Answer:
(a) [tex]x\³ - 6x - 6[/tex]
(b) Proved
Step-by-step explanation:
Given
[tex]r = $\sqrt[3]{2} + \sqrt[3]{4}$[/tex] --- the root
Solving (a): The polynomial
A cubic function is represented as:
[tex]f = (a + b)^3[/tex]
Expand
[tex]f = a^3 + 3a^2b + 3ab^2 + b^3[/tex]
Rewrite as:
[tex]f = a^3 + 3ab(a + b) + b^3[/tex]
The root is represented as:
[tex]r=a+b[/tex]
By comparison:
[tex]a = $\sqrt[3]{2}[/tex]
[tex]b = \sqrt[3]{4}$[/tex]
So, we have:
[tex]f = ($\sqrt[3]{2})^3 + 3*$\sqrt[3]{2}*\sqrt[3]{4}$*($\sqrt[3]{2} + \sqrt[3]{4}$) + (\sqrt[3]{4}$)^3[/tex]
Expand
[tex]f = 2 + 3*$\sqrt[3]{2*4}*($\sqrt[3]{2} + \sqrt[3]{4}$) + 4[/tex]
[tex]f = 2 + 3*$\sqrt[3]{8}*($\sqrt[3]{2} + \sqrt[3]{4}$) + 4[/tex]
[tex]f = 2 + 3*2*($\sqrt[3]{2} + \sqrt[3]{4}$) + 4[/tex]
[tex]f = 2 + 6($\sqrt[3]{2} + \sqrt[3]{4}$) + 4[/tex]
Evaluate like terms
[tex]f = 6 + 6($\sqrt[3]{2} + \sqrt[3]{4}$)[/tex]
Recall that: [tex]r = $\sqrt[3]{2} + \sqrt[3]{4}$[/tex]
So, we have:
[tex]f = 6 + 6r[/tex]
Equate to 0
[tex]f - 6 - 6r = 0[/tex]
Rewrite as:
[tex]f - 6r - 6 = 0[/tex]
Express as a cubic function
[tex]x^3 - 6x - 6 = 0[/tex]
Hence, the cubic polynomial is:
[tex]f(x) = x^3 - 6x - 6[/tex]
Solving (b): Prove that r is irrational
The constant term of [tex]x^3 - 6x - 6 = 0[/tex] is -6
The divisors of -6 are: -6,-3,-2,-1,1,2,3,6
Calculate f(x) for each of the above values to calculate the remainder when f(x) is divided by any of the above values
[tex]f(-6) = (-6)^3 - 6*-6 - 6 = -186[/tex]
[tex]f(-3) = (-3)^3 - 6*-3 - 6 = -15[/tex]
[tex]f(-2) = (-2)^3 - 6*-2 - 6 = -2[/tex]
[tex]f(-1) = (-1)^3 - 6*-1 - 6 = -1[/tex]
[tex]f(1) = (1)^3 - 6*1 - 6 = -11[/tex]
[tex]f(2) = (2)^3 - 6*2 - 6 = -10[/tex]
[tex]f(3) = (3)^3 - 6*3 - 6 = 3[/tex]
[tex]f(6) = (6)^3 - 6*6 - 6 = 174[/tex]
For r to be rational;
The divisors of -6 must divide f(x) without remainder
i.e. Any of the above values must equal 0
Since none equals 0, then r is irrational
Duke works on commission basis. He gets 20% of his total sales. If he
earned $55 last week, what were his sales for the week?
Answer:
$275
Step-by-step explanation:
Find his sales by dividing 55 by 0.2:
55/0.2
= 275
So, his sales for the week were $275
Find the missing side or angle. Round to the nearest tenth. C=95° a=5 c=6 A=[?]°
Answer:
Missing angle A = 56.1° (approximately)
Step-by-step explanation:
<A = arcsin(a×sin(C)/c)
= arcsin(5×sin 95/6)
= 56.1154084° ≈ 56.1°
Answered by GAUTHMATH
Deepak wrote out the steps to his solution of the equation StartFraction 5 Over 2 minus 3 x minus 5 plus 4 x equals negative StartFraction 7 Over 4 EndFraction – 3x – 5 + 4x = –. Which step has an incorrect instruction? Step 1 Step 2 Step 3 Step 4
Answer:
Step 1 is incorrect.
