The polynomial function of least degree with only real coefficients will be; y = x³ - 8 · x² + 22 · x - 20.
What is polynomial ?Algebraic expressions called polynomials include constants and indeterminates. Polynomials can be thought of as a type of mathematics.
The statement indicates that the polynomial has real coefficients having zeros 3 and 1 - 32 and a lead coefficient of 1.
By algebra of quadratic equations, equations with real coefficients with complex roots are α + i β and α - i β. Then we get;
y = 1 · (x - 3) · (x - 3 - i) · (x - 3 + i)
y = (x - 3) · [x² - 3 · x - i · x - 3 · x + i · x + (3 + i) · (3 - i)]
y = (x - 3) · (x² - 6 · x + 9 - i²)
y = (x - 3) · (x² - 6 · x + 10)
y = x³ - 6 · x² + 10 · x - 2 · x² + 12 · x - 20
y = x³ - 8 · x² + 22 · x - 20
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Answer:
[tex]p(x)=x^3-5x^2+16x-30[/tex]
Step-by-step explanation:
Given information:
Degree 3 polynomial with real coefficients.Zeros: 3 and (1 - 3i).Lead coefficient of 1.For any complex number [tex]z = a+bi[/tex] , the complex conjugate of the number is defined as [tex]z^*=a-bi[/tex].
If f(z) is a polynomial with real coefficients, and z₁ is a root of f(z)=0, then its complex conjugate z₁* is also a root of f(z)=0.
Therefore, if p(x) is a polynomial with real coefficients, and (1 - 3i) is a root of p(x)=0, then its complex conjugate (1 + 3i) is also a root of p(x)=0.
Therefore, the polynomial in factored form is:
[tex]p(x)=a(x-3)(x-(1-3i))(x-(1+3i))[/tex]
As the leading coefficient is 1, then a = 1:
[tex]p(x)=(x-3)(x-(1-3i))(x-(1+3i))[/tex]
Expand the polynomial:
[tex]\implies p(x)=(x-3)(x-(1-3i))(x-(1+3i))[/tex]
[tex]\implies p(x)=(x-3)(x-1+3i)(x-1-3i)[/tex]
[tex]\implies p(x)=(x-3)(x^2-x-3xi-x+1+3i+3ix-3i-9i^2)[/tex]
[tex]\implies p(x)=(x-3)(x^2-x-x-3xi+3ix+1+3i-3i-9i^2)[/tex]
[tex]\implies p(x)=(x-3)(x^2-2x+1-9(-1))[/tex]
[tex]\implies p(x)=(x-3)(x^2-2x+10)[/tex]
[tex]\implies p(x)=x^3-2x^2+10x-3x^2+6x-30[/tex]
[tex]\implies p(x)=x^3-2x^2-3x^2+10x+6x-30[/tex]
[tex]\implies p(x)=x^3-5x^2+16x-30[/tex]
Find the coordinates of the other endpoint of the segment, given its midpoint and one endpointand y.)midpoint (-7.-21), endpoint (-13.-15)
Ok, we are going to use the midpoint formula
M = ( (x1 + x2) / 2 , (y1 + y2) / 2 )
(-7,-21)=((x1-13)/2 , (y1-15)/2 )
Break up this formula into two equations.
(x1-13)/2=-7 and (y1-15)/2=-21
Solve for x1 and y1 from the equations. So:
x1=(-7*2)+13
x1=(-14)+13=-1
y1=(-21*2)+15=(-42)+15=-27
So the other endpoint is (-1, -27).
Questlon 5 Refer to the figure. HJ I JE. HII IE. HJ HI J H E Complete the explanation to show triangle EJH is congruent to triangle EIH. The two triangles given are _____triangles. The leg and hypotenuse of triangle EJH are congrue hypotenuse of triangle EIH. By the ______ Theorem the third side ma triangles are congruent by the____ Triangle Congruence Theorem.choice 1.acute,obtuse or right angleschoice 2.corresponding parts of congruent triangles, pythagorean,or side-angle-side triangle congruence.choice 3. side-side-side, side-angle-side,or angle-side-angle
In the given figure, we have two triangles △EJH and △EIH
We are given the following information
[tex]\begin{gathered} \bar{HJ}\perp\bar{JE} \\ \bar{HI}\perp\bar{IE} \\ \bar{HJ}\cong\bar{HI} \end{gathered}[/tex]This means that these two triangles are "Right Triangles"
Therefore.
