Mathematics, 02.11.2020 09:30 estheradame547

Give a polynomial of degree 3 that has zeros of 18, 10i, and −10i, and has a value of −3434 when x=1. Write the polynomial in standard form axn+bxn−1+….

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The polynomial in standard form is: $\mathbf{P(x) = \frac{2}{101}x^3 - \frac{36}{101}x^2 +\frac{200}{101}x - \frac{3600}{101}}$

The zeros of the polynomial are 18, 10i and -10i.

So, the polynomial is:

$\mathbf{P(x) = a(x - 18)(x - 10i)(x -(-10i))}$

Open bracket

$\mathbf{P(x) = a(x - 18)(x - 10i)(x +10i)}$

Apply difference of two squares

$\mathbf{P(x) = a(x - 18)(x^2 - (10i)^2)}$

Expand

$\mathbf{P(x) = a(x - 18)(x^2 - 100i^2)}$

In complex numbers, i^2 = -1.

So, we have:

$\mathbf{P(x) = a(x - 18)(x^2 - 100(-1))}$

Expand

$\mathbf{P(x) = a(x - 18)(x^2 + 100)}$

From the question, P(1) = -34.

So, we have:

$\mathbf{a(1 - 18)(1^2 + 100) = -34}$

$\mathbf{a(- 17)(1 + 100) = -34}$

Divide both sides by -17

$\mathbf{a(1 + 100) = 2}$

$\mathbf{a(101) = 2}$

Make (a) the subject

$\mathbf{a = \frac{2}{101}}$

Substitute $\mathbf{a = \frac{2}{101}}$ in $\mathbf{P(x) = a(x - 18)(x^2 + 100)}$

$\mathbf{P(x) = \frac{2}{101}(x - 18)(x^2 + 100)}$

Expand

$\mathbf{P(x) = \frac{2}{101}(x^3 - 18x^2 +100x - 1800)}$

Open bracket

$\mathbf{P(x) = \frac{2}{101}x^3 - \frac{36}{101}x^2 +\frac{200}{101}x - \frac{3600}{101}}$

Hence, the polynomial in standard form is: $\mathbf{P(x) = \frac{2}{101}x^3 - \frac{36}{101}x^2 +\frac{200}{101}x - \frac{3600}{101}}$

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Answer from: jfjhbc

Given:

Polynomial of degree 3 that has zeros of 18, 10i, and −10i, and has a value of −34 when x=1.

To find:

The polynomial in standard form.

Solution:

If c is a zero of a polynomial then (x-a) is a factor of that polynomial.

18, 10i, and −10i are zeroes, so (x-18), (x-10i) and (x+10i) respectively are the factors of given polynomial.

So, the three degree polynomial is

P(x)=a(x-18)(x-10i)(x+10i)

where, a is a constant.

P(x)=a(x-18)(x^2-(10i)^2)

P(x)=a(x-18)(x^2-100i^2)

P(x)=a(x-18)(x^2-100(-1)) $[\because i^2=-1]$

P(x)=a(x-18)(x^2+100)

P(x)=a(x^3-18x^2+100x-1800) ...(i)

The value of P(x) is -34 at x=1.

-34=a((1)^3-18(1)^2+100(1)-1800)

-34=a(1-18+100-1800)

-34=1717a

Divide both sides by 1717.

$\dfrac{-34}{1717}=a$

$\dfrac{-2}{101}=a$

Put this value in (i).

$P(x)=\dfrac{-2}{101}(x^3-18x^2+100x-1800)$

$P(x)=\dfrac{-2}{101}x^3+\dfrac{36}{101}x^2-\dfrac{200}{101}x+\dfrac{3600}{101}$

Therefore, the required polynomial is $P(x)=\dfrac{-2}{101}x^3+\dfrac{36}{101}x^2-\dfrac{200}{101}x+\dfrac{3600}{101}$ .

Answer from: Quest

answer: if you weigh 60 pounds on earth, on the moon you will weigh about 10 pounds. if you weigh 120 pounds on earth, you will weigh about 20 pounds on the moon. since weight is caused by the mass of two objects that are near each other, the bigger the objects, the more the force of gravity.

step-by-step explanation: if you weigh 60 pounds on earth, on the moon you will weigh about 10 pounds. if you weigh 120 pounds on earth, you will weigh about 20 pounds on the moon. since weight is caused by the mass of two objects that are near each other, the bigger the objects, the more the force of gravity.

Answer from: Quest

.632

step-by-step explanation:

6 * 1/10 = 3/5

3*0.01 = 0.03

3/5 + 0.03 = .63

2*0.002 = .002

.63 + 0.002 = .632

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