Step-by-step explanation:
The given expression is :
[tex]\dfrac{5}{2}-3x-5+4x=\dfrac{-7}{4}[/tex]
In step 1, Deepak rearranged the terms in order to simplify like terms together that is followed in step 2.
o, rearranging of the terms of an expression is commutative property and not distributive property. The distributive property is generally applicable to multiplication and is given as:
[tex]x(y+z)=xy+yz[/tex]
The commutative property is given as:
x+y = y +a
Therefore, adding terms by changing their order is commutative property of addition and not distributive property. Step 1 instruction is wrong.
Answer:
step1
Step-by-step explanation: yes sirr
What is the length of DK?
A) 8.5
B) 12.7
C) 7.0
D) 3.5
Answer:
A
8.5 is the answer
If you test is going on so
all the best!
Someone tell me where everyone is going right please !!
Answer:
1. 3rd option
2. 4th option
3. 1st option
4. 2nd option
5. 2nd option
Step-by-step explanation:
1.
on changing the signs of an equality the sign gets reveresd
-6 < -2x
6 > 2x
3 > x
2.
2x - 3 > 11 - 5x
adding 5x and 3 to both sides and combining like terms(2x + 5x) + (-3 + 3) > (11 + 3) + (-5x + 5x)
7x > 14
x > 2
put of all the options x = 4 is a value greater than 2 thus satisfying the condition.
3.
6k + 10. 5 = 3k + 12
subtracting 3k and 10.5 from both the sides while combining like terms(6k - 3k) + (10.5 - 10.5) = (3k - 3k) + (12 - 10.5)
3k = 1.5
k = 0.5
4.
y + 3 = -y + 9
adding y and subtracting 3 from both sides(y + y) + (3 - 3) = (-y + y) + (9 - 3)
2y = 6
y = 3
5.
-9x + 1 = -x + 17
adding x and subtracting 1 from both sides
(-9x + x) + (1 -1) = (-x +x) + (17 -1)
-8x = 16
multiplying the equation by -1
8x = -16
diving both sides by 8
x = -2
Answer:
[tex]1. \: \: x < 3[/tex]
[tex]2. \: \: 2[/tex]
[tex]3. \: \: k = 0.5[/tex]
[tex]4. \: \: y = 3[/tex]
[tex]5. \: \: x = 12[/tex]
Step-by-step explanation:
[tex]1. \: \: - 2(5 - 4x) < 6x - 4 \\ - 10 + 8x < 6x - 4 \\ - 10 + 2x < - 4 \\ 2x < 6 \\ x < 3[/tex]
[tex]2. \: \: 2x - 3 > 11 - 5x \\ 7x - 3 > 11 \\ 7x > 14 \\ x = 2[/tex]
[tex]3. \: \: 6k + 10.5 = 3k + 12 \\ 3k + 10.5 = 12 \\ 3k = 1.5 \\ k = 0.5[/tex]
[tex]4. \: \: y + 3 = - y + 9 \\ 2y + 3 = 9 \\ 2y = 6 \\ y = 3[/tex]
[tex]5. \: \: - 9x + 1 = - x + 17 \\ - 8x + 1 = 17 \\ - 8x = 16 \\ x = - 2[/tex]
Hope it is helpful....write a function rule for the following data
If line n bisects CE, find CD.
If it bisect, then using the Line Bisector Theorem, it must be equal.
So x+6=4x-21
get like terms on 1 side
x+6=4x-21
-x. +21
27= 3x
x=9
plug the x in
CD= 9+6
CD=15
The value of CD such that line n is bisecting line segment CE is 15.
