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Rama09 [41]
3 years ago
8

Find the polynomial of minimum degree, with real coefficients, zeros at x=4+4i and x=2, and y-intercept at 64

Mathematics
1 answer:
Mice21 [21]3 years ago
8 0

Answer:

\displaystyle -x^3+10x^2-48x+64

Step-by-step explanation:

We want to find the minimum-degree polynomial with real coefficients and zeros at:

x= 4+4i\text{ and }  x = 2

As well as a <em>y-</em>intercept of 64.

By the Complex Root Theorem, if <em>a</em> + <em>b</em>i is a root, then <em>a</em> - <em>b</em>i is also a root.

So, a third root will be 4 - 4i.

The factored form of a polynomial is given by:

P(x)=a(x-p)(x-q)...

Where <em>a</em> is the leading coefficient and <em>p</em> and <em>q</em> are the zeros. More factors can be added if necessary.

Substitute:

P(x)=a(x-(2))(x-(4+4i))(x-(4-4i))

Since we want the minimum degree, we won't need to add any exponents.

Expand the second and third factors:

\displaystyle \begin{aligned} (x-(4+4i))(x-(4-4i))&=(x-4-4i)(x-4+4i) \\ &= x(x-4-4i)-4(x-4-4i)+4i(x-4-4i)\\ &=x^2-4x-4ix-4x+16+16i+4ix-16i-16i^2\\ &= x^2-8x+32\end{aligned}

Hence:

P(x)=a(x-2)(x^2-8x+32)

Lastly, we need to determine <em>a</em>. Since the <em>y-</em>intercept is <em>y</em> = 64, this means that when <em>x</em> = 0, <em>y</em> = 64. Thus:

64=a(0-2)(0^2-8(0)+32)

Solve for <em>a: </em>

-64a=64\Rightarrow a=-1

Our factored polynomial is:

P(x)=-(x-2)(x^2-8x+32)

Finally, expand:

\displaystyle \begin{aligned} P(x) &=-(x^2(x-2)-8x(x-2)+32(x-2)) \\&=-(x^3-2x^2-8x^2+16x+32x-64)\\&=-(x^3-10x^2+48x-64)\\&= -x^3+10x^2-48x+64\end{aligned}

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