An item is regularly priced at $60. Jane bought it at a discount of 30% off the regular price. How much did Jane pay?
$
X
5

Answer:

[tex]f(x)=x^3-12x^2+37x+130[/tex]

Step-by-step explanation:

So I'm assuming when you provided n=3, that means the degree is 3? So the first thing to know, is you can express a polynomial using it's factors as: [tex]f(x)=a(x\pm b)(x\pm b)(x\pm c)[/tex] where the sign of each factor depends on the sign of the factor... the point is it can be either. Notice the a? This usually will determine the stretch/compression of the polynomial, since sometimes the factors will have a coefficient for the x, but in this case I'm assuming all the coefficients of x are 1. So the next thing that is vital to know is that imaginary solutions come in conjugates. This means that if you have a zero at: [tex]a-bi \text{ then }a+bi \text{ is also a zero}[/tex]. So this means you have the 3 zeroes at x=-2, x=7+4i, x=7-4i. These are all the zeroes of the polynomial, since the Fundamental Theorem of Algebra states that a polynomial with degree n will have n solutions, which can be complex or real.

So the factor isn't going to be written as (x-2), it's going to be (x+2) since when you plug in -2 as x, it makes x+2 equal to 0.

The same thing applies for the two imaginary factors, so you're going to have the other two factors as (x-(7+4i)) and (x-(7-4i). You can instead think of it as ((x-7)+4i) and ((x-7)-4i) and you can use the difference of squares identity: [tex](a-b)(a+b)=a^2-b^2[/tex] where in this case a=x-7 and b=4i

So this gives you the equation: [tex]((x-7)-4i)((x-7)+4i) = (x-7)^2-(4i)^2[/tex]

Which becomes: [tex](x^2-14x+49)-16(-1) = x^2-14x+49+16 = x^2-14x+65[/tex]

So this gives us one of the factors: [tex]x^2-14x+65[/tex]

Now plug this in with the other factor and we get the equation:

[tex]f(x) = a(x+2)(x^2-14x+65)[/tex]

Now plug in 1 as x and make f(x) = 156, to solve for a

[tex]156=a(1+2)(1^2-14(1)+65)\\156=a(3)(1-14+65)\\156=a(3)(52)\\156=152a\\a=1[/tex]

So in this case, a=1, so that last step wasn't necessary, although I would check each time just in case a =/= 1.

Original equation

[tex]f(x) = (x+2)(x^2-14x+65)[/tex]

Multiply

[tex]f(x)=(x^3-14x^2+65x)+(2x^2-28x+130)\\[/tex]

Combine like terms:
[tex]f(x)=x^3+(-14x^2+2x^2)+(65x-28x)+130[/tex]

add like terms:

[tex]f(x)=x^3-12x^2+37x+130[/tex]

Answer:

[tex]f(x)=x^3-12x^2+37x+130[/tex]

Step-by-step explanation:

Complex Conjugate Theorem

For a polynomial p(x) with real coefficients, the complex zeros occur in conjugate pairs.  So if (a + bi) is a zero, then its conjugate (a - bi) is also a zero.

Given zeros:

-2 and (7 + 4i)

As one of the given zeros is a complex number, (7 - 4i) is also a zero.

Write each zero as part of a factor and multiply them together, adding a leading coefficient [tex]a[/tex]:

[tex]\begin{aligned}f(x) & = a(x+2)(x-(7+4i))(x-(7-4i))\\& = a(x+2)(x-7-4i)(x-7+4i)\\& = a(x+2)(x^2-7x+4ix-7x+49-28i-4ix+28i-16i^2)\\& = a(x+2)(x^2-14x+49-16i^2)\end{aligned}[/tex]

Remember that [tex]i^2=-1[/tex], therefore:

[tex]\begin{aligned}f(x) & = a(x+2)(x^2-14x+49-16(-1))\\& = a(x+2)(x^2-14x+49+16)\\& = a(x+2)(x^2-14x+65)\end{aligned}[/tex]

To find the value of the leading coefficient (a), use the given [tex]f(1)=156[/tex] :

