Locating Zeros of Polynomial Function:
Approximate the real zeros of f(x)=x^3-4x+6 to the nearest tenth

Answer:

23

Step-by-step explanation:

The notation (f∘g)(1) means f(g(1)). We can first find g(1), then use that value as the argument to the f function.

g(1) = 2·1-6 = -4

f(-4) = -4(-4)+7 = 16+7 = 23

Line segment of length k is divided into 3 equal parts.

so first segment is 0-k/3 and third segment is 2/3k-k

so mid-pt of 1st = k/6 and 3rd = 5/6k

so the distance in between = 5/6k-k/6 = 4/6k = 2/3k

ans is A

The distance between the midpoints of the first segment and the third segment is 2k/3. Hence, option A is the right choice.

The mid-point of a line segment is the point from which the distance to both ends of the line segment is equal.

In the question, we are given a line segment of length k units, which is divided into 3 equal parts.

We are asked to find the distance between the midpoints of the first and third segments.

Firstly, we divide the line segment at points k/3 and 2k/3, to get three equal parts of lengths k/3 each.

Now, the mid-point of the first segment = (0 + k/3)/2 = k/6.

The mid-point of the third segment = (2k/3 + k)/2 = 5k/6

Therefore, the distance between the midpoints of the first segment and the third segment is (5k/6 - k/6) = 4k/6 = 2k/3. Hence, option A is the right choice.

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View the image to see how I solved this problem.

There is an obvious factor of x², so we can reduce this to the problem of finding zeros of a quadratic by factoring that out. Then, we can determine the zeros of the quadratic factor to complete the finding of zeros for the function.

A factor of x² can be factored out, leaving ...

... f(x) = x²(x² +x +4)

The latter factor has only complex zeros. It can be rewritten by completing the square.

... x² +x +4 = (x² +x +1/4) +(3 3/4) = (x +1/2)² +(3 3/4)

So, the real zeros are where x² = 0, at x = 0. The complex zeros are where the above expression is zero, ...

... (x +0.5)² +3.75 = 0

... (x +0.5)² = -3.75

... x + 0.5 = √-3.75 = ±0.5i√15

... x = -0.5(1 ±i√15)

The function zeros are 0 (multiplicity 2) and -0.5±0.5i√15.

28/33 × 100% = 0.84848... × 100% ≈ 85%

Michael got about 85% of the available points.

_____

Since we're not told the number of points per question, we don't actually have enough information to answer the question.

To find out the percentage of correct answers, divide the number of correct answers by the total number of questions and multiply by 100. Michael got 85% of the questions correct on his math test.

To find out what percent of the questions Michael answered correctly, we need to divide the number of correct answers by the total number of questions and multiply by 100.

Here's the step-by-step calculation:

So, Michael got 85% of the questions correct on his math test.

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The surface area of an open cylinder can be expressed as 2πrh + πr^2. The cost of building the open cylinder can be calculated using the surface area and the cost per square foot. If the volume of the cylindrical container is known, the cost of building the cylinder can be expressed in terms of the cylinder's radius.

(a) The surface area of an open cylinder consists of the sum of the lateral surface area and the area of the bottom. The lateral surface area is given by the formula 2πrh, and the area of the bottom is πr^2. Therefore, the surface area of the open cylinder, S, can be expressed as:

S = 2πrh + πr^2

(b) The cost of building the open cylinder can be calculated by multiplying the surface area by the cost per square foot. Let C be the cost of building the cylinder:

C = 6(2πrh + πr^2) + 9πr^2

(c) If the volume of the cylindrical container is 500 cubic feet, we can express the cost of building the cylinder, C, in terms of the cylinder's radius, r. We already know that the volume of a cylinder is given by the formula V = πr^2h, so we can solve for h and substitute it into the equation for C:

C(r) = 6(2πr(V/πr^2) + πr^2) + 9πr^2

(d) If the volume of the cylindrical container is V cubic feet, we can express the cost of building the cylinder, C, in terms of the cylinder's radius, r:

C(r) = 6(2πr(V/πr^2) + πr^2) + 9πr^2

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Surface area S of the open cylinder is 2πrh + πr². The cost C to build the cylinder is 12πrh + 9πr². Given the volume V, cost C(r) can be expressed as 12V/r + 9πr².