Choice 1 = right angles
When the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent.
Therefore,
Choice 2 = side-angle-side triangle congruence
Choice 3 = side-angle-side
Answer: choice 1 -Right Angle
Choice 2 -Pythagorean
Choice 3- Side-Side-Side
Step-by-step explanation:other guy is completely wrong lol
help pleasesjjsnsbsbbbsbs
Student asking the same question for third time in less than ten minutes. Can't help him or her out with additional information to complete the exercise.
Closing the session!
Liz attended class every day since she started school as a kindergarten. She said she has been in school for about 1,000 days.What numbers could be the actual number of school days if Liz rounded to the nearest ten?4 grade student
Solution.
Given that Liz attended class every day since she started school as a kindergarten and that she said she has been in school for about 1000 days (by rounding up the actual number of days to the nearest ten)
The actual number is a number that when rounded up, we would arrive at 1000.
This number falls between the numbers 995 and 1004.
Answer: Any of the numbers below could be the actual number:
995, 996, 997, 998, 999, 1000, 1001, 1002, 1003, 1004
evaluate the following function for f(-2) .f(x)=3x+12
Given :
[tex]f(x)=3x+12[/tex]WE need to find the value of f(-2)
So, substitute with x = -2
[tex]f(-2)=3\cdot-2+12=-6+12=6[/tex]So, the value of f(-2) = 6
for each of the following polynomial functions, write the equation of a different polynomial function that has the same key characteristic. explain your thinking.
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Draw the given graph.
STEP 2: Get the function plotted on the graph.
[tex]undefined[/tex]5. The domain of f(x) = -2x + 1 is {-4, -1, 0, 2}. Find the range.
Explanation:
The function is f(x) = -2x + 1
Domain = {-4, -1, 0, 2}
Note that the domain is a set of of all the values of x ( i.e. the independent variable)
The range is a set of the corresponding value of f(x) for each value of x in the domain.
For x = -4
f(-4) = -2(-4) + 1 = 8 + 1
f(-4) = 9
f(-1) = -2(-1) + 1 = 2 + 1
f(-1) = 3
f(0) = -2(0) + 1 = 0 + 1
f(0) = 1
f(2) = -2(2) + 1 = -4 + 1
f(2) = -3
Therefore the set of all the values above which is the range will be given as:
Range = { 9, 3, 1, -3}
Each n in the model represents the same value. 136.5 What is the value of n? there are 7 N's in the problem
The value of n must be 136.5/7
n = 136.5/7
n = 19.5
Result n = 19.5
Classify the triangle formed by the side lengths as right, acute, or obtuse. 11 13 23
The triangle of side lengths 11, 13 and 23 is obtuse triangle.
Since none of the sides are equal.
simplify 5(3c-4d)-8c
Answer:
7c - 20d
Step-by-step explanation:
5(3c - 4d) - 8c ← distribute parenthesis by 5
= 15c - 20d - 8c ← collect like terms
= 7c - 20d
what percentage of students scored before 70-90 points on the exam? Round your answer to the nearest tenth of a percent?
We want to find the percentage of students that scored between 70-90 points on the examn. Also, we know that there are a total of 71 students, so we have to count the number of students who got between 70-90 points.
We see them represented on the histogram as the two largest bars, and we obtain:
[tex]\begin{gathered} 21\text{ students scored between 70-80 points} \\ 22\text{ students scored between 80-90 points} \end{gathered}[/tex]So, the total of students that scored between 70-90 points is 21+22=43.
For finding the percentage, we make the quotient between the number of students with a score of 70-90 and the total of the students in the class.
[tex]=\frac{43}{71}\cdot100=60.6[/tex]This means that approximately a 60.6% of the class students scored between 70 and 90 points.