[tex]\begin{aligned}f(1) & = 156\\\implies a(1+2)(1^2-14(1)+65) & = 156\\a(3)(52) & = 156\\156a & = 156\\\implies a & = 1\end{aligned}[/tex]

Therefore, the polynomial in factored form is:

[tex]f(x)=(x+2)(x^2-14x+65)[/tex]

Finally, expand the brackets:

[tex]\begin{aligned}f(x) & =(x+2)(x^2-14x+65)\\& = x^3-14x^2+65x+2x^2-28x+130\\& = x^3-14x^2+2x^2+65x-28x+130\\& = x^3-12x^2+37x+130\end{aligned}[/tex]

Learn more about complex numbers here:

https://brainly.com/question/26344541

https://brainly.com/question/28032896

The answer choice which must be true regarding the linear function is; The set must have a constant additive rate of change.

Since a linear function typically takes the slope-intercept form; f(x) = mx +c.

It therefore follows that the equation must have a constant slope m, which is the described additive constant rate of change.

It therefore follows that, the answer choice which is true regarding the linear function is therefore; The set must have a constant additive rate of change.

Read more on rate of change;

https://brainly.com/question/1509153

#SPJ1

The ordered pairs which is the solution of the system of Inequalities is;

x > 0 and y < 2

We are given the inequalities;

x + 2y > 4    ----(1)

3x - y < 2     -----(2)

Subtract 2y from both sides in eq 1 to get;

x > 4 - 2y   ----(3)

Put 4 - 2y for x in eq 2 to get;

3(4 - 2y) - y < 2

12 - 4y - y < 2

12 - 5y < 2

Subtract 12 from both sides to get;

-5y < -10

Divide both sides by -5 to get;

y < 2

Put 2 for y in eq 3 to get;

x > 4 - 2(2)

x > 0

Read more about Inequality Ordered Pairs at; https://brainly.com/question/1528681

#SPJ1

Step-by-step explanation:

the slope of a line is always the factor of x, when the line is in the form

y = ...

so,

3x + 3y = 18

3y = -3x + 18

y = -x + 6

the slope is -1 = -1/1

as the slope is the ratio of y coordinate change / x coordinate change.

a perpendicular slope switches x and y upside-down and flips the sign.

so, in our case the perpendicular slope is

1/1 = 1

Volume is a three-dimensional scalar quantity. The volume of concrete that will have to be poured into the pool deck is 6.0185 cubic yards.

A volume is a scalar number that expresses the amount of three-dimensional space enclosed by a closed surface.

Since an inch is equal to 1/12 of a foot. Therefore, the thickness of the pool deck is,

3 inches = 0.25 foot

If a pool deck of 650 square feet is to be laid with concrete 3" thick, then the volume of concrete that will be needed is,

Volume of concrete = 650 ft² × 0.25 ft = 162.5 ft³

Now, one yard is equal to 3 feet, therefore,

1 cubic foot = 1/27 cubic yards

162.5 cubic feet = 6.0185 cubic yards

Hence, the volume of concrete that will have to be poured into the pool deck is 6.0185 cubic yards.

Learn more about Volume:

https://brainly.com/question/13338592

#SPJ1

[tex]\huge\text{Hey there!}[/tex]

[tex]\huge\textbf{Equation:}[/tex]

[tex]\mathbf{7 \dfrac{2}{3} + \dfrac{4}{7}}[/tex]

[tex]\huge\textbf{Solving:}[/tex]

[tex]\mathbf{7 \dfrac{2}{3} + \dfrac{4}{7}}[/tex]

[tex]\mathbf{= \dfrac{7\times3+2}{3} + \dfrac{4}{7}}[/tex]

[tex]\mathbf{= \dfrac{21 + 2}{3} + \dfrac{4}{7}}[/tex]

[tex]\mathbf{= \dfrac{23}{3} + \dfrac{4}{7}}[/tex]

[tex]\mathbf{= \dfrac{173}{21}}[/tex]

[tex]\mathbf{\approx 8 \dfrac{5}{21}}[/tex]

[tex]\huge\textbf{Answer:}[/tex]

[tex]\huge\boxed{\mathsf{8 \dfrac{5}{21}}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

~[tex]\frak{Amphitrite1040:)}[/tex]

Answer:

[tex]8\frac{5}{21}[/tex]

Step-by-step explanation:

[tex]7\frac{2}{3} +\frac{4}{7} = 7\frac{14}{21} +\frac{12}{21} = 8\frac{5}{21}[/tex]

The equation of the line that passes through (-10,-7) and (-5,-9) is y=-2.5x-31.

