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]
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Part A: How many sides does a parallelogram have?
Part B: How are the opposite angles of parallelograms related?
Part C: If mA = 115° and m/B = 65°, determine m/C and m/D. Show your work.
A) A parallelogram has 4 sides.
B) The opposite angles of parallelogram are congruent.
C) [tex]\angle A \cong \angle C, \angle B \cong \angle D\\\\\implies m\angle C=115^{\circ}, m\angle D=65^{\circ}[/tex]
what is the median of the data set? 9, 3, 10, 12, 4, 5, 12, 2
Answer:
8
Step-by-step explanation:
12+4=16
16 divided by 2= 8
A fivey sequence is a sequence of positive integers which adds up to 5. For example, 2, 2, 1 and 2, 1, 2 are two different fivey sequences. How many fivey sequences are there?
The total number of possible fivey sequences are; 6
How to solve a sequence?We are told we are dealing with a fivey sequence which means that all the terms of the sequence must be positive integers and they must add up to 5.
Now, the possible numbers that can be used are;
1, 2 and 3.
If we start with 3, number of possible sequences is; 1
If we start with1, number of possible sequences is; 3 ways
If we start with 2, number of possible sequences is; 2 ways
Thus;
Total number of fivey sequences is 1 + 2 + 3 = 6 possible sequences
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A sports team sold a total of 293 adult and child tickets to a recent game. Adult tickets cost $25.50 and child tickets cost $8. The team collected $6019 from ticket sales. Using any method, calculate how many of each ticket (adult and child) were sold.
The number of tickets for adults is 210 and the number of tickets for children is 83.
What is a linear equation?It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.
If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.
Let x be the number of tickets for adults and y be the number of tickets for children.
From the problem:
x + y = 293
25.50x + 8y = 6019
After solving with the substitution method:
[tex]\rm 25.5\left(293-y\right)+8y=6019[/tex]
y = 83
x = 210
Thus, the number of tickets for adults is 210 and the number of tickets for children is 83.
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Find the mean of the data in the dot plot below. students size of each of mr. sirac's classes 21 20 22 23 24 25 26 28 27 number of students show calculator report a stuck? watch a video or use a hint.
The mean of the data about students size of Mr. Sirac classes is 24.
Given students size of each of Mr. Sirac'sclasses 21,20,22,23,24,25,26,28,27 and we have to find the mean of the data.
We know that mean is the basically sum of numbers divided by the number. It is also known as average.
Mean =∑X/n where n is number of values and ∑X is sum of all the values.
∑X=21+20+22+23+24+25+26+28+27
=216
n=9
Mean is calculated by dividing sum by n.
Mean =216/9
=24
Hence the mean of the data about students size of each of Mr. Sirac classes is 24.
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what is the equation of the line through the points (3,2),(8,5),(13,8), and (18,11)
The equation of the line through the points is y = 3/5 + 1/5.
First, we find the Slope using the formula: m = y2 - y1/x2 - x1
You may choose any two points but I will use these two for no reason at all.)
Points: (3,2) and (8,5)
m = y2 - y1/x2 - x1
m = 5 - 2/8 - 3
m = 3/5
Now that we have the slope, we can now find the Slope-Intercept Form by using the Slope Form Formula. (need at least one point and slope.)
Formula: y - y1 = m(x - x1)
Point: (3,2)
What is the slope intercept form?y - y1 = m(x - x1)
y - 2 = 3/5(x - 3)
y - 2 = 3/5x - 9/5
y - 2 = 3/5x - 9/5
y = 3/5x - 9/5 + 10/5 (2 can be is any fraction that consists of the numerator multiplied twice by the denominator, such as 2/1, 4/2, 8/4, etc.)
y = 3/5 + 1/5
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In the given figure a + b + c = 295° , fins the value of a , b , c and d....
Answer:
a and c = 115 and b and d = 65
Step-by-step explanation:
Hope this helps :)
........................................
Answer:
vii. x = 70° viii. x = 60°
Step-by-step explanation:
vii.
The purple line drawn is parallel to the lines PQ and ST.