Decide whether the word problem represents a linear or exponential function. Circle either linear or exponential. Then, write the function formula.
a. The given table is
Notice, the value of x increases at equal intervals of 1
Also, the value of y increases at an equal interval of 3
This means for the y values the difference between consecutive terms is 3
Also, for the x values, the difference between consecutive terms is 1
Hence, the table represents a linear function
The general form of a linear function is
[tex]y=mx+c[/tex]Where m is the slope
From the interval increase
[tex]m=\frac{\Delta y}{\Delta x}=\frac{3}{1}=3[/tex]Hence, m = 3
The equation becomes
[tex]y=3x+c[/tex]To get c, consider the values
x = 0 and y = 2
Thi implies
[tex]\begin{gathered} 2=3(0)+c \\ c=2 \end{gathered}[/tex]Hence, the equation of the linear function is
[tex]y=3x+2[/tex]b. The given table is
Following the same procedure as in (a), it can be seen that there is no constant increase in the values of y
Hence, the function is not linear
This implies that the function is exponential
The general form of an exponential function is given as
[tex]y=a\cdot b^x[/tex]Consider the values
x =0, y = 3
Substitute x = 0, y = 3 into the equation
This gives
[tex]\begin{gathered} 3=a\times b^0 \\ \Rightarrow a=3 \end{gathered}[/tex]The equation become
[tex]y=3\cdot b^x[/tex]Consider the values
x =1, y = 6
Substitute x = 1, y = 6 into the equation
This gives
[tex]\begin{gathered} 6=3\cdot b^1 \\ \Rightarrow b=\frac{6}{3}=2 \end{gathered}[/tex]Therefore the equation of the exponential function is
[tex]y=3\cdot2^x[/tex]c. The given table is
As with (b) above,
The function is exponential
Using
[tex]y=a\cdot b^x[/tex]When
x = 0, y = 10
This implies
[tex]\begin{gathered} 10=a\cdot b^0 \\ \Rightarrow a=10 \end{gathered}[/tex]The equation becomes
[tex]y=10\cdot b^x[/tex]Also, when
x = 1, y =5
The equation becomes
[tex]\begin{gathered} 5=10\cdot b^1 \\ \Rightarrow b=\frac{5}{10} \\ b=\frac{1}{2} \end{gathered}[/tex]Therefore, the equation of the exponential function is
[tex]y=10\cdot(\frac{1}{2})^x[/tex]Multiply. Write your answer in decimal form: (8 x 10^2)(4 x 10^2)
Answer:
32×10⁴
Step-by-step explanation:
open the bracket
8×4×10^(2+2)
32×10⁴
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Aunt Eloise’s house is always 20°C. She has just made a fresh cup of tea (tea is made with boiling water and water boils at 100°C) five minutes after she made the tea her mad scientist nephew came in, stuck a thermometer in the cup and announced that the tea was now only 70°C. She had gotten involved with her book and forgot to have even a sip of her tea. Now she won’t drink it because it isn’t piping hot anymore.Write and equation that models this problem and use it to predict the temperature of the tea 20 minutes after it was taken off the stove.
Given:
a.) She has just made a fresh cup of tea (tea is made with boiling water and water boils at 100°C)
b.) Five minutes after she made the tea her mad scientist nephew came in, stuck a thermometer in the cup, and announced that the tea was now only 70°C.
c.)
The Oldest rocks on Earth are about 4 x 10^9 years old. For which of these ages could this be an approximation?
A. 3,862,100,000 years
B. 3.849999999x10^9 years
C. 0.000000004 years
D.4,149,000,000 years
E.3.45x10^9 years
A set of four numbers that begins with the number 32 is arranged fromsmallest to largest. If the median is 35, which of the following could possiblybe the set of numbers?a) 32, 32, 36, 38b) 32, 35, 38, 41c) 32, 34, 36, 39d) 32, 36, 40, 44
Given the word problem, we can deduce the following information:
1. A set of four numbers that begins with the number 32 is arranged from
smallest to largest.
2. The median is 35.
To determine the possible set of numbers of which the median is 35, we first note that median is the number separating the other half of the ordered data sample from the lower half.
