Write a polynomial of least degree with rational coefficients and with the root - 19-5 square root 2.

Write your answer using variable x and in standard form with leading coefficients of 1.

y = x² + 38x + 311

Step-by-step explanation:

If a polynomial has rational coefficients and irrational roots, then the roots must be conjugate pairs.

y = (x - (-19-5√2)) (x - (-19+5√2))

y = (x + 19+5√2) (x + 19-5√2)

Use FOIL to distribute:

y = x² + x (19-5√2) + x (19+5√2) + (19+5√2) (19-5√2)

y = x² + 19x - 5√2 x + 19x + 5√2 x + (19+5√2) (19-5√2)

y = x² + 38x + (19+5√2) (19-5√2)

Use FOIL to distribute the last term:

y = x² + 38x + 19² - 95√2 + 95√2 - 50

y = x² + 38x + 361 - 50

y = x² + 38x + 311