Pre calculus 6a. Consider the equation x^5 - 3x^4 + mx^3 + nx^2+ ox + q = 0, where m, n, P, q € R.The equation has three distinct real roots which can be written as log2a, log2b and log2C.The equation also has two imaginary roots, one of which is di where dE R.Show that abc = 8.6b. The values a, b, and C are consecutive terms in a geometric sequence. Show that one of the real roots is equal to 1.6c. Given that q = 8d^2, find the other two real roots.
Answers
We have a fifth degree polynomial:
[tex]x^5-3x^4+mx^3+nx^2+px+q=0[/tex]This polinomial has 3 real roots, that can be expressed as: log2(a), log2(b) and log2(c).
Also, it has two imaginary roots, one of which is di (they have to be conjugate, so the other imginary root is -di).
We have to show that abc = 8.
If we consider the information given, we have some information about all the roots.
We can rewrite the polynomial in factorized form as:
[tex]\begin{gathered} (x-\log _2a)(x-\log _2b)(x-\log _2c)(x-di)(x+di)=0 \\ (x-\log _2a)(x-\log _2b)(x-\log _2c)(x^2+d^2)=0 \end{gathered}[/tex]As the polynomial is defined for real numbers, we can write a polynomial with only the real roots as:
[tex](x-\log _2a)(x-\log _2b)(x-\log _2c)=0[/tex]Then, we can relate the roots as:
[tex]\begin{gathered} 2^{(x-\log _2a)(x-\log _2b)(x-\log _2c)}=2^0 \\ 2^{(x-\log _2a)}\cdot2^{(x-\log _2b)}\cdot2^{(x-\log _2c)}=1 \\ \frac{2^x}{2^{\log_2a}}\cdot\frac{2^x}{2^{\log_2a}}\cdot\frac{2^x}{2^{\log_2a}}=1 \\ \frac{2^{3x}}{a\cdot b\cdot c}^{}=1 \\ abc=2^{3x} \\ abc=2^3\cdot2^x \\ abc=8\cdot2^x \end{gathered}[/tex]Related Questions
There are 100 centimeters in a meter .63 centimeters is what fraction of a mater?
Answers
Since 1 meter = 100 cm
Then to change from cm to meter, divide by 100
Since there is 63 cm, then divide it by 100
[tex]\frac{63}{100}[/tex]The fraction of a meter is 63/100
Find the inverse y = (x - 2)³ +6
Ox-6+2=y
Ox-6+2=y
Ox-6+2=y
Answers
Answer:
y = [∛(x-6)] - 2
Step-by-step explanation:
To find the inverse :
Solve for xSwap x and ySolve for x using balancing method or solve and ping method:
y= (x-2)³+6
y - 6 = (x+2)³
∛(y-6) = x+2
[∛(y-6)] - 2 = x
x = [∛(y-6)] - 2
Swap x and y :
y = [∛(x-6)] - 2
This is the inverse of y = (x - 2)³ +6
Hope this helped and have a good day
If f(x) = x² + 5 and g(x) = 3x, find (f o g)(x) and (g o f)(x) .
Answers
Answer:
[tex](f \circ g)(x) = 9x^2 + 5\\\\\\(g \circ f)(x) = 3x^2 + 15[/tex]
Step-by-step explanation:
We are given
[tex]f(x) = x^2+5\\\\g(x) = 3x\\\\(f\circ g)(x) = f(g(x))\\\\[/tex]
To find this, wherever you see an x in f(x) substitute the expression in g(x)
[tex](f\circ g)(x) = f(g(x))\\\\= f(3x)\\\\= (3x)^2 + 5\\\\=9x^2 + 5\\\\\\[/tex]
To find [tex](g \circ f)(x) = g(f(x))\\\\[/tex]
Wherever there is an x in the expression for g(x) substitute that x with the expression in f(x)
[tex](g \circ f)(x) = g(f(x))\\\\\\= g(x^2 + 5) = 3(x^2 + 5) \\\\= 3x^2 + 15[/tex]
Which set of measurement could be the interior angle measures of triangle.A. 10°, 10°, 160°B. 15°, 75°, 90°C. 20°, 80°, 100°D. 35°, 35°, 105°E. 60°, 60°,60°
Answers
The sum of the interior angles of a triangle must add 180°.
