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pashok25 [27]
2 years ago
12

Write the equation for a 3rd degree polynomial function that has roots at x = 1–3i and x = 2. The y‑intercept is (0,10). Write a

n equation for this function in factored form with real values.
Mathematics
1 answer:
HACTEHA [7]2 years ago
5 0

Answer:

\displaystyle P(x)=-\frac{1}{2}(x^2+2x+10)(x-2)

Step-by-step explanation:

<u>Factored Form Of Polynomials </u>

If we know the roots of a polynomial as

\alpha _1,\alpha _2,\alpha _3

the polynomial can be expressed in factored form as

a(x-\alpha _1)(x-\alpha _2)(x-\alpha _3)

We are given two of the three roots of the polynomial:

\alpha _1=1-3i

\alpha _2=2

The other root must be the conjugate of the complex root:

\alpha _3=1+3i

Recall the product of two complex conjugate numbers is

(1-3i)(1+3i)=1^2-(3i)^2=1+9=10

The required polynomial is

P(x)=a\left[ x-(1-3i)\right ]\left[ x-(1+3i)\right ](x-2)

P(x)=a(x^2+2x+10)(x-2)

This is the factored form of the polynomial where only real numbers appear

We need to find the value of a, such as

P(0)=10

P(0)=a(0^2+2(0)+10)(0-2)=10

-20a=10

Thus the value of a is

\displaystyle a=-\frac{1}{2}

The expression of the required polynomial is

\boxed{P(x)=-\frac{1}{2}(x^2+2x+10)(x-2)}

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