What is a line segment?A line section that can connect two places is referred to as a segment.
The line is here! It extends endlessly in both directions and has no beginning or conclusion.
Given that line, n is bisecting the line segment CE at D
CD = DE
Given CD = x + 6 and DE = 4x - 21
x + 6 = 4x - 21
4x - 21 = x + 6
4x - x = 6 + 21
3x = 27
x = 9
Therefore CD = 9 + 6 = 15
Hence "The value of CD such that line n is bisecting line segment CE is 15".
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1. What is the solution of the system of equations?
y = -3x + 8
y = -5x - 2
Answer:
This system of equations has no solution since the first one says:
x
2
−
y
=
8
while the second one gives us:
x
2
−
y
=
−
(
y
−
x
2
)
=
−
0
=
0
and
8
≠
0
Step-by-step explanation:
Helpppppp and explain tooo thank you :)
Answer:
{5, 6, 7}
Step-by-step explanation:
When we have a given relation, the domain is the set of inputs, and the range as the set of the outputs.
so for a function f(x), and a domain {a. b. c}
The range is:
{f(a), f(b), f(c)}
In this case, we have:
f(x) = x + 6
and the domain is {-1, 0, 1}
Then the range is:
{ f(-1), f(0), f(1) }
{-1 + 6, 0 + 6, 1 + 6}
{5, 6, 7}
The correct option is the third one.
What is the solution to the equation One-fourth x + 2 = negative StartFraction 5 Over 8 EndFraction x minus 5?
x = negative 8
x = negative 7
pls hurry
x = 7
x = 8
Answer:
c = 24
Step-by-step explanation:
Find the height h of the triangle.
Answer:
A, 12 times the square root of 3
Step-by-step explanation:
24 sqaured plus 12 squared=432
square root of 432=20.9
12 times the sqaure root of 3=20.9
Convert 43.81 ounces to grams
16 ounces = 1 pound
1 kilogram ≈ 2.2 pounds
Round your answer to the nearest hundredth
Answer:
1241.99
Step-by-step explanation:
1241.993 grams is the original answer but rounded to the nearest hundred is 1241.99
Can someone help me with this math homework please!
Answer:
Check off both of the boxes
Step-by-step explanation:
In your reponse you included that most ancient societies didn't have symbols for math operations and didn't have symbols for variables/unknown quantities.
Hope it helps (●'◡'●)
1. If f(x) = 6x + 13x + 2x - 5 and f(-1) = 0, find the
factors of f(x)
Answer:
-26
Step-by-step explanation:
Insert the f(-1) into the equation.It'll be f(-1) = 6(-1) + 13(-1) + 2(-1) - 5After you times all of it. You'll get 7-13-2-5Solve it and you'll get -26Solve 3x – ly= 11 and -2x – 4y=-26 by elimination
If anyone can help me with this it’d be appreciated
Answer:
(5, 4 )
Step-by-step explanation:
Given the 2 equations
3x - y = 11 → (1)
- 2x - 4y = - 26 → (2)
Multiplying (1) by - 4 and adding to (2) will eliminate the y- term
- 12x + 4y = - 44 → (3)
Add (2) and (3) term by term to eliminate y
- 14x + 0 = - 70
- 14x = - 70 ( divide both sides by - 14 )
x = 5
Substitute x = 5 into either of the 2 equations and solve for y
Substituting into (1)
3(5) - y = 11
15 - y = 11 ( subtract 15 from both sides )
- y = - 4 ( multiply both sides by - 1 )
y = 4
solution is (5, 4 )
A study of homeowners in the 5th congressional district in Maryland found that their annual household incomes are normally distributed with a mean of $41,182 and a standard deviation of $11,990 (based on data from Nielsen Media Research). If an advertising campaign is to be targeted at those whose household incomes are in the top 20%, find the minimum income level for this target group.