C : 20 + 80 + 100 = 200 NO
D : 35+35+105 = 175 NO
A : 10+ 10+160 = 180 Yes
B : 15 +75 + 90 = 180 Yes
E :60+60+60 = 180 Yes
find the percent of change. Round to the nearest whole percent original:26 new:30
Answers
We want to find the percentage change;
We can do that using the formula;
[tex]\text{ \%P}=\frac{New-Original}{\text{Original}}\times100\text{\%}[/tex]Given:
[tex]\begin{gathered} \text{Original = 26} \\ \text{New = 30} \end{gathered}[/tex]Substituting the given values;
[tex]undefined[/tex]Graph the line that has an x-intercept at (3, 0) and a y-intercept at (0, 1). What is the slope of this line
Answers
Answer:
[tex] - \frac{1}{3} [/tex]
Step-by-step explanation:
The slope of a line passing through two points is given by
[tex]\boxed{\text{Slope} = \frac{y_1 - y_2}{x_1 - x_2} }[/tex]
where [tex](x_1,y_1)[/tex] is the 1st coordinate and [tex](x_2,y_2)[/tex] is the 2nd coordinate
Given: (3, 0) and (0, 1)
Slope
[tex] = \frac{0 - 1}{3 - 0} [/tex]
[tex] = - \frac{1}{3} [/tex]
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Suppose the scores of students on an exam are normally distributed with a mean of 208 and a standard deviation of38. According to the empirical rule, what percentage of students scored between 170 and 246 on the exam?Answer:%
Answers
We going to use the normal distribution graph
First we got a mean of 208 and a standard deviation of 38
Now
In the middle of the graph, we have the mean, then every section is added/subtracts an std to the mean. In our case with one std we get the values that we are looking.
Answer: The percentage of students scored between 170 and 246 on the exam is 68%.
(11^9) over (15^6) to the 4th power
A 11^6 + 4/15^6 +4
B 11^36/15^24
C 11^9x4/15^6x4
D 11/13/15^10
Answers
The expression 11⁹ / 15⁶ to the fourth power is equivalent to 11³⁶ / 15²⁴
How to solve an equationAn equation shows the relationship between two or more numbers and variables.
Giving the exponents:
11⁹ / 15⁶ to the fourth power
Representing this equation in exponent form:
(11⁹ / 15⁶)⁴
Applying the law of exponents:
(11⁹)⁴ / (15⁶)⁴
= 11³⁶ / 15²⁴
The equation (11⁹ / 15⁶)⁴ gives 11³⁶ / 15²⁴
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During a sale, Neil found digital cameras on sale for $204 that had previously cost $600. What percentage is the discount? Write your answer using a percent sign (%).
Answers
We need to find the percentage of the discount over a camera that had previously cost $600 and now is on sale for $204.
In order to find that percentage, we need to find the total discount (previous cost minus cost on sale) and then divide the result y the previous cost.
The total discount is:
[tex]\$600-\$204=\$396[/tex]Then, dividing this result by the previous cost, we obtain:
[tex]\frac{\$396}{\$600}=0.66[/tex]Now, we can write it as a percentage:
[tex]0.66=\frac{66}{100}=66\%[/tex]Answer: 66%
Select the postulate that is illustrated for the real numbers.
25 + 0 = 25
A The commutative postulate for multiplication
B The multiplication inverse
C The addition inverse postulate
D Commutative postulate for addition
E The distributive postulate
F Additive identity
G Multiplication by one
Answers
Answer:
Additive Identity
Step-by-step explanation:
Hello!
Adding 0 to any number will give you that number itself, as proven by adding 0 to 25, which gave us 25.
This is the identity property of addition, as the output would be identical to the sum of the non-zero terms.
We can also experiment with these:
21 + 4 + 0 = 21 + 4 = 257(3 + 0) = 7(3) = 21simplify (3^-2)^4 A 1/3^2B 3^2C 1/3^8D 3^8
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Ok, so
We're going to simplify the following expression:
[tex](3^{-2})^4[/tex]Remember that if we have two exponents elevating each other in the same base, they multiply.