A line is a one-dimensional shape that is straight, has no thickness, and extends in both directions indefinitely. The equation of a line is given by,

y =mx + c

where,

x is the coordinate of the x-axis,

y is the coordinate of the y-axis,

m is the slope of the line, and

c is the y-intercept.

The slope of the line is,

Slope, m = (-10+5)/(-7+9) = -5/2 = -2.5

Substitute any one of the points in the equation of the line,

-7 = -2.5(-10) + C

-7 = 25 + C

C = -31

Hence, the equation of the line that passes through (-10,-7) and (-5,-9) is y=-2.5x-31.

Learn more about Equation of Line:

https://brainly.com/question/21511618

#SPJ1

Answer: the answer is y = - 2/5x - 11

Step-by-step explanation:

it’s correct on khan academy

Considering the slopes of the given lines, they are parallel lines.

The slope, given by change in y divided by change in x, determines if the lines are parallel, perpendicular, or neither, as follows:

The first line, in standard form, is given by:

5y = x + 5

y = 0.2x + 1.

The slope is of m = 0.2.

For the second line, we have that:

25y = 5x - 75

y = 0.2x - 3

The slope is of m = 0.2.

Same slope, hence the lines are parallel.

More can be learned about the slope of a line at brainly.com/question/12207360

#SPJ1

Answer:

I'm not too sure but:

L = (-2,-3)

M = (-9,-3)

K = (-2,-8)

J = (-9,-8)

Step-by-step explanation:

What we know:

We know that we go through the x axis then the y axis hence the saying "Along the corridor, Up the stairs".

Due to us knowing this we use where the letters (L, M, K & J) are placed on the diagram and use the "Along the corridor, Up the stairs" technique to figure out the answer.

This means that L is (-2,-3), M is (-9,-3), K is (-2,-8) & J is (-9,-8).

Yet again as I said at the top i'm not too sure but this is what I think is correct.

I hope it helps! :)

Answer:

11

Step-by-step explanation:

since 30÷45 = 2/3 then

The space taken by a small book is equal to

the space taken by 2/3 a large book.

…………………………………………………

Calculating the number of large book corresponding to 23 small books :

= 23×(2÷3) + 3

= 18.333333333333 (large book)

Then ,Kevin can pack the shelf with (at most) : 30 - 19 = 11 more large books

The coordinates of the mid-point between points B and C is given by:

(-5, -10).

The mid-point between two points is the halfway point between them, and is found using the mean of the coordinates.

The points in this problem are given as follows:

B(-40, -30), C(30, 10)

Hence the coordinates of the mid-point are:

More can be learned about the mid-point at https://brainly.com/question/10956693

#SPJ1

the probability of rolling 4 or a number less than 6 is P = 5/6

The standard number cube has the outcomes {1, 2, 3, 4, 5, 6}

We want to find the probability of rolling 4 or a number less than 6. (4 is a number less than 6).

The outcomes that meet that condition are:

{1, 2, 3, 4, 5}

So 5 out of 6 outcomes meet the condition, then the probability is:

P(4 or less than 6) = 5/6

If you want to learn more about probability:

https://brainly.com/question/25870256

#SPJ1

Answer:

Option B

Step-by-step explanation:

Slope

And

These are acceleration

Y intercept is (0,2) as x coordinate is 0

This is initial velocity 2m/s

The runner stopped at 8s

Answer:

[tex]\frac{1}{2}[/tex]

Step-by-step explanation:

plug 1 in for t

[tex]F(t)=4(\frac{1}{2^3^t})\\ F(1)=4(\frac{1}{2^3^*^1})\\ F(1)=4(\frac{1}{2^3})\\ F(1)=4(\frac{1}{8})\\ F(1)=\frac{4}{8} \\F(1)=\frac{1}{2}[/tex]

The measure of TV from the figure is 38 units

The given diagram in question is a kite. From the figure, the expression below is true;

6x - 3 = 39

Determine x

6x = 42

x= 7

TV = 2(2x+5)

TV = 2(2(7)+5)

TV = 2(19)

TV = 38 units

Hence the measure of TV from the figure is 38 units

Learn more on kite here: https://brainly.com/question/23279609

#SPJ1

Answer:

180

Step-by-step explanation:

Let x be the first integer.