∴ ∠ a + 130° = 180° [alternate angles]
⇒ ∠ a = 50°
x = a + 20° [Co-interior angles with angle [tex]x[/tex]]
⇒ x = 50° + 20°
⇒ x = 70°
viii.
The yellow line drawn and the line AB are parallel.
∴ n° + 130° = 180° [Co-interior angles]
n = 50°
The green line and line DE are parallel.
∴ m° + 110° = 180° [Co-interior angles]
m = 70°
n + m + x = 180° [Angles on a straight line]
⇒ 50° + 70° + x = 180°
⇒ x = 60°
Answer:
vii) x = 70° viii) x = 60°
Step-by-step explanation:
Please refer to the attached photos for better understanding (Apologies for the terrible drawing.)
vii) Angle RSV + Angle RST = 180° (Sum of angles in a straight line)
Angle RSV + 130° = 180°
Angle RSV = 180° - 130°
= 50°
Angle RVS + Angle RSV + Angle SRV = 180° (Sum of angles in a triangle)
Angle RVS + 50°+ 20° = 180°
Angle RVS = 180° - 70° = 110°
Angle RVX + Angle RVS = 180° (Sum of angles in a straight line)
Angle RVX + 110° = 180°
Angle RVX = 180° - 110° = 70°
Angle x = Angle RVX = 70° (Corresponding Angles)
viii) Angle FDG + Angle CDE = 180° (Sum of angles in a straight line)
Angle FDG + 110° = 180°
Angle FDG = 180° - 110°
= 70°
Angle BGC = Angle FDG = 70° (Corresponding Angles)
Angle CBG + Angle ABC = 180° (Sum of angles in a straight line)
Angle CBG + 130° = 180°
Angle CBG = 180 - 130°
= 50°
Angle x + Angle CBG + Angle BGC = 180° (Sum of angles in a triangle)
Angle x + 50° + 70° = 180°
Angle x = 180° - 120° = 60°
A fish tank has a length of 12 cm a width of 15 cm and a depth of 25 cm find the volume of the fish tank.. any indian here..
Answer:
our volume is 4500cm³
Step-by-step explanation:
note: the following will be assuming that this fish tank is rectangular, as most fish tanks are
To find the volume of a rectangular structure, we multiply :
length × width × height
here, we know our
length to be 12 cm
width to be 15 cm
and height/depth to be 25 cm
so, we can follow our volume formula (length × width × height) using these values
length × width × height = volume
12 × 15 × 25 = 4500
so, our volume is 4500cm³
(also called 4500cubic centimeters)
(we write volume as cubed--units³)
hope this helps! have a lovely day :)
What is the solution to the system of equations?
Answer:
(b) (-3, 2, -5)
Step-by-step explanation:
The solution to the system of equations will satisfy every equation. If it fails to satisfy any equation, it is not a solution.
StrategyThe system of equations does not seem to lend itself to simple solution by substitution or elimination. Hence, a reasonable strategy for finding the answer is to try the offered choices. Of course, a calculator can be used to find the answer almost as quickly.
CheckUsing the third equation, we can check the answer choices fairly easily. That equation involves the least number of arithmetic operations. Substituting for (x, y, z), we have ...
a) 3(1) -(11) -(5) = -13 ≠ -6
b) 3(-3) -(2) -(-5) = -6 . . . . . . a potential solution
c) 3(1) -(8) -(0) = -5 ≠ -6
d) 3(-1) -(3) -(4) = -10 ≠ -6
The only viable choice is (-3, 2, -5).
Check
In the other two equations, we have for this solution, ...
2(-3) -2(2) +(-5) = -15 . . . . works in the first equation
6(-3) -3(2) -(-5) = -18 . . . . works in the second equation
Whats the correct answer answer asap
Answer:
C
Step-by-step explanation:
it just is
PLEASE HELP!!!!!!
20 POINTS
The graph which has the same end behaviour as the function given; f(x) = (10x³+x² +3)/(2x²+1) is; a linear graph whose slope is 5.
