Factοring is the prοcess οf reversing the distributive prοperty sο that a pοlynοmial can be written as the prοduct οf simpler pοlynοmials is true.
What is factοring?Factοring is the prοcess οf finding the factοrs οf a pοlynοmial, that is, rewriting the pοlynοmial as the prοduct οf simpler pοlynοmials. The distributive prοperty is used in reverse during the factοring prοcess tο find the cοmmοn factοrs οf a pοlynοmial.
Fοr example, cοnsider the pοlynοmial expressiοn [tex]2x^2 + 6x[/tex]. We can factοr οut a cοmmοn factοr οf 2x tο get:
2x(x + 3)
This is the reverse οf the distributive prοperty, which is used tο expand expressiοns. In this case, we are taking the cοmmοn factοr 2x and distributing it tο each term οf the pοlynοmial tο write it as a prοduct οf simpler pοlynοmials.
Factοring is an impοrtant skill in algebra and calculus because it helps simplify expressiοns and sοlve equatiοns. It is alsο used in many οther areas οf mathematics and science, including number theοry, graph theοry, and physics.
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Need urgently!! Fast asap
Answer:
1a) x=40°
2a) x=30°
Step-by-step explanation:
Use the fact that the sum of the angles of a triangle is always 180°.
So for 1a), x + x+40 + x+20 = 180
Simplify to
3x + 60 = 180
3x = 120
x = 40
apply the same logic to all the others.
For circles, the sum of the angles is 360°, so for 2b)
5x+3x+4x = 360 =>
12x = 360
x = 30°
Find the principal.
Annual rate of interest
=
6. 5
%
=6. 5%equals, 6, point, 5, percent
Period
=
4
=4equals, 4 years
Total interest
=
1222
=1222equals, 1222 rupees
Principal
=
=equals
rupees
The principal is $467.69 rupees (rounded to two decimal places).
The concept of simple interest is a type of interest calculation based on a fixed percentage rate applied to an initial amount of money, called the principal, for a specific period of time
To find the principal, we can use the formula:
Total Interest = (Principal x Rate x Time) / 100
Where,
Rate = Annual Rate of Interest
Time = Period in years
Plugging in the given values, we get:
1222 = (Principal x 6.5 x 4) / 100
Simplifying the equation:
1222 = (26 x Principal) / 10
Multiplying both sides by 10/26:
Principal = $467.69
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The sides of a triangle are 43,96 , and 89. Use the Pythagorean Theorem to determine if the triangle is right, acute, or obtuse.
Using Pythagorean theorem, the triangle is an acute triangle.
What is Pythagorean theorem?
The Pythagorean theorem is a fundamental result in Euclidean geometry that describes the relationship between the three sides of a right-angled triangle.
In equation form, this can be written as:
[tex]c^2 = a^2 + b^2[/tex]
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides (also known as the legs) of the right-angled triangle.
If the triangle is not a right triangle, then the inequality [tex]c^2 < a^2 + b^2[/tex]holds for an acute triangle and [tex]c^2 > a^2 + b^2[/tex] holds for an obtuse triangle.
Using this formula, we can check if the given triangle is right, acute, or obtuse:
a = 43, b = 96, and c = 89
[tex]c^2 = 89^2 = 7921[/tex]
[tex]a^2 + b^2 = 43^2 + 96^2 = 1849 + 9216 = 11065[/tex]
Since [tex]c^2 < a^2 + b^2[/tex], we can see that:
[tex]c^2 < a^2 + b^2[/tex]
So the triangle is an acute triangle.
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suppose that a new employee starts working at $7.22 per hour and receives a 4% raise each year. After time t, in years, his hourly wage is given by the equation y=$7.03(1.03)^t. find the amount of time after which he will be earning $10.00 per hour.
After approximately 9.95 years, the employee will be earning $10.00 per hour.
What is property of logarithms?
The properties of logarithms are a set of rules that can be used to manipulate logarithmic expressions, including the product, quotient, power, and change of base properties.
We are given that the employee's hourly wage after t years is given by the equation [tex]y=7.03(1.03)^t[/tex]. We need to find the amount of time t after which he will be earning $10.00 per hour. So we set y = $10.00 and solve for t as follows:
$10.00 = $[tex]7.03(1.03)^t[/tex]
Divide both sides by $7.03:
1.4246 = [tex]1.03^t[/tex]
Take the natural logarithm of both sides:
ln(1.4246) = ln([tex]1.03^t[/tex])
Using the property of logarithms that ln([tex]a^b[/tex]) = b * ln(a), we can simplify the right-hand side:
ln(1.4246) = t * ln(1.03)
Solve for t:
t ≈ 9.95 years
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A population of 250 wild turkeys decreases by 2. 2% per year. At the end of 8 years, there will be approximately 209 turkeys in the population
Each year, the wild turkey population, which was once 250, drops by 2.2%. There will be around 209 turkeys left after 8 years.
Wild turkey populations are declining at a 2.2% annual pace. The initial population of 250 turkeys is anticipated to drop to around 209 turkeys after 8 years. Using the exponential decay formula, we can determine the rate of decline annually: P is the population at a certain moment, P0 is the starting population, r is the rate of reduction expressed as a decimal, and t is the time in years. P = P0 (1 - r)t. The answer to the r equation is r = (1 - (P/P0)(1/t)) = 0.022. As a result, the population is declining by 2.2% annually.