Let y be the 2nd integer.

Given info from the question,

x = 5% x y

x = 0.05y - equation 1

x + y = 63 - equation 2

Lets now substitute equation 1 into equation 2.

0.05y + y = 63

1.05y = 63

y = 63 / 1.05

= 60

Now we substitute y into equation 1.

x = 0.05(60)

= 3

Now since we know both x and y,

we can find the product of them.

xy = 60 * 3 = 180

[tex]\Large\maltese\underline{\textsf{A. \space What is Asked}}[/tex]

Find the slope of the line that contains the points (-8,-2) and (-17,4)

[tex]\Large\maltese\underline{\textsf{B.\space This problem has been solved!}}[/tex]

If we are given two points, we can easily find the slope of the line that passes through the points. With points (y₂, y₁) and (x₂, x₁), the slope is [tex]\bf{\dfrac{y2-y1}{x2-x1}[/tex].

Similarly, with points (-8,-2) and (-17,4) the slope is.

[tex]\bf{\dfrac{4-(-2)}{-17-(-8)}}[/tex] | as we can see, it is possible to simplify this further

[tex]\bf{\dfrac{4+2}{-17+8}}[/tex] | simplify

[tex]\bf{\dfrac{6}{-9}[/tex] | reduce

[tex]\bf{-\dfrac{2}{3}}[/tex]

[tex]\rule{300}{1.7}[/tex]

[tex]\bf{Result:}[/tex]

                    [tex]\bf{=-\dfrac{2}{3}}[/tex]

[tex]\boxed{\bf{aesthetic\not101}}[/tex]

Using the equation of slope of a line, the slope of the line that have the given coordinates is -2/3

The slope of a line represents the measure of how steep the line is, or the rate at which the line rises or falls as you move along it. It is calculated as the ratio of the vertical change (change in y-coordinates) to the horizontal change (change in x-coordinates) between any two points on the line.

If you have two points on a line, (x₁, y₁) and (x₂, y₂), the slope (m) can be calculated using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Substituting the values into the formula above;

m = 4 - (-2) / -17 - (-8)

m = 4 +  2 / -17 + 8

m = 6 / -9

m = -2/3

Learn more on slope of a line here;

https://brainly.com/question/16949303

#SPJ2

Answer:

f(-2) = 3, f(0, -3) = -1

Answer:

1.39b + 2.49m = 50.73

1.29b + 2.69m = 51.63

Step-by-step explanation:

Systems of linear equations are used when we have 2 equations that both contain 2 unknown variables. Since we are trying to find the price of bread and milk, we can let b = num of bread and m = num of milk.

Betty spends $50.73 buying b loaves of bread and m gallons of milk, so the first equation would be 1.39b + 2.49m = 50.73. Betty spends $51.63 buying b loaves of bread and m gallons of milk at Store B, so the second equation is 1.29b + 2.69m = 51.63.

Answer:

  d.  =SQRT(ABS(A2))

Step-by-step explanation:

A spreadsheet will not tell you the square root (or half power) of a negative number. It only performs math using real numbers.

Trying to take a square root of a negative number will result in a #NUM! error being shown in the result cell. Assuming you want the root of the magnitude of the numbers in column A, the value you want the root of is ABS(A2).

Using a dollar sign on a cell reference renders that portion of the cell reference fixed when the formula is copied. Referencing cell A$2 will mean you will get 399 copies of the square root of the contents of cell A2 in cells B2 through B400.

