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lubasha [3.4K]
3 years ago
5

Part A: Create a third-degree polynomial in standard form. How do you know it is in standard form? (5 points)

Mathematics
1 answer:
Svetlanka [38]3 years ago
4 0

Answer:

(See explanation for further details)

Step-by-step explanation:

a) Let consider the polynomial p(x) = 5\cdot x^{3} +2\cdot x^{2} - 6 \cdot x +17. The polynomial is in standards when has the form p(x) = \Sigma \limit_{i=0}^{n} \,a_{i}\cdot x^{i}, where n is the order of the polynomial. The example has the following information:

n = 3, a_{0} = 17, a_{1} = -6, a_{2} = 2, a_{3} = 5.

b) The closure property means that polynomials must be closed with respect to addition and multiplication, which is demonstrated hereafter:

Closure with respect to addition:

Let consider polynomials p_{1} and p_{2} such that:

p_{1} = \Sigma \limits_{i=0}^{m} \,a_{i}\cdot x^{i} and p_{2} = \Sigma \limits_{i=0}^{n}\,b_{i}\cdot x^{i}, where m \geq n

p_{1}+p_{2} = \Sigma \limits_{i=0}^{n}\,(a_{i}+b_{i})\cdot x^{i} + \Sigma_{i=n+1}^{m}\,a_{i} \cdot x^{i}

Hence, polynomials are closed with respect to addition.

Closure with respect to multiplication:

Let be p_{1} a polynomial such that:

p_{1} = \Sigma \limits_{i=0}^{m} \,a_{i}\cdot x^{i}

And \alpha an scalar. If the polynomial is multiplied by the scalar number, then:

\alpha \cdot p_{1} = \alpha \cdot \Sigma \limits_{i = 0}^{m}\,a_{i}\cdot x^{i}

Lastly, the following expression is constructed by distributive property:

\alpha \cdot p_{1} = \Sigma \limits_{i=0}^{m}\,(\alpha\cdot a_{i})\cdot x^{i}

Hence, polynomials are closed with respect to multiplication.

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Alex_Xolod [135]

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Step-by-step explanation:

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3 years ago
Joe earns $11.50/hr for the first 40 hours in the week, time and a half for the next 8 hours band double time for any additional
algol13

Answer:

$690

Step-by-step explanation:

To know how much money Joe earned, we need to split his 52 hours of work in three parts:

40 hours earning 11.5 per hour

8 hours earning 17.25 per hour (one time and a half of his regular payment)

The last 4 hours earning 23 per hours (double time earnings)

So, to know the total earning, we multiply each block of hours for its respective value per hour, and them sum them all:

Total earning = 40*11.5 + 8*17.25 + 4*23 = $690

7 0
3 years ago
Please ASAP
topjm [15]

Answer:

  (a) x +5y = 22

  (b) p = 11, q = 5

Step-by-step explanation:

<h3>(a)</h3>

The derivative of a function tells you the slope of its curve at every point. Then the slope of C at x=2 is ...

  dy/dx = 2³ +2(2) -7 = 5

The normal to the curve at the point of interest will have a slope that is the opposite reciprocal of this: -1/5. Then the point-slope equation of the normal line can be written as ...

  y -k = m(x -h) . . . . line with slope m through point (h, k)

  y -4 = -1/5(x -2) . . . . line with slope -1/5 through point (2, 4)

  5y -20 = -x +2 . . . . multiply by 5

  x +5y = 22 . . . . . . add x+20 to put in standard form

The graph shows curve C and the desired normal line.

__

<h3>(b)</h3>

The power rule for derivatives tells you ...

  (d/dx)(a·x^n) = a·n·x^(n-1)

This relation gives you two ways to find the values of p and q.

  a) using the exponent of the term

  b) using the coefficient of the term

Either way, the values are ...

  p = 10 +1 = 11

  q = 4 +1 = 10/2 = 5

6 0
2 years ago
James calculated the height of a cylinder that has a volume of 324 pi cubic inches and a radius of 12 in his work is shown below
wel

Answer:

13.5cm

Step-by-step explanation:

V=Bn

Step 1: 3245 - 12h

Step 2: 324 = 24 sh

324 24

Step 3: 245 245

Step 4: h-13.5

inches

What is the first error that James made when calculating the height of the cylinder?

In step 1, he substituted into the volume formula incorrectly.

In step 2, he calculated 122 incorrectly. It should be 144 rather than 24.

In step 4, the should have canceled, making the correct answer 13.5 cm.

James calculated the height of the cylinder correctly

7 0
3 years ago
Read 2 more answers
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