polynomial equations and inequalities

Question

determine the equation of

a. the cubic function with zeros 1,-2,-5 and y-intercept 10

b. the quadratic function with roots at 2 & -1 and a double root at 3, and through (4,-10)

Answer 1

If a polynomial equation has a zero or root c, then the corresponding factoring (x - c).

The pattern for a cubic equation is as follows:

y = a(x - c)(x - d)(x - e)

For zeros 1, -2, -5, the initial setup becomes

y = a(x - 1)(x + 2)(x + 5).

To find a, use the y-intercept coordinates

(0, 10).

10 = a(0 - 1)(0 + 2)(0 + 5). Solving for a, the result is a = -1.

Resulting cubic equation:

y = -x^3 - 6x^2 -3x + 10..

Since part b has 4 roots, it is actually a quartic polynomial rather than a quadratic, which has 2 roots.

Follow the strategy as shown above to find the result.

Answer 2

a. We looking for cubic function in factored form f(x) = a(x - 1)9x - (-2))(x - (-5)) = a(x - 1)(x + 2)(x + 5). To get a we use y - intercept f(0) = 10; a(0 - 1)(0 + 2)(0 + 5) = 10; - 10a = 10, a = - 1

f(x) = - (x -1)(x + 2)(x + 5) = - x³ - 6x² _{- 3x + 10}

_{b. Fourth degree polynomial function with roots 2 & -1 and double root at 3 is g(x) = a(x - 2)(x + 1)(x - 3)}² _{To get a we use point (4, - 10): g(4) = - 10, a(4 - 2)(4 + 1)(4 - 3)}² _{= - 10, 10a = -10, a = - 1}

_{g(x) = -(x - 2)(x+ 1)(x - 3)}²_{; g(x) = - x}⁴_{+ 7x}³ _{- 13x}² _{- 3x + 18}