So, this is equivalent to write:
[tex](3^{-2})^4=(3)^{-8}[/tex]Now, we can rewrite the last expression in this new one:
[tex](3^{-2})^4=(3)^{-8}=\frac{1}{3^8}[/tex]R= {y | y is an integer and -5 ≤ y ≤ -4}
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The R= {y | y is an integer and -5 ≤ y ≤ -4} is null set because there is no integers between -5 and -4.
The set is
R= {y | y is an integer and -5 ≤ y ≤ -4}
An integer is a combination of number zero, all the positive numbers and negative numbers without any fraction.
Any set that does not contains any numbers or elements is called null set, which is also called empty set or void set. Null set can be denoted as {} or ∅.
The set is R= {y | y is an integer and -5 ≤ y ≤ -4}
There is no integers between -5 and -4, therefore it is a null set
Hence, The R= {y | y is an integer and -5 ≤ y ≤ -4} is null set because there is no integers between -5 and -4.
The complete question is :
Write the set in roaster form R= {y | y is an integer and -5 ≤ y ≤ -4}
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An invester invested a total of 3,300 in two mutual funds. One fund earned a 5% profit while the other earned a 2% profit. If the investor's total profit was $126, how much was invested in each mutual fund?
Answers
Given
Total investment of $3,300
5% profit and 2% profit totaling $126
[tex]\begin{gathered} \text{Let} \\ x\text{ be the investment on 5\% profit} \\ y\text{ be the investment on 2\% profit} \end{gathered}[/tex]The equations therefore will be
[tex]\begin{gathered} x+y=3300\text{ based on the total amount of investment} \\ 0.05x+0.02y=126\text{ based on the investor's total profit} \end{gathered}[/tex]Use substitution method using the first equation
[tex]\begin{gathered} x+y=3300 \\ y=3300-x \\ \\ \text{Substitute }y\text{ to the second equation} \\ 0.05x-0.02y=126 \\ 0.05x-0.02(3300-x)=126 \\ 0.05x-66-0.02x=126 \\ 0.05x-0.02x=126-66 \\ 0.03x=60 \\ \frac{0.03x}{0.03}=\frac{60}{0.03} \\ \frac{\cancel{0.03}x}{\cancel{0.03}}=\frac{60}{0.03} \\ x=2000 \end{gathered}[/tex]Now that we have solve for x, substitute it to the first equation to get the value of y
[tex]\begin{gathered} x+y=3300 \\ 2000+y=3300 \\ y=3300-2000 \\ y=1300 \end{gathered}[/tex]Therefore, the amount invested in mutual fund that earned 5% was $2000, and the amount invested that earned 2% was $1300.
When Ross solved the equation 12 = 3x, he made a mistake. Hiswork is shown below. What mistake did Ross make?3x = 123x - 3= 12-3x=9
Answers
The mistake Ross has made was in line 2, and it is that we can only combine like terms, that is, terms that have the same variable at the same power. So
[tex]3x-3\ne x,[/tex]We cannot performe this operation.