The formula that will give you the square root of the magnitude of the adjacent cell in column A will be

  =SQRT(ABS(A2)) . . . . . . . . . formula for cell B2, copied to cells below

An ordinal number is a number that specifies how something ranks in respect to other numbers. The correct option is B.

An ordinal number is a number that specifies how something ranks in respect to other numbers, such as first, second, third, and so on. This order or sequence might be determined by size, importance, or chronology.

The example that illustrates the ordinal number is "Gracie told Bradly she would be the first one to go down the slide".

Hence, the correct option is B.

Learn more about Ordinal Numbers:

https://brainly.com/question/14327859

#SPJ1

Since the grade of the numerator and the denominator is the same, then the limit exists and is distinct from 0. The limit of the expression is 4/7.

In this problem we must apply some algebraic handling and some known limits to determine whether the limit exists or not. The limit exists if and only if the result exists.

[tex]\lim_{x \to \infty} \frac{4\cdot x - 1}{7\cdot x + 3}[/tex]

[tex]\lim_{x \to \infty} \frac{4\cdot x - 1}{7\cdot x + 3} \cdot \frac{x}{x}[/tex]

[tex]\lim_{x \to \infty} \frac{4 - \frac{1}{x} }{7 + \frac{3}{x} }[/tex]

[tex]\lim_{x \to \infty} \frac{4}{7}[/tex]

4/7

Since the grade of the numerator and the denominator is the same, then the limit exists and is distinct from 0. The limit of the expression is 4/7.

To learn more on limits: https://brainly.com/question/12207558

#SPJ1

The sequences are classified, respectively, as:

Geometric, Arithmetic, Neither.

A geometric sequence is a sequence in which the result of the division of consecutive terms is always the same, called common ratio q.

In the first sequence, we have that:

[tex]q = \frac{45}{15} = \frac{15}{5} = \frac{5}{\frac{5}{3}} = \frac{\frac{5}{3}}{\frac{5}{9}} = 3[/tex]

Hence it is a geometric sequence.

In an arithmetic sequence, the difference between consecutive terms is always the same, called common difference d.

In the second sequence, we have that:

d = -4 - 1 = 1 - 6 = 6 - 11 = 11 - 16 = -5

Hence it is an arithmetic sequence.

[tex]\frac{4}{3} \neq \frac{3}{2}, 1 - \frac{1}{2} \neq 2 - 1[/tex]

Hence it is neither arithmetic nor geometric.

More can be learned about geometric and arithmetic sequences at https://brainly.com/question/11847927

#SPJ1

Answer:

The histogram

The histogram is a popular graphing tool. It is used to summarize discrete or continuous data that are measured on an interval scale. It is often used to illustrate the major features of the distribution of the data in a convenient form.

Hope it helps :)

Answer:

D. Histograph can be used to display an interval of values from a data set

Answer:

23 years old

Step-by-step explanation:

Let x = age of younger sibling.

Then, the age of the older sibling is x + 9.

x + x + 9 = 55

2x = 46

x = 23

Answer: 23 years old

Answer:

The answer is

[tex]a \sin(theta) [/tex]

Step-by-step explanation:

The solution is in the image

Using the continuity concept, it is found that the function is continuous for all real values.

A function f(x) is continuous at x = a if it is defined at x = a, and:

[tex]\lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x) = f(a)[/tex]

In this problem, we are given a piece-wise function, hence we have to look at the points where the definition of the function changes. In this problem, it can only be discontinuous at x = 0, which we have to verify.

Then:

The 3 values are equal, hence the function is continuous at x = 0 and for all real values.

More can be learned about the continuity of a function at https://brainly.com/question/24637240

#SPJ1

Answer:

28

Step-by-step explanation:

Givens

Game 1: 30

Game 2: 18

Game 3: 28

Game 4: x

Solution

(30 + 18 + 28 + x) / 4 = 26                 Multiply both sides by 4

4* (30 + 18 + 28 + x) / 4 = 26 * 4       The 4s on the left cancel

30 + 18 + 28 + x = 104                       Add the 3 other numbers together

76 + x = 104                                       Subtract 76 from both sides.

76-76+x = 104-76                              The 76s on the left disappear.

x = 28

Answer

in the next game he must score 28 points to average 26