The conjugate of the given zero 2 - 4i is 2 + 4i, which must also be a zero of the polynomial f(x) due to the Complex Conjugate Root Theorem.
Given the zero 2 - 4i of the polynomial function f(x) = [tex]x^4 - 4x^3 + 21x^2 - 4x + 20,[/tex] we know by the Complex Conjugate Root Theorem that if a polynomial has real coefficients, then any complex zeros must occur in conjugate pairs. Therefore, the conjugate of 2 - 4i, which is 2 + 4i, must also be a zero of the polynomial function.
As a result, the polynomial can be factored to include the factors (x - (2 - 4i)) and (x - (2 + 4i)). When these factors are multiplied out, they will give you a quadratic factor of the polynomial, which when multiplied with the remaining factors, will result in the original polynomial f(x). The two factors, when multiplied together, yield a quadratic with real coefficients, which means any remaining zeros must also satisfy this quadratic equation.
solve the equation 14.50-p=53 please help
find the slope of the line through points (1,8) and (2,7). write the answer in simplest form.
Which graph represents the function of f(x) = 9x 2 -36/3x+6 Please help
The given function is a quadratic equation, not a linear one, and so it would be represented by a parabolic curve on a graph, not a straight line.
Explanation:The function you're asking about, f(x) = 9x² - 36/3x + 6, is a quadratic equation. Quadratic equations are represented by parabolas on the graph not a straight line. Parabolas usually have a shape like a U (or an upside down U, depending on the equation).
The equation you've given should result in a parabolic graph. The straight line equations and graphs you’re referring to, like y = 9+3x, would not represent your function. The line graph with a y-intercept of 9 and a slope of 3 is not a representation of your function.
Also, it's important to note that the equation you provided, f(x) = 9x² - 36/3x + 6, can be simplified to f(x) = 9x² - 12x + 6.
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what is the correct name for the angle shown
Answer: The correct name is (B) ∠S.
Step-by-step explanation: We are given to find the correct name for the angle shown in the figure.
We can see in the figure that the rays RS and ST are intersecting at the point S. Therefore, RS and ST are making an angle at the point 'S'.
So, the name of the angle will be ∠S or ∠RST or ∠TSR.
First option is ∠R. This is not correct.
Second option is ∠S. This is the correct option.
Third option is ∠RTS. This is also not correct.
Fourth option is ∠SRT. This is not correct.
Thus, the correct option is (B) ∠S.
How many ways are there to arrange the first five letters of the alphabet?
Answer:
120 ways
Explanation:
Since there are no repeating letters, and there are 5 total letters, there are 5!= 120 ways to arrange them. In other words, there are 5 slots to place the first letter in, then 4 slots for the second letter, 3 for the third, 2 for the fourth, and then the last letter goes in the 1 slot that is left.
I hope this is correct and if it is I hope it helps. I hope you have a great day and remember it doesn't matter what other people say or think about you, you are still and always will be strong, smart, and an amazing person. Have a wonderful day.
I am happy to help with anything.
John received $0.76 change from a purchase in the drugstore. if he received eight coins, and five of the coins are the same denomination, how many quarters did he receive?
5 nickels , 2 quarters, and 1 pennies = 8 coins.
he received 2 quarters
Write the equation of a hyperbola with vertices (0, -4) and (0, 4) and foci (0, -5) and (0, 5).
Find the volume of a cylinder with a diameter of 8 inches and a height that is three times the radius round to the nearest hundredth
volume for cylinder = pi x r^2 x h
radius = half the diameter = 8/2 =4
height = 3 times the radius = 3*4 =12
using 3.14 for pi
volume = 3.14 x 4^2 x 12 = 602.88 cubic inches
What is the sum of the measures of the exterior angles of any convex polygon
Answer:
360 degrees is sum of the measures of the exterior angles of any convex polygon.
Step-by-step explanation:
Convex Polygon:
A convex polygon is a polygon with all its interior angle less than 180 degrees.Thus, all the vertices of the polygon will be away from the interior of the polygon.If all the angles are not less than 180 degrees then it becomes concave polygon.Example: Triangle is a convex polygon with three sides.Exterior angle sum Property:
If we extend the side of the polygon, then the angle formed by the extended side and polygon side is an exterior angle.The sum of all the exterior angles if a convex polygon is 360 degrees.If this convex polygon is a regular polygon, then the sum is 360 degrees and each exterior angle is equal to [tex]\frac{360}{n}[/tex], where n is the number of sides in regular polygon.The sum of the measures of the exterior angles of any convex
polygon is 360°.