Now, we check the median of each choices:
For a) 32, 32, 36, 38:
[tex]Median=\frac{32+36}{2}=34[/tex]For b) 32, 35, 38, 41:
[tex]Median=\frac{35+38}{2}=36.5[/tex]For c) 32, 34, 36, 39
[tex]Median=\frac{34+36}{3}=35[/tex]For d) 32, 36, 40, 44:
[tex]Median=\frac{36+40}{2}=38[/tex]Therefore, the answer is: c) 32, 34, 36, 39
What is the average rate of change over the interval [1,2]Type the numerical value for your answer as a whole number, decimal, or fractionMake sure answers are completely simplified
To calculate the average rate of change over the interval [1,2] we need to identify the points in the extremes of the interval.
This points are (1,50) and (2,25).
We calculate the average rate of change as the slope:
[tex]r=\frac{y_2-y_1}{x_2-x_1}=\frac{25-50}{2-1}=-\frac{25}{1}=-25[/tex]Answer: the rate of change over [1,2] is -25.
In a game, Billy must roll two dice. One die is astandard six-sided number die, and the otherdie has a different color on each side (red,blue, green, orange, yellow, and purple). Whatis the probability that Billy rolls a 3 and agreen?A 162% czB 12D1WIN
The probability of getting a 3 is:
[tex]P=\frac{1}{6}[/tex]The probability of getting green is:
[tex]P=\frac{1}{6}[/tex]Therefore the probability of getting a 3 and a green is:
[tex]P=\frac{1}{6}\cdot\frac{1}{6}=\frac{1}{36}[/tex]Hence the answer is C.
If p(x) is a polynomial function where p(x) = 3(x + 1)(x - 2)(2x-5)a. What are the x-intercepts of the graph of p(x)?b. What is the end behavior (as x→ ∞, f(x)→?? and as x→ -∞, f (x)→ ??) of p(x))?c. Find an equation for a polynomial q(x) that has x-intercepts at -2, 3⁄4, and 7.
Hello there. To solve this question, we have to remember some properties about polynomial functions.
Given the polynomial function
[tex]p(x)=3(x+1)(x-2)(2x-5)[/tex]We want to determine:
a) What are the x-intercepts of the graph of p(x)?
For this, we have to determine the roots of the polynomial function p(x). In this case, we have to determine for which values of x we have
[tex]p(x)=0[/tex]Since p(x) is written in canonical form, we find that
[tex]p(x)=3(x+1)(x-2)(2x-5)=0[/tex]A product is equal to zero if at least one of its factors is equal to zero, hence
[tex]x+1=0\text{ or }x-2=0\text{ or }2x-5=0[/tex]Solving the equations, we find that
[tex]x=-1\text{ or }x=2\text{ or }x=\dfrac{5}{2}[/tex]Are the solutions of the polynomial equation and therefore the x-intercepts of p(x).
b) What is the end-behavior of p(x) as x goes to +∞ or x goes to -∞?
For this, we have to take the limit of the function.
In general, for polynomial functions, those limits are either equal to ∞ or -∞, depending on the degree of the polynomial and the leading coefficient.
For example, a second degree polynomial function with positive leading coefficient is a parabola concave up and both limits for the function as x goes to ∞ or x goes to -∞ is equal to ∞.
On the other hand, an odd degree function usually has an odd number of factors (the number of x-intercepts in the complex plane) hence the limits might be different.
In this case, we have a third degree polynomial equation and we find that, as the leading coefficient is positive and all the other factors are monoic, that
[tex]\begin{gathered} \lim_{x\to\infty}p(x)=\infty \\ \\ \lim_{x\to-\infty}p(x)=-\infty \end{gathered}[/tex]That is, it gets larger and larger when x is increasing arbitrarily, while it get smaller and smaller as x is decreasing.
c) To find the equation for a polynomial q(x) that has x-intercepts at -2, 3/4 and 7.
The canonical form of a polynomial of degree n with x-intercepts at x1, x2, ..., xn and leading coefficient equals a is written as
[tex]f(x)=a\cdot(x-x_1)(x-x_2)\cdots(x-x_n)[/tex]So in this case, there are infinitely many polynomials satisfying this condition. Choosing a = 1, we find that q(x) is equal to
[tex]\begin{gathered} q(x)=(x-(-2))\cdot\left(x-\dfrac{3}{4}\right)\cdot(x-7) \\ \\ \boxed{q(x)=(x+2)\cdot\left(x-\dfrac{3}{4}\right)\cdot(x-7)} \end{gathered}[/tex]These are the answers to this question.