Answer:
The minimum income level for this target group is of $51,253.6.
Step-by-step explanation:
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Mean of $41,182 and a standard deviation of $11,990
This means that [tex]\mu = 41182, \sigma = 11990[/tex]
Find the minimum income level for this target group.
The 100 - 20 = 80th percentile, which is X when Z has a p-value of 0.8, so X when Z = 0.84.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.84 = \frac{X - 41182}{11990}[/tex]
[tex]X - 41182 = 0.84*11990[/tex]
[tex]X = 51253.6[/tex]
The minimum income level for this target group is of $51,253.6.
The graph of a line goes down and to the right when:
A. there is no coefficient of x.
B. the coefficient of x is 0.
c. the coefficient of xis positive.
D. the coefficient of x is negative.
Answer:
The answer is D.
Step-by-step explanation:
When a graph of a line goes down and to the right, it shows that as the value of x increases, the value of y decreases. This represents a negative coefficient for x.
The graph is a drawn out version of the question.
Te graph of a line goes down and to the right when the coefficient of x is negative, option (D) is correct.
What is a straight line?A straight line is a combination of endless points joined on both sides of the point.
The slope 'm' of any straight line is given by:
[tex]\rm m =\dfrac{y_2-y_1}{x_2-x_1}[/tex]
y = mx + c
We have:
The graph of a line goes down and to the right when.
The orientation of the line depends on the slope.
If the slope of the line is positive, the line will go up.
If the slop of the line is negative, the line will go down.
Thus, the graph of a line goes down and to the right when the coefficient of x is negative, option (D) is correct.
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hello! I'm having a bit of trouble trying to solve this, could anyone help explain it to me?
[tex]\displaystyle\ 60^2 =36(36+x)\\\\1296+36x=3600\\\\36x=2304\\\\\boxed{x=64}[/tex]
Step By Step Explanation:
[tex] {60}^{2} = 36(36 + x)[/tex]Step 1Calculate 60 to the power of 2 and get 3600.
[tex]3600 = 36(36 + x)[/tex]Step 2Use the distributive property to multiply 36 by 36 + ×.
[tex]3600 = 1296 + 36 \times [/tex]Step 3Swap sides so the all variable terms are on the left hand side.
[tex]1296 + 36 \times = 3600[/tex]Step 4Subtract 1296 from both sides.
[tex]36 \times = 3600 - 1296[/tex]Step 5Subtract 1296 from 3600 to get 2304.
[tex]36 x = 2304[/tex]Step 6Divide both side by 36.
[tex] \times = \frac{2304}{36} [/tex]Step 7Divide 2304 by 36 to get 64
My Answer is
[tex]\color{GREEN} \times = 64[/tex]Which operation will solve the following word problem? Jeff earns $14.00 per hour, Tom earns half as much as Jeff. How much does Tom earn per hour?
a.) Multiplication
b.) Subtraction
c.) Addition
d.) Division
Answer:
Multiplication
Amount Tom earns per hour = $7
Step-by-step explanation:
Amount Jeff earns per hour = $14
Amount Tom earns per hour = 1/2 × $14
Note: 1/2 = 0.5
Amount Tom earns per hour = 1/2
× $14
= 0.5 × $14
= $7
Amount Tom earns per hour = $7
What is the answer to this question
Answer:
26 m is the answer to the question
Solve for the dimensions of the area model below and then write an equation showing how the area as a sum is equivalent to the area as a product.