What is a Convex polygon?This is the polygon that is the boundary of a convex set and has
all the interior angles to be less than 180°.
The interior angles is 180(n-2) degrees and each exterior angle
is supplementary to its interior angle thereby resulting in the
sum of the exterior angles to be 360°.
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If f(x)=5x-7 and g(x)√x-2 which statement is true?
-5 is in the domain or
-5 is not in the domain
-5 is in the domain of the function f(x)=5x-7, but not in the domain of the function g(x)√x-2. The domain of a function is the set of all acceptable input values.
Explanation:The question is asking whether -5 is in the domain of the given functions, f(x)=5x−7 and g(x)√x−2. The domain of a function is the set of all permissible inputs or arguments, i.e. x-values that can be inserted into the function.
In the case of f(x)=5x-7, the domain is all real numbers, as we can substitute any real number into 'x' and obtain a valid result. Hence, -5, indeed, is in the domain of the function f(x). However, for the function g(x)√x-2, the domain is all non-negative real numbers since we can't take the square root of a negative number in the real number system, hence -5 is not in the domain of g(x).
Thus, the statement '-5 is not in the domain' would be true if we are referring to g(x). However, if we are referring to f(x), the statement would be false because -5 is in the domain.
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Given the function g(x) = 4(3)x, Section A is from x = 1 to x = 2 and Section B is from x = 3 to x = 4.
A) Find the average rate of change of each section. (4 points)
which equation of a circle represents the graph?
The equation of circle will be;
⇒ x² + y² = 25
What is Circle?
The circle is a closed two dimensional figure , in which the set of all points is equidistance from the center.
Given that;
The circle is shown in figure.
Centre of circle = (0, 0)
Radius of circle = 5
Now,
We know that;
The standard form of circle with radius 'r' and center (h, k) is,
⇒ (x - h)² + (y - k)² = r²
Hence, The equation of circle with radius '5' and center (0, 0) is,
⇒ (x - 0)² + (y - 0)² = 5²
⇒ x² + y² = 25
Thus, The equation of circle will be;
⇒ x² + y² = 25
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A student reader is making $5.90 per hour and gets a $0.52 raise. what is the percent increase? (round to the nearest tenth of a percent.)
How do i graph the line y=2/3x+ 490 using the slope intercept method?
Graph of the line [tex]y = \frac{2}{3} x+490[/tex] is drawn with slope [tex]m = \frac{2}{3}[/tex] and y-intercept 490 and x-intercept = -735.
What is slope?
"Slope of the line is defined as the ratio of difference in the increment in y coordinates to the difference in increment in x-coordinates."
Formula used
standard equation of line
y = mx + c
m = slope of the line
c = y- intercept
According to the question,
Given equation of line,
[tex]y = \frac{2}{3} x+490[/tex]
Compare it with standard equation of line y = mx + c we get,
Slope [tex]m = \frac{2}{3}[/tex]
y-intercept = 490
Calculate x-intercept by putting y=0
x-intercept = -735
Now plot it on the graph as shown and join the points with slope [tex]m = \frac{2}{3}[/tex]
Hence, graph of the line [tex]y = \frac{2}{3} x+490[/tex] is drawn with slope [tex]m = \frac{2}{3}[/tex] and y-intercept 490 and x-intercept = -735.
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Pythagoras is planting a rose garden in the center of his rectangular yard, which is 20 feet long and 8 feet wide. He wants to dig up an area in the shape of an equilateral triangle with sides 5 feet each. After he digs up the rose-bed section, approximately how much grass will be left? Round to the nearest foot.
Answer:
The approximate remaining grass area is 144 square feet.
Explanation:
To find the area of the rectangular yard, we use the formula:
Area of rectangle [tex]= \text{length} \times \text{width} \)[/tex]
Given that the length is 20 feet and the width is 8 feet, we have:
Area of Rectangle [tex]= 20 \times 8 = 160 \text{ square feet} \)[/tex]
Next, we find the area of the equilateral triangle. Since we know the length of each side (5 feet) and an equilateral triangle can be divided into two congruent 30-60-90 triangles, we can use the formula for the area of an equilateral triangle:
Area of equilateral triangle [tex]= \frac{\sqrt{3}}{4} \times (\text{side length})^2 \)[/tex]
Plugging in the value of the side length (5 feet), we get:
Area of equilateral triangle [tex]= \frac{\sqrt{3}}{4} \times (5)^2 \)[/tex]
Area of equilateral triangle [tex]= \frac{\sqrt{3}}{4} \times 25 \)[/tex]
Area of equilateral triangle [tex]= \frac{25\sqrt{3}}{4} \text{ square feet} \)[/tex]
Now, to find the remaining grass area after Pythagoras digs up the rose-bed section, we subtract the area of the equilateral triangle from the total area of the rectangle:
Remaining grass area [tex]= \text{Area of rectangle} - \text{Area of equilateral triangle} \)[/tex]
Remaining grass area [tex]= \ 160 - \frac{25\sqrt{3}}{4} \)[/tex]
Using a calculator, we find:
Remaining grass area [tex]\approx 144.3 \text{ square feet} \)[/tex]
Rounded to the nearest foot, the approximate remaining grass area is 144 square feet.