Which graph has the same end behaviour as the function given?The function given can further be simplified by means of polynomial division in which case;
although not a perfect division; the quotient of the division is; (5x-0.5) and evidently, the function which would have the same end behaviour as the given function is a linear function whose slope is 5.
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What is the common difference for this arithmetic sequence?
-5, -1, 3, 7, 11, ...
Answer:
4
Step-by-step explanation:
The common difference is found by subtracting the 2nd term from the 1st term. In this example we can say 11 is the 2nd term and 7 is the 1st term. 11 - 7 = 4.
Solve the equation for
The equation solved for w is w = 6g - 40
How to solve the equation?The complete question is in the attached image
The equation is given as:
g = 1/6(w + 40)
Multiply both sides by 6
6g = w + 40
Subtract 40 from both sides
w = 6g - 40
Hence, the equation solved for w is w = 6g - 40
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Graph the solution set of this inequality: 3x -2y ≥ 12
The slope m = 3/2 and the y-intercept = ( 0, -6 ).
The graph is a solid line and shaded area below the boundary line since y is less than (3/2)x - 6.
What is the Graph the solution set of the inequality?Given that:
3x -2y ≥ 12
First we write the inequality in slope intercept form y = mx + b.
Solve for y
3x -2y ≥ 12
-2y ≥ 12 - 3x
Now, divide each term by -2
y ≤ -6 + 3x/2
We re-arrange
y ≤ (3/2)x - 6
Using the slope intercept form y = mx + b. we can see that;
m = 3/2
b = -6
Hence, the slope m = 3/2 and the y-intercept = ( 0, -6 ).
The graph is a solid line and shaded area below the boundary line since y is less than (3/2)x - 6.
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A nurse must administer 180 micrograms atropine sulfate. This drug is available in solution form. The concentration of the atropine sulfate solution is 500 micrograms per milliliter. How many milliliters should be given? Simplify your answer.
Answer:
0.36ml
Step-by-step explanation:
Note: micrograms = [tex]10^{-6} g[/tex]
Given the concentration,
[tex]500*10^{-6} g = 1ml\\1*10^{-6}g =\frac{1}{500} ml\\180*10^{-6} g=(\frac{1}{500} *180)ml\\180micrograms = 0.36ml[/tex]
Given that Log 10^2=0.301 and log10^3= 0.477, find the value of log10^6
This value is approximate
=======================================================
Work Shown:
[tex]\log_{10}(2) \approx 0.301\\\\\log_{10}(3) \approx 0.477\\\\\log_{10}(6) = \log_{10}(2*3)\\\\\log_{10}(6) = \log_{10}(2)+\log_{10}(3) \ \text{ see note below}\\\\\log_{10}(6) \approx 0.301 + 0.477\\\\\log_{10}(6) \approx 0.778\\\\[/tex]
Note: I used the rule that log(A*B) = log(A)+log(B) which works for any valid log base.
Step-by-step explanation:
Since the question has already been answered, I'd like to add something new, and explain why: [tex]log_b(a*c)=log_ba+log_bc[/tex]
So let's just say that:
[tex]x=log_ba[/tex] and that [tex]y=log_bc[/tex]
This means that: [tex]b^x=a[/tex] and that [tex]b^y=c[/tex].
So if we were to multiply the two, a and c. You get
[tex]a*c=b^{x+y}[/tex]
This is due to the exponent identity that: [tex]b^a*b^c=b^{a+c}[/tex]
So if you rewrite this in logarithmic form you get:
[tex]log_b{ac}=x+y[/tex]
and remember what x and y are equal to? that's right, it's the logarithms
so now you substitute the logarithms back in and get
[tex]log_b{ac} = log_ba+log_bc[/tex]
PLS HELP ILL GIVE 5 STARS
The entrance to a business is 4 feet above the ground. The business is required to build an accessibility ramp with a slope of 1/12. How far from the entrance will the ramp need to begin
Answer:
4² + b² = 12²
16 + b² = 144
b² = 144 - 16
b² = 128
b = √128
b = 11.3 feet
Step-by-step explanation:
The pie is cut into 15 equal slices
Shade 1/5 of the pie
What is the truth value for the following conditional statement?
p: true
q: true
p → q
T T → F
T T → T
F T → T
T F → T
Based on the conditional statement given, the truth value can be found to be T T → T.