Full question:
For each multiplication expression, sketch an area model. Label the dimensions and the area of each part. Then write an equation showing that the area as a product equals the area as a sum. a. (x+1)(x+2), b. 3(2x+5), c. (2x-3)(x+2), d. (x-1)(y-1), e. -2y(y+3), f. (-x+1)(3x+y-4)
Answer and explanation:
a. (x+1)(x+2)= x×x+x×2+1×x+1×2
The dimensions (length and width) is x+1 and x+2
b. 3(2x+5) = 3×2x+3×5
The dimensions is 3 and 2x+5
c. (2x-3)(x+2)= 2x×x+2x×2-3×x-3×+2
The dimensions are 2x-3 and x+2
d. (x-1)(y-1)= x×y+x×-1-1×y-1×-1
Dimensions are x-1 and y-1
e. -2y(y+3)= -2y×y-2y×3
Dimensions are -2y and y+3
f. (-x+1)(3x+y-4)= -x×3x-x×y-x×-4+1×3x+1×y+1×-4
Dimensions are -x+1 and 3x+y-4
The formula for the circumference of a circle is R = c/2(pi)
Find the radius of a circle that has a circumference of 16(pi)
A) r = 4
B) r = 8
C) r = 12
D) r = 16
what rule changes input to output numbers
Answer:
C. 3x - 5
Step-by-step explanation:
The inputs are the values of x and the outputs are the values of y.
checking all options by placing any output as x and seeing if it gives the same output as in the table.
All options except C do not give the same output as in the table (ill show the working for option C you can do for yourself the rest ^^")
Let's take x = 4
placing it in 3x - 5
output (y) = 3(4) - 5
= 12 - 5
= 7
same as the output given in the table so the answer is C.
A sweater is on sale for $45.00, down from $75.00. What is the
percent of discount of the sweater?
A. 50%
B. 30%
C. 55%
D. 40%
Answer:
D
Step-by-step explanation:
The percentage of discount on the sweater will be 40%. The correct option is D.
What is the percentage?The percentage is defined as representing any number with respect to 100. It is denoted by the sign %. The percentage stands for "out of 100." Imagine any measurement or object being divided into 100 equal bits.
Given that a sweater is on sale for $45.00, down from $75.00. The percentage discount will be calculated as:-
Percentage discount = (75 - 45) / 75
Percentage discount = ( 30 / 75 )
Percentage discount = 0.4 x 100 = 40%
Therefore, the percentage of discount on the sweater will be 40%. The correct option is D.
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PLEASE HELP! PLEASE SHOW WORK
Use the following expression to answer this three part question:
f(x) = 2x2 + 4x − 6
Part A: What are the x-intercepts of the graph of f(x)? Show your work.
Part B: Is the vertex of the graph of f(x) going to be a maximum or minimum? What are the coordinates of the vertex? Justify your answers and show your work.
Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph.
Answer:
Step-by-step explanation:
The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function.
Which correctly completed the statement given below?
Answer:
C = <CDB, SAS
You can prove side BD because of the reflexive property (using two of the same sides to prove something) and you are given the other side and the angle. So, since you have a side, angle, and side in that order, you can prove it with SAS.
HELPPPPPPPPPPPPPPPP SOMEONE !!!What is the rate of change of the amount earned with respect to hours worked for this function?
hours per dollar
hours per dollar
3 dollars per hour
13 dollars per hour
Answer:
$13 per hour
Step-by-step explanation:
This is because 26÷2=13, 65÷5=13
Help im stuck on this problem
Answer:
J.
[tex]{ \bf{m \angle A}} = { \bf{180 \degree -48 \degree }} \\ { \tt{m \angle A = 132 \degree}}[/tex]
If an item weighs 45kilograms what is its weight in pounds
Evaluate i^15 i^12
Show work
Answer: 15 = -i & 12=1
use the pattern : i, -1, -i, 1
Step-by-step explanation:
i = [tex]\sqrt{-1}[/tex]
[tex]i^{2}[/tex] = [tex]\sqrt{-1}[/tex]
[tex]i^{3}[/tex] = [tex]\sqrt{-1}[/tex]
[tex]i^{4}[/tex] = [tex]\sqrt{-1}[/tex]
the pattern just repeats from here
5 = i
6 = -1
7 = -i
8 = 1
9 = i
10 = -1
11 = -i
12 = 1
13 = i
14 = -1
15 = -i
16 = 1