How long will it take until the diver enters the water? How do you know?
Given:
The equaiton that model height of the driver jumps from the ledge.
[tex]h=-t^2+8t+115[/tex]Requried:
We need to find the time taken by driver to enter the water.
Explanation
Which equation represents a line which is perpendicular to the line y=-5/4x-4?A. 4y−5x=−4B. 5x+4y=−8C. 4x−5y=15D.4x+5y=40
The slope of a line, m, comes in the equation as the coefficient of x.
In the given equation, m= -5/4. Two perpendicular lines have slopes that are the negative reciprocals of each other.
So, the slope of the perpendicular line will be +4/5.
Between the given options, letter c will be:
4x-5y=15
-5y=15-4x (divided by -5)
y=4/5x-3
Letter C
Which expression is equal to -2i(4 - i)?
Answer:
-8i + 2i^2
I am new but I hope this helps you
Answer:
-8i+2i²
this looks like an equation in factorization
* Use the digits 0, 2, and 5 to write all of the three-digit numbers that fit each
description. You can repeat digits in a number.
multiples of 2
The 3-digit multiples of 2 using 0, 2, and 5 are:
250502520What are multiples?A multiple in science is created by multiplying any number by an integer. In other words, if b = na for some integer n, known as the multiplier, it can be said that b is a multiple of a given two numbers, a and b. This is equivalent to stating that b/a is an integer if an is not zero. In mathematics, multiples are the results of multiplying an integer by a given number. Multiples of 5 include, for instance, 10, 15, 20, 25, 30, etc. Numerous 7s include 14, 21, 28, 35, 42, 49, etc.So, 3-digit multiples of 2 using the digits 0, 2, and 5 are:
3 digits multiples of 2:
250502520Therefore, the 3-digit multiples of 2 using 0, 2, and 5 are:
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Please help, disregard the option I chose because I'm not sure it's right :)
Consider that the graph of f(x) is the graph of a cubic function, that is, the graph of a 3 degree polynomial. If you apply first derivative to such a polynomial, the result is another polynomial of degree 2.
Now, take into account that the graph of a polynomial of degree 2 is a parabola. The parabola can open up or down. It depends of the leadding coefficient of the polynomial. In this case, due to the graph of f(x), the leadding coefficient is positive, which means that the parabola of f'(x) opens up.
Hence, you can conclude that the graph of f'(x) is option C.
Write an equation in point slope form for the line given the slope of 4,and a point on the line (1,2)
[tex]\begin{gathered} \text{ the equation of a line in slope-point form is} \\ y=mx+b,\text{ we know that m=4, and that (1,2) is on the line, so} \\ 2=4(1)+b \\ 2=4+b \\ b=2-4 \\ b=-2 \\ \\ \text{ Thus, the equation has the form} \\ y=4x-2 \\ \end{gathered}[/tex]
Answer:
[tex]y-2=4(x-1)[/tex]
Step-by-step explanation:
Pre-SolvingWe are given that a line has a slope (m) of 4, and that it contains the point (1,2).
We want to write the equation of this line in point-slope form.
Point-slope form is given as [tex]y-y_1=m(x-x_1)[/tex], where m is the slope and [tex](x_1, y_1)[/tex] is a point, hence the name.
Since we are already given the slope, we can immediately plug it into the formula.
Substitute 4 for m.
[tex]y-y_1=4(x-x_1)[/tex]
Now, let's label the values of (1,2) to avoid confusion while substituting.
[tex]x_1=1\\y_1=2\\[/tex]
Substitute these values into the formula.
[tex]y-2=4(x-1)[/tex]
Topic: point-slope form
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Find the second endpoint of the segment that has an endpoint at (9,5) and its midpointat (4, 2).
it
8 more than the product of 12 and 11.
Please Help
Determine the sum of the infinite geometric series
1/2-1/3+2/9-…
A. -1/2
B. the sum cannot be determined
C. 1/3
D. 3/10
We (B) cannot determine the sum of the given infinite geometric series (1/2-1/3+2/9-…).