How do we find the truth value?
The truth value can be found by using the Boolean Table which is shown as:
p q p → q
T T T
T F F
F T T
F F T
From the table, we can then see that if both p and q are true, then p → q is true as well.
We will then have a truth value of T T → T.
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In 2003, a school population was 1219. By 2009 the population had grown to 1987.
1) How much did the population grow between the year 2003 and 2009?
students
2) How long did it take the population to grow from 1219 students to 1987 students?
years
3) What is the average population growth per year?
P =
students/year
4) What was the population in the year 2000?
students
5) Find an equation for the population, P, of the school t years after 2000.
:
6) Using your equation, predict the population of the school in 2012.
students
1) Between the years 2003 and 2009, 768 students joined the school.
2) It took 6 years for the population to grow from 1219 students to 1987 students.
3) The average population growth per year is 128 students per year.
P = 128 students per year
4) The population of students in the year 2000 was a total of 835 students.
5) 1219 + (p3) = 1987
6) 1987 + (p3) = 2371
P3 = 384
Which transformations could have occurred to
AABC to AA"B"C"?
O a rotation and a dilation
• a rotation and a reflection
• a reflection and a dilation
• a translation and a dilation
Step-by-step explanation:
•a reflection and a dilation
function g(x) : Five more than the opposite of 3 times the square of a number increased by 4 times the number
Answer:
[tex]g(x)= -3x^{2} +4x +5[/tex]
Step-by-step explanation:
Five more = Addition of 5
Opposite = (-)
3 times = Multiplication by 3
Square = To the power (exponent) of two
4 times the number = Multiplication by 4
The number = unknown number = x
There are 3 people at a party. If each person must shake hands with every other person at the party exactly once,how many handshakes will there be ?
Which of the following is the equivalent expression for (x + 2)(x - 2)? LIKE PLS HELP
If graphed on the same grid, which of the following could be the graph of y = 2.5x2?
If graphed on the same grid, the diagrammatic representation of the parabola gives an upward curve.
What is the graph of a function?The graph of the function is a diagrammatic representation of the slope, x-intercept, and y-intercept of the given function.
From the information given:
y = 2.5x²
Here,
The domain of the given function varies from -∞ to +∞. The range is the value of x greater than or equation to 0x-intercept = (0,0)
y-intercept = (0,0)
Therefore, if graphed on the same grid, the diagrammatic representation of the parabola gives an upward curve.
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this i algebra pic click this question
Answer:The answer is 27.04
Step-by-step explanation:
The solution is in the attached
Need help with this one step by step please? A.
500 meters
B.
360 meters
C.
63 meters
D.
1,000 meters
Answer:
B. 360 meters
Step-by-step explanation:
The triangles shown are similar, so their corresponding parts are proportional.
SetupThere are a couple of useful proportions here. We can equate the ratio of horizontal segments to the ratio of diagonal segments:
(600 m)/(250 m) = ??/(150 m)
Or, we can equate the ratios of the diagonal segment to the corresponding horizontal segment:
??/(600 m) = (150 m)/(250 m)
SolutionEither way, when we multiply the equation by the denominator of ??, we get the result ...
?? = (600)(150/250) m = 360 m
The distance from F to G is 360 meters.
Each marble bag sold by Kaitlin's Marble Company contains 5 blue marbles for every 4 orange marbles. If a bag has 28 orange marbles, how many blue marbles does it contain?
Answer:
35
Step-by-step explanation:
28 Orange Marbles divided by 4 Orange Marbles =
7
5 Blue Marbles for every 7 Orange Marbles =
35
An auditorium has 288 seats distributed evenly in 9 rows. Find the unit rate of seats
per row?
Answer:
32 seats per row
Step-by-step explanation:
Total number of seats in auditorium = 288
Number of rows in which seats evenly distributed = 9
Number of seats per row = 288/9= 32