Solve the literal equation to isolate V2
P1V1=P2V2
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Answer:
Step-by-step explanation:
Find the value of x.1714X = 93x = 93x= 3x = 485
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Given a right angle triangle:
The hypotenuse of the triangle = 17
The legs of the triangle are the sides 14 and x
We will find x using the Pythagorean theorem as follows:
[tex]x^2+14^2=17^2[/tex]Solve for x:
[tex]\begin{gathered} x^2=17^2-14^2=93 \\ \\ x=\sqrt[]{93} \end{gathered}[/tex]So, the answer is the first option
[tex]x=\sqrt[]{93}[/tex]what is the x and y intercept of -8x+6y=24
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we have the following:
[tex]\begin{gathered} -8x+6y=24 \\ 6y=8x+24 \\ y=\frac{8}{6}x+\frac{24}{6} \\ y=\frac{4}{3}x+4 \end{gathered}[/tex]therefore, x intercept:
[tex]\begin{gathered} y=0 \\ 0=\frac{4}{3}x+4 \\ \frac{4}{3}x=-4 \\ x=\frac{-4\cdot3}{4} \\ x=-3 \\ (-3,0) \end{gathered}[/tex]y intercept:
[tex]\begin{gathered} x=0 \\ y=\frac{4}{3}\cdot0+4 \\ y=4 \\ (0,4) \end{gathered}[/tex]The answer is:
x-intercept: (-3,0)
y-intercept: (0,4)
Y-int (0,4)
What is the distance between points A (7,3) and B (5,-1)?1) √102) √123) √144) √20
Answers
Given the points:
[tex]\begin{gathered} A(7,3) \\ B(5,-1) \end{gathered}[/tex]You can use the formula for calculating the distance between two points:
[tex]d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Where the points are:
[tex]\begin{gathered} (x_1,y_1)_{} \\ (x_2,y_2) \end{gathered}[/tex]In this case, you can set up that:
[tex]\begin{gathered} x_2=x_B=5 \\ x_1=_{}x_A=7 \\ y_2=y_B=-1 \\ y_1=x_A=3_{} \end{gathered}[/tex]Then, substituting values into the formula and evaluating, you get:
[tex]d_{AB}=\sqrt[]{(5-7)^2+(-1-3)^2}=\sqrt[]{(-2)^2+(-4)^2}=\sqrt[]{4+16}=\sqrt[]{20}[/tex]Therefore, the answer is: Option 4.
Compute the following probabilities. What is the probability that a randomly selected person recieved a flu vaccination 117/194
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The probability a random selected person will receive a flu is 0.6031
What is Probability?Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty.
To solve this problem, we just simply need to just find the decimal value of the given fraction.
[tex]\frac{117}{194} = 0.6031[/tex]
The probability of the event occurring is 0.6031
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complete question
Compute the following probabilities. What is the probability that a randomly selected person received a flu vaccination 117/194. Write the probability value in decimal.
Which ratio is less than 7/15?9/152/53/524/45
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SOLUTION:
Case:
Given:
Required:
Method:
Final answer:
A sample of people who recently hired an attorney yielded the following information about their attorneys.
Number of People
Would You Recommend Your Attorney to a Friend?
Yes
No
Not sure
Three people who provided information for the table were selected at random. Determine the probability that the first two would not recommend their attorney and the third is not sure if he or she would recommend their attorney.
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As assistant manager of a soccer specialty store you have been asked to review the inventory costs associated with the store's best selling soccer shoe. The information you have available concerning this shoe follows: Average Demand =18 pairs per week; Standard Deviation of demand = 6 pairs Lead time for ordering shoes= 1 week; Ordering cost = $32 per order Cost of a pair = $90; Carrying cost per pair per year = 10% of cost of per pair Service level =98%, Z =2.05; The store operates 50 weeks per year.
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The annual demand when the manager reviews the stock is 900 units.
What is the annual demand?Mean demand, d = 18 units per week
Standard deviation of weekly demand, σd = 6 units per week
Lead Time, LT = 1 week
Order Cost, S = $32 per order
Unit Cost, C = $90 per unit
Holding Cost, H = 10% of C = $9 per unit per year
Service level, SL = 98%
z score = NORMSINV(SL) = NORMSINV(98%) = 2.05
Operating weeks = 50 weeks per year
The annual Demand, D will be:
= 50*18
= 900 units
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Solve for x. Then find m
(8x+4)°
(10x-6)°
Both lines are intersecting and the two equations are vertical pairs
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For the vertical angles, the value of x is found as 5. The measure of the angle ∠QRT = 44° for the two intersecting lines.
What is referred as the vertical angles?When two lines intersect at a point, vertical angles are formed. They are always on equal footing. In other words, four angles are formed anytime two lines pass or intersect. We can see that two opposite angles are equal, and these are referred to as vertical angles.For the given pair of angles in the question.
Two lines are intersecting to form two equations are vertical pairs.
∠QRT = ∠VRS (vertical angles)
Put the values.
8x + 4 = 10x - 6
Simplifying.
2x = 10
x = 5
Put the values of 'x' in the angle.