Building a Cylindrical Container without a Top

The problem involves calculating the surface area and cost of a cylindrical container with no top. Here are the detailed steps:

(a) Surface Area Calculation

The surface area S of a cylindrical container with radius r and height h (with no top) includes:

The lateral surface area: 2πrh

The bottom area: πr²

Thus, the total surface area S is given by:

S = 2πrh + πr²

(b) Cost Calculation

The cost C to build the cylindrical container is calculated as:

Cost of lateral sides: $6 per square foot

Cost of the bottom: $9 per square foot

Therefore, the cost C is:

C = 6(2πrh) + 9(πr²) = 12πrh + 9πr²

(c) Cost in Terms of Radius r (Volume = 500 cubic feet)

Given the volume V = 500 cubic feet, we have:

Volume V = πr²h = 500

Rearranging for h, we get:

h = 500 / (πr²)

Now, substitute h in the cost equation:

C(r) = 12πr(500 / πr²) + 9πr² = 6000/r + 9πr²

(d) Cost in Terms of Radius r (Volume = V cubic feet)

For a general volume V, we have:

Volume V = πr²h = V

Rearranging for h, we get:

h = V / (πr²)

Now, substitute h in the cost equation:

C(r) = 12πr(V / πr²) + 9πr² = 12V/r + 9πr²

Final answer:

The function y=2x+13 takes on a positive value for x > -6.5 and a negative value for x < -6.5.

Explanation:

The function y=2x+13 represents a linear equation. To find the values of x for which the function takes on a positive value, we need to solve the inequality 2x+13 > 0.

Subtracting 13 from both sides gives 2x > -13.

Dividing both sides by 2, we get x > -6.5.

Therefore, for x values greater than -6.5, the function will be positive.

To find the values of x for which the function takes on a negative value, we need to solve the inequality 2x+13 < 0. Subtracting 13 from both sides gives 2x < -13.

Dividing both sides by 2 and flipping the inequality sign, we get x < -6.5.

Therefore, for x values less than -6.5, the function will be negative.

The two points on the graph of the function - 7y + 8 = 21x - 6 are;

⇒ (0, 2) and (2, -4)

What is Coordinates?

A pair of numbers which describe the exact position of a point on a cartesian plane by using the horizontal and vertical lines is called the coordinates.

Given that;

The function is,

⇒ - 7y + 8 = 21x - 6

Now,

We can draw the graph of the function - 7y + 8 = 21x - 6 as shown in figure.

And, The points on the graph are;

⇒ (0, 2) and (2, -4)

Thus, The two points on the graph of the function - 7y + 8 = 21x - 6 are;

⇒ (0, 2) and (2, -4)

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Answer:

2

Step-by-step explanation:

DC = [tex]\frac{5}{2}[/tex] = 2.5

since the triangles are similar then the ratios of corresponding sides are equal

[tex]\frac{AC}{AD}[/tex] = [tex]\frac{BC}{DE}[/tex]

[tex]\frac{7.5+DC}{7.5}[/tex] = [tex]\frac{8}{6}[/tex] (cross-multiply )

45 + 6DC = 60

6DC = 60 - 45 = 15 ( divide both sides by 6 )

DC = [tex]\frac{5}{2}[/tex]

The length of fence she will need to divide the garden is 9.22 m.

Rectangle is a quadrilateral (has four sides and four angles) in which both sides are parallel and opposite to each other.

Pythagoras theorem is used to show the relationship between the sides of a right angled triangle.

From the diagram, let x represent the length of the fence, hence using Pythagoras:

The length of fence she will need to divide the garden is 9.22 m.

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In y years, the loss is ...

... loss = 15y

Then in 6 years, the loss will be ...

... loss = 15·6 = 90

The loss of students for those 6 years will be 90 students.

All the numbers shown are real.

The answers to problems 4, 7, 11 are irrational, because they are square roots of numbers that are not perfect squares.

The remaining numbers are rational because they are terminating decimals or the square root of a square number.

Answer:

x can take any value less than 3.

Step-by-step explanation:

This is an inequality.

[tex]2x < 6[/tex]

Lets solve the inequality for x

[tex]x < 3[/tex]

This means x can take any value less than 3.

We can show this by using a number line.

Notice how number 3 is not fully shaded, this means x cannot be 3. And all the numbers less than 3 to infinity is shaded, meaning x can take any value in that range.

The relation will not be a function if the to ordered pairs have the same first value. That will happen when

... 2k+3 = 3k-1

... 4 = k . . . . . . add 1-2k

The relation will not be a function when k=4.

Tony because he spent $19 while Mario spent $18. To explain, you would multiply 3 by $4 because Tony bought 3 of the $4 packs of pencils. Therefore, he would have spent $12. In addition, add $7 to $12, you get $19. On the other hand, Mario bout 4 of the $6 binders. That would mean he spent $24 on binders However, he had a $6 coupon which means he can take $6 off the total price of $24. That would mean subtracting $6 from $24. That would get Mario $18

Answer:

Your answer is 16√3 units2

Step-by-step explanation:

Consider the ordering

... -2 < -1

Now consider the ordering of their absolute values:

... 1 < 2

_____

Hopefully, you see that changing the sign reflects the sequence across the origin, so that the ordering is reversed when the signs are changed.