Find the directrix of the parabola y = (1/8)(x – 5)2 - 3.
Answer:
y= -5
Step-by-step explanation:
rewrite as standard form [tex]4.2(y-(3-))=(x-2)^{2}[/tex]
(h.k) = (2, -3), P=2
y= -3 -P
y= -3 -2
y= -5
There are exactly four positive integers $n$ such that \[\frac{(n + 1)^2}{n + 23}\] is an integer. Compute the largest such $n$.
To find the largest positive integer $n$ such that $rac{(n + 1)^2}{n + 23}$ is an integer, we need to find the largest perfect square that is less than or equal to $n + 23$. The largest such $n$ is $41$.
Explanation:To find the largest positive integer $n$ such that $rac{(n + 1)^2}{n + 23}$ is an integer, we can start by considering the numerator. The expression $(n + 1)^2$ is a perfect square, so it will always be divisible by $n + 23$ if $n + 23$ is also a perfect square. Therefore, we need to find the largest perfect square that is less than or equal to $n + 23$.
Let's consider some examples. If $n + 23 = 25$, then $n = 2$. If $n + 23 = 36$, then $n = 13$. If $n + 23 = 49$, then $n = 26$. If $n + 23 = 64$, then $n = 41$. Therefore, the largest value of $n$ such that $rac{(n + 1)^2}{n + 23}$ is an integer is $41$.
There are no positive integers $n$ that satisfy the given condition, and we cannot compute the largest such $n$.
To find the largest positive integer $n$ such that $\frac{(n + 1)^2}{n + 23}$ is an integer, we need to consider the factors of the numerator and denominator.
Let's expand the numerator, $(n + 1)^2$, using the binomial expansion formula:
$(n + 1)^2 = n^2 + 2n + 1$
Now, we divide this expression by $n + 23$ and express the result as an integer:
$\frac{n^2 + 2n + 1}{n + 23}$
We can use polynomial long division to divide $n^2 + 2n + 1$ by $n + 23$:
- (n - 21)
__________
n + 23 | n^2 + 2n + 1
- (n^2 + 23n)
_____________
-21n + 1
- (-21n - 483)
_____________
484
We obtain a remainder of 484. In order for the fraction to be an integer, the remainder must be 0. However, in this case, the remainder is not 0, which means that $\frac{(n + 1)^2}{n + 23}$ is not an integer for any positive integer $n$.
Therefore, there are no positive integers $n$ that satisfy the given condition, and we cannot compute the largest such $n$.
Complete question :-
There are exactly four positive integers $n$ such that \[\frac{(n + 1)^2}{n + 23}\] is an integer. Compute the largest such $n$.
Mike forgot to replace the cap on a bottle of room freshener. The room freshener began to evaporate at the rate of 15% per day. If the original amount of room freshener in the bottle was 40 grams, which of the following graphs best represents the amount of room freshener f(x) that would remain in the bottle after x days?
graph of function f of x equals 30 multiplied by 0.88 to the power of x
graph of exponential function going down from left to right in quadrant 1 through the point 0, 40 and approaching the x axis
graph of function f of x equals 50 multiplied by 0.82 to the power of x
graph of exponential function going down from left to right in quadrant 1 through the point 0, 4 and approaching the x axis
The person under is wrong but there chart is right. 100%-15%=85%
85%=.85
F(x)= 40(.85)^x
Write that in desmos graphing and you will get the chart.
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The photo printed has length of 6 imams width of 4. A smaller picture is printed the ratio of the new picture to the size of the first picture is 1.5 to 2. What's the length and width of the new picture?
The new picture will have a length of 4.5 inches and a width of 3 inches.
There is asking about the size of a scaled-down photo in comparison to the original photo that has a length of 6 inches and a width of 4 inches. Given the ratio of the new picture to the size of the first picture is 1.5 to 2, we need to calculate the new dimensions.