∠QRT = 8x + 4
∠QRT = 8×5 + 4
∠QRT = 44
Thus, the measure of the angles ∠QRT is found as 44°.
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What is the relationship between Za and Zb?BCbYXDChoose 1 answer:Vertical anglesComplementary anglesSupplementary anglesNone of the above
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Relationship between
are not vertical angles because vertical angles are opposite by vertex
are not complementary because they dont sum 90°
are not suplementary because they dont sum 180
then right option is none of the above
sal's orange stand made that show a relationship between the total number of oranges and the number of boxes they sell. Each box has the same number of oranges. Sal made an error in one of the boxes.
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Determine the number of oranges in every box.
For row number 1:
[tex]\frac{13}{2}=6.5[/tex]For row number 2:
[tex]\frac{65}{5}=13[/tex]For row number 3:
[tex]\frac{91}{7}=13[/tex]For row number 4:
[tex]\frac{130}{10}=13[/tex]For row number 5:
[tex]\frac{156}{12}=13[/tex]Since number of oranges in one box in every row except (Row number 1) is 13. In row number 1, number of oranges in one box is 6.5. So Sal's error is in row number 1.
Answer A: Row number 1
PART B:
As number of orange in one box in every row (Except row 1) is 13. So every box in row number 1, must contain 13 oranges. There are 2 boxes in row number 1.
Determine the number of oranges in row number 1.
[tex]\begin{gathered} N=13\cdot2 \\ =26 \end{gathered}[/tex]So correct number of oranges in row number 1 is 26.
Answer B: 26.
need help asap !!!!!!!
Answers
Answer:
B see my photo. hope it helps
Answer:
-3/2
Step-by-step explanation:
-2/3 × 1/6 = -12/3
3/4 × -12/3 = -36/12 = -3
-3 × 1/2 = -3/2
which expression is equivalent to [tex] \sqrt{15} \times \sqrt{12} [/tex]
Answers
Consider that √15 and √12 can be written as follow:
√15 = √(3*5)
√12 = √(3*4)
Moreover, √15*√12 = √(15*12). Then, you can write:
√(15*12) = √(3*5*3*4) = √(3^2*2^2*5) = 3*2√5 = 6√5
Hence, the equivalent expression is 6√5
1. Perform the indicated operations.
(8x2 - 16x - 9) - (14x² - 17x + 9)
r 1 Assessment FINAL Version
-6x²+x-18
6x² - x
6x²+x+18
-6x²-x-18
Answers
The final answer after subtraction is - 6x² + x - 18.
What is quadratic equation?
The polynomial equation whose highest degree is two is called a quadratic equation or sometimes just quadratics. It is expressed in the form of:
ax² + bx + c = 0
where x is the unknown variable and a, b and c are the constant terms.
What is subtraction?
Subtraction represents the operation of removing objects from a collection. The minus sign signifies subtraction −.
Given,
(8x2 - 16x - 9) - (14x² - 17x + 9)
Subtract
= 8x² - 16x - 9 - 14x² + 17x + 9)
= 8x² - 16x - 9 - 14x² + 17x - 9
= - 6x² + x - 18
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Question 16 of 22What is the product of (2x+3x - 1) and (3x+5)?A. 6x3 +19x2 +12x-5B. 6x3 + 10x2 +15x-5C. 6x3 +9x? - 3x-5D. 6x3 +19x2 - 12x+5
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Given: (2x+3x - 1) and (3x+5)
The product of them will be as follows:
[tex]undefined[/tex]if I habe 1000cat a 300cat bowl how many cats have to share
Answers
- The first figure corresponds to the clue#4. Because this figure has a the form of a pyramid.
- The second figure has the following area:
A = 2(4 cm)(5 cm) + 2(2 cm)(4 cm) + 2(5 cm)(2 cm)
A = 40 cm² + 16cm² + 20cm²
A = 76 c²
Hence, the second figure corresponds to clue#3
- The third figure is a triangular prism. Then, it corresponds to clue#2
- The four figure is composed by 6 squares. Its total area is the area of one square multipled by 6. Then, it correspondds to clue#1