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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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2 years ago
What is the value of theta for the acute angle in a right triangle?
balu736 [363]

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5 0
3 years ago
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34.23 times one year is ....
kenny6666 [7]

Answer:

410.76

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3 years ago
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Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 21 in. by 12 in. by
a_sh-v [17]

Answer:

Dimension of the box is 16.1\times 7.1\times 2.45

The volume of the box is 280.05 in³.

Step-by-step explanation:          

Given : The open rectangular box of maximum volume that can be made from a sheet of cardboard 21 in. by 12 in. by cutting congruent squares from the corners and folding up the sides.

To find : The dimensions and the volume of the box?

Solution :

Let h be the height of the box which is the side length of a corner square.

According to question,

A sheet of cardboard 21 in. by 12 in. by cutting congruent squares from the corners and folding up the sides.

The length of the box is L=21-2h

The width of the box is W=12-2h

The volume of the box is V=L\times W\times H

V=(21-2h)\times (12-2h)\times h

V=(21-2h)\times (12h-2h^2)

V=252h-42h^2-24h^2+4h^3

V=4h^3-66h^2+252h

To maximize the volume we find derivative of volume and put it to zero.

V'=12h^2-132h+252

0=12h^2-132h+252

Solving by quadratic formula,

h=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

h=\frac{-(-132)\pm\sqrt{132^2-4(12)(252)}}{2(12)}

h=\frac{132\pm72.99}{24}

h=2.45,8.54

Now, substitute the value of h in the volume,

V=4h^3-66h^2+252h

When, h=2.45

V=4(2.45)^3-66(2.45)^2+252(2.45)

V\approx 280.05

When, h=8.54

V=4(8.54)^3-66(8.54)^2+252(8.54)

V\approx -170.06

Rejecting the negative volume as it is not possible.

Therefore, The volume of the box is 280.05 in³.

The dimension of the box is

The height of the box is h=2.45

The length of the box is L=21-2(2.45)=16.1

The width of the box is W=12-2(2.45)=7.1

So, Dimension of the box is 16.1\times 7.1\times 2.45

6 0
3 years ago
What is f+0.3&lt;-1.7 please help
satela [25.4K]
F<-2.  You just subtract 0.3 to -1.7
5 0
3 years ago
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