The scale factor from the original to the new photo is 1.5 / 2, which simplifies to 0.75. Thus, we multiply the original dimensions by 0.75 to find the dimensions of the new picture:
Length of new picture = 6 inches * 0.75 = 4.5 inches
Width of new picture = 4 inches * 0.75 = 3 inches
Please help me I'm so confused and it's due at 7:30
80 POINTS!
Create a factorable polynomial with a GCF of 2y. Rewrite that polynomial in two other equivalent forms. Explain how each form was created. (give the polynomial with the GCF of 2 AND give two more forms and explain how you got it.
Katie earned $54.30 in interest from two accounts. One account Paid 4.1% apr and the other paid 3.2% apr. if she had a total of $1500 in the two accounts, How much was in each account?
A first grader measures 1 meter high how much bigger is a first grader compared to the height of a book
To solve this problem, we would be needing the actual height or size of the book which is not given in the problem. However for the sake of calculation let us assume that the average height of a book is 8.5 inches.
Then the next step we have to do is to convert the height of the first grader from units of meter to units of inches. We know that:
1 meter = 39.37 inches
Therefore the height of the first grader is 39.37 inches.
Taking the ratio with the bigger number in the numerator (height of the 1st grader / height of the book):
ratio = 39.37 inches / 8.5 inches
ratio = 4.63
Therefore this means that the first grader is about 4.63 times taller than the book.
What is the area the shaded triangle?
Answer:
96
Step-by-step explanation:
The area of the shaded triangle will be 96 mm² so option (B) is correct.
What is a triangle?A triangle is a 3-sided shape that is occasionally referred to as a triangle. There are three sides and three angles in every triangle, some of which may be the same.
The sum of all three angles inside a triangle will be 180° and the area of a triangle is given as (1/2) × base × height.
As per the given two inscribed triangles.
The bigger one is the right-angle triangle.
The area of the right-angle triangle will be as,
A = 1/2 x 8 x (6 + 24)
A = 4 x 30= 120 mm².
The area of the smaller right-angle triangle,
a = 1/2 x 8 x 6 = 24 mm²
The area of the shaded triangle = 120 - 24 = 96 mm².
Hence "The area of the shaded triangle will be 96 mm²".
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If $10,000 is invested in an account at the rate of 6.25% compounded monthly, then the amount after 15 years is approximately
The amount after 15 years is approximately $26,971.67.
Explanation:To calculate the amount after 15 years, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amountP = the principal amount (initial investment)r = annual interest rate (in decimal form)n = number of times interest is compounded per yeart = number of yearsIn this case, P = $10,000, r = 0.0625 (6.25% as a decimal), n = 12 (monthly compounding), and t = 15. Substituting these values into the formula:
A = $10,000(1 + 0.0625/12)^(12*15) = $26,971.67 (approximately).
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Evaluate the following expression using the values given: Find 3x − y − 3z if x = −2, y = 1, and z = −2. −13 −1 1 13
The correct answer is -1
Prove that for any x in the reals 5 divides x if and only if 5 divides x squared
Rewrite in simplest radical form 1/x^3/6.
Can anyone solve this.
At what angle do the angle bisectors of two same side interior angles intersect in construction with two parallel lines and a transversal?
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Which of the following expressions is not equal to p^2?
A.
p^-3 * p^5 * p^-2 * p^2
B.
p^-9 * p^0 * p^2 * p^5
C.
p^-3 * p^-1 * p * p^5
D.
p^-5 * p^9 * p^-2 * p^0
Option B, [tex]p^{-9} *p^{0} * p^{2} *p^{5}[/tex] is not equal to [tex]p^{2}[/tex].
Here,
We have to find the expression is not equal to p^2.
What is Property of addition in power of number?
This property states that when multiplying two powers with the same base, we add the exponents.
Now,
Let an expression [tex]x^{a} x^{b}[/tex]
Then it can be written as, [tex]x^{a} x^{b} = x^{a+b}[/tex]
So, Option B is given as,
[tex]p^{-9} *p^{0} * p^{2} *p^{5} = p^{-9+0+2+5} = p^{-2}[/tex]
Which is not equal to [tex]p^{2}[/tex].
Similarly, we check all options.
We get only option B is not equal to [tex]p^{2}[/tex].
Hence,
Option B, [tex]p^{-9} *p^{0} * p^{2} *p^{5}[/tex] is not equal to [tex]p^{2}[/tex].
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