What is infinite geometric series?A geometric series is one where each pair of consecutive terms' ratios is a fixed function of the summation index. The ratio is a rational function of the summation index in a more general sense creating what is known as a hypergeometric series.The result of an infinite geometric sequence is an infinite geometric series. There would be no conclusion to this series. The infinite geometric series has the general form a₁ + a₁r + a₁r² + a₁r³ +..., where r is the common ratio and a1 is the first term.So, the sum of 1/2-1/3+2/9-…
We can easily observe that the terms of the following given series are not in a series or in a particular sequence.Then, it is not possible to find the sum of this given series.Therefore, we (B) cannot determine the sum of the given infinite geometric series (1/2-1/3+2/9-…).
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Kaylee drove 160 miles in 5 hours. If she continued at the same rate, how far would she travel in 17 hours?
The distance covered by Kaylee in 17 hours at the same rate is 544 miles.
According to the question,
We have the following information:
Distance covered by Kaylee = 160 miles
Time taken by Kaylee = 5 hours
We know that the following formula is used to find the speed:
Speed = distance/time
Speed = 160/5 mile/hour
Speed = 32 miles/hour
Now, we have to find the distance when time taken is 17 hours and the speed is the same.
Now, from the formula of speed, we can find the distance:
Distance = speed*time
Distance = 32*17
Distance = 544 miles
Hence, the distance covered by Kaylee in 17 hours is 544 miles.
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for each triangle identify a base and corresponding height use them to find the are
A)
For this tringle we can turn the figure like this:
now we have two right triangles and we can calulate the base of the first triangle with the sin law
[tex]\begin{gathered} \frac{\sin (90)}{3}=\frac{sin(a)}{2.5} \\ \sin (a)=\frac{2.5\sin (90)}{3} \\ \sin (a)=0.8 \\ a=\sin ^{-1}(0.8)=53º \end{gathered}[/tex]the angle b is going to be:
[tex]\begin{gathered} 180=90+53+b \\ b=180-90-53 \\ b=37 \end{gathered}[/tex]Now the base is going to be:
[tex]\begin{gathered} \frac{\sin(90)}{3}=\frac{\sin(37)}{\text{base}} \\ \text{base}=\frac{3\sin (37)}{\sin (90)}=1.8 \end{gathered}[/tex]and the base of the secon triangle is going to be:
[tex]\text{base}2=7.2-1.8=5.4[/tex]And the area of the triangles is going to be:
[tex]A1=\frac{base\times2.5}{2}=\frac{1.8\times2.5}{2}=2.25[/tex][tex]A2=\frac{base2\times2.5}{2}=\frac{5.4\times2.5}{2}=6.75[/tex]so in total the area is going to be:
[tex]A1+A2=2.25+6.75=9[/tex]B)
the procedure is similar, first we turn the tiangle like this:
the angle a is going to be:
[tex]\begin{gathered} \frac{\text{sin(a)}}{4.8\text{ }}=\frac{\sin (90)}{6} \\ \sin (a)=\frac{4.8\sin (90)}{6}=0.8 \\ a=\sin ^{-1}(0.8) \\ a=53º \end{gathered}[/tex]the angle b is going to be:
[tex]\begin{gathered} 180=90+53+b \\ b=180-90-53 \\ b=37º \end{gathered}[/tex]now the base is going to be:
[tex]\begin{gathered} \frac{\sin (37)}{base}=\frac{sen(90)}{4.8} \\ \text{base}=\frac{4.8\sin (37)}{\sin (90)} \\ \text{base}=2.8 \end{gathered}[/tex]and the base of the other triangle will be:
[tex]\text{base}2=5-2.8=2.2[/tex]And the area of the triangles will be:
[tex]\begin{gathered} A1=\frac{base\times4.8}{2}=\frac{2.8\times4.8}{2}=6.72 \\ A2=\frac{base2\times4.8}{2}=\frac{2.2\times4.8}{2}=5.28 \end{gathered}[/tex]And the total area will be:
[tex]A1+A2=6.72+5.28=12[/tex]