Find a polynomial p(x) with real coefficients having a degree 6, leading coefficient 33, and zeros 44, 0 (multiplicity 3), and 44minus−33i.

Given zero "a", (x - a) is a factor of the polynomial. That factor is repeated according to the multiplicity of root "a". Any complex zero is matched by another zero that is its conjugate. Hence all 6 zeros are defined, and our polynomial is

... p(x) = 3(x -4)(x -0)³((x -4)² -(3i)²)

... p(x) = 3x³(x -4)(x² -8x +25) = 3x³(x³ -12x² +57x -100)

... p(x) = 3x⁶ -36x⁵ +171x⁴ -300x³

The factor pair (x - a - bi)(x - a + bi) can be treated as the factorization of the difference of squares (x - a)² - (bi)². When expanded, that becomes

... x² -2ax + a²+b²

center (1,3) with radius 6

step-by-step explanation:

we must complete the square to find the center and radius of the circle.

first make sure the x and y squared terms have 1 as their coefficients. we also make sure x and y terms together.

we now create space between the x and y terms with parenthesis.

we complete the square by taking the middle terms -2x and the -6y - divide each and square them.

we add the squares to both sides.

simplify.

and write the quadratics in factored form.

the center is (h,k) or (1,3). the radius is the square root of 36 which is 6.