² so 1x² - 8x + 25 or just x² - 8x + 25.

Now, to find the other zeros, solve x² - 8x + 25 = 0 using the quadratic formula.