Use the given root to find the remaining roots

Question

q(x)=x^4-6x^3+3x^2+12x-10

given root: x=√2

Answer 1

Hello Mary J.

Use synthetic division with the √2 root.

√2¦ 1 ******** -6 ******** 3 ******** 12 ******** -10

***************√2 ***** 2-6√2 *** -12+5√2 **** 10

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

**** 1 ******-6+√2 **** 5-6√2 *****5√2 *********0

Since √2 is a root, so is its conjugate, -√2. Use this on the result from the first go.

-√2¦ 1 ******** -6+√2 ******** 5-6√2 ******** 5√2

***************** -√2 *********** 6√2 ********-5√2

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

***** 1 ********** -6 ************* 5 ************ 0

This last line yields a quadratic, x² - 6x + 5. Set it equal to zero, factor, and solve for the last two roots, i.e.

x² - 6x + 5 = 0

→ (x - 5)(x - 1) = 0

→ x = 5 and x = 1.

So the roots that were not given were

x = -√2, 5, and 1.

I hope the above helps, and have a great evening.

Michael Ehlers