There are a variety of methods of performing this subtraction, and corresponding different methods of regrouping.

If you're taught (as I was) to do the subtraction right-to-left, then the first regrouping you need to do is when you try to subtract 500 from 100. You must regroup the 8 thousands and 1 hundred to 7 thousands and 11 hundreds. Then, when you subtract 5 hundreds, you end with 7 thousands and 6 hundreds.

The next regrouping you need to do is when you try to subtract 6 ten-thousands from 5 ten-thousands. You must regroup the 2 hundred-thousands and 5 ten-thousands to 1 hundred-thousand and 15 ten-thousands. Then, when you subtract 6 ten-thousands, you end with 1 hundred-thousand and 9 ten-thousands.

The end result is

... 258,197 - 64,500 = 193,697

_____

If you use an abacus or soroban or similar tool to help you keep track of the numbers, you were likely taught to do the subtraction left-to-right. In this case, the first regrouping  comes when you want to subtract 6 ten-thousands. Practitioners of this method know that -6 = -10 +4, so the number represented on the tool becomes (2-1) hundred-thousands and (5+4) ten-thousands plus the rest of the initial number, or 198,197 after subtracting the 6 ten-thousands.

The subtraction proceeds until you find you need to subtract 500 from 100. At this point, the tool is representing the partial result as 194,197. Again, if you practice this method, you know that -5 = -10 +5, so you reduce the thousands digit by 1 (to 3) and add 5 to the hundreds digit to get 193,697.

_____

An attempt is made to show the regroupings in the attachment. In each case there are two of them. However, working left-to-right, the result of the first subtraction of 6 ten-thousands is 19 ten-thousands, so you never actually write down anything else. Of course, if you're using an abacus or soroban, you don't write down anything—you simply change the position of the beads on the tool.

7/3 = 2 1/3.  Rounded to the nearest hundredth, we get 2.33.

you can use a calculator and plug n 989x49 which equals 48461

989 times 49 is 48461

Hope this helps :)

If you add the two purchases, you find 9 hotdogs and 9 hamburgers cost 24.75. One hotdog and one hamburger will cost 1/9 of that total, or 2.75.

... One hotdog and one hamburger costs $2.75.

_____

This is what your question asks. Usually, we want to know the individual costs of the items. If we multiply this total by 5, then we have the cost of 5 hotdogs and 5 hamburgers: 13.75. This $1.75 more than the cost of 5 hotdogs and 4 hamburgers, so the cost of a hamburger must be $1.75

... One hotdog costs $1.00

... One hamburger costs $1.75

depth = 3.3 cm

the volume (V ) of a cuboid is

V = Ah ( A is the area and h the depth )

A = 20 × 28 = 560 cm², thus

h = [tex]\frac{V}{A}[/tex] = [tex]\frac{1843}{560}[/tex] ≈ 3.3 cm

D

to divide 2 fractions

• leave the first fraction

• change division to multiplication

• turn the second fraction upside down

• simplify if possible

= - [tex]\frac{3}{8}[/tex] × - [tex]\frac{7}{6}[/tex] = [tex]\frac{3(7)}{8(6)}[/tex] = [tex]\frac{7}{16}[/tex]

A plot of the points shows you the triangle is an acute triangle.

_____

All angles are less than 90°.

x = 5 or x = - 1

equate f(x) and g(x)

x² - 7x + 3 = - 3x + 8 ( subtract (- 3x + 8 ) from both sides )

x² - 4x - 5 = 0 ← in standard form

(x - 5)(x + 1) = 0

equate each factor to zero and solve for x

x - 5 = 0 ⇒ x = 5

x + 1 = 0 ⇒ x = - 1

Note that the graph shows 8 dots with associated x-values 1 through 8.  The domain is the set of integers {1, 2, 3, 4, 5, 6, 7, 8}.  The range is the set of all y-values associated with this domain, a bit harder to read off the graph.  One by one, I will approximate the y-value associated with each x value and present these 8 y-values as the range:  {7.5, 9, 12, 15. 18, 22, 23}.

E is located at (-2,-4)

F is located at (2,2)

To find the midpoint, add the 2 X values together and divide by two, then do the same with the y values:

X = -2 + 2 = 0/2 = 0

Y = -4 + 2 = -2/2 = -1

The midpoint is located at (0,-1)

340 miles because you do 123-21 divided by 0.30.

I'm so smart!! :D

The only answer choice involving powers of prime numbers is ...

... 2^8