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Please please help me with this!
Answer:
288π in³
Step-by-step explanation:
V-sphere = 4/3πr³
D = 12in r = D/2 = 12/2 = 6 in
V = (4/3)(6)³π = (4/3)(216)π = 288π in³
Angle Relationships & PT Quiz Review Are the angles are complementary, supplementary, or neither?
1. m<1=91°, m <2 = 89°
2: m< 3 = 17°, m<4 = 73°
3. m< 5 = 124° m <6 = 66°
4. m <7 = 33° m< 8 = 148°
5. m< 9 = 52° m <10 = 38°
pleasee help fast
The cubic function has three roots. Verify
that x=2 is one of them and find the other two.
a) Verified
b) 4 + √7 and 4 - √7
Given a cubic function with three roots, one of the roots is given as x = 2. The other two roots can be found by using the factor theorem. Using factor theorem, we have(f(x) = x³ - 10x² + 27x - 18)(x - 2) is a factor of f(x)if and only if f(2) = 0.So, we have f(2) = 2³ - 10(2²) + 27(2) - 18 = 0. Therefore, (x - 2) is a factor of f(x) and we can find the other two roots by factorizing the cubic function. Using long division method, we get f(x) = (x - 2)(x² - 8x + 9) Now, to find the other two roots, we solve the quadratic equation x² - 8x + 9 = 0 using the quadratic formula. x = (-b ± √(b² - 4ac))/2a. We have a = 1, b = -8 and c = 9. Substituting the given values in the above formula, we get x = (8 ± √(8² - 4(1)(9)))/2= (8 ± √28)/2= 4 ± √7. Therefore, the three roots of the cubic function are 2, 4 + √7 and 4 - √7.
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1. Explain what Marc did in steps 4 and 5.
2. Why did he do this?
3. Create your own radical equation and explain how to solve it.
4. Is there an extraneous solution to your equation?
John goes to the casino and plays a slot machine. The probability that he wins on the first spin is 2/5. For all subsequent spins, the probability of John winning will be 5/6 if John wins in the preceding round, and the probability of John winning will be 1/5 if John did not win in the preceding round.(a) John plays 3 rounds. Find the probability that he wins on the third round, given that he only won two rounds of the three. (4 marks)(b) Let X denote the number of rounds John need to play before he finally wins for the first time. Comment on the suitability of modelling X after the geometric distribution. Compute P(X = 5). (3 marks)(c) John visits the casino on 20 separate days, he played exactly 10 rounds on each day. Let Y denote the number of days (out of 20) that John does not win anything. State any necessary assumptions required in order to suitably model Y after the binomial distribution. State clearly the parameters of this binomial distribution as well. (4 marks)(d) Assume that your assumptions in Question 2(c) hold. Compute E(Y) and Var(Y ). (4 marks)
For (a), the probability that John wins on the third round given that he only won two rounds of the three is 5/6.
For (b), the suitability of modelling X after the geometric distribution is appropriate because the geometric distribution is the probability distribution of the number of Bernoulli trials needed to get one success. The probability of John winning on the fifth round is $(\frac{2}{5})^4(\frac{5}{6})=\frac{50}{648}$
For (c), any necessary assumptions required in order to suitably model Y after the binomial distribution is that the trials (i.e. rounds of play) are independent, and that the probability of success is the same for each trial. The parameters of this binomial distribution are: n = 10, p = 1/5.
For (d), assuming the assumptions in Question 2(c) hold, the expected value of Y (E(Y)) is 8 and the variance of Y (Var(Y)) is 2.4.
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The Gross Domestic Product (GPD) of a country in the first quarter of 2014 was $1.1395x10^7. Rewrite the GDP in standard notation.
113,950,000,000 ([tex]1.1395x10^7[/tex] is scientific notation, so you need to move the decimal to the left 7 places and add the relevant commas to get 113,950,000,000 in standard notation).
What is notation?Notation is a method of representing information, usually in a concise and organized way. It can be used to represent mathematical equations, algorithms, music, dance, and other concepts. Notation is often used as a shorthand for communicating ideas to others, as it allows for a more precise and efficient way of conveying information. Notation also allows for the precise and consistent representation of a concept, allowing for greater accuracy and understanding.
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Consider the following theorem. Theorem: For any integer , is odd if and only if + 1 is even.
Construct a proof for the theorem by selecting sentences from the following scrambled list and putting them in the correct order. Note that each statement will be used at most once (or not at all).
Since 2 | 2( + 1), 2 | ( + 1).
Suppose is odd.
Therefore, + 1 is even.
Suppose + 1 is even.
Therefore, + 1 is odd.
Thus, for some integer , + 1 = 2 + 1.
Thus, for some integer , = 2.
Thus, for some integer , = 2 + 1.
Therefore, is odd.
Thus, for some integer , + 1 = 2.
Now + 1 = (2 + 1) + 1 = 2 + 2 = 2( + 1).
Now = (2 - 1) = 2( - 1) + 1 .
Proof:
(⇒)
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---Select---
(⇐)
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The answer of the given question based on the theorem For any integer , is odd if and only if + 1 is even the answer is for any integer , is odd if and only if + 1 is even.
What is Theorem?A theorem is statement in mathematics that has been proven to be the true based on logical reasoning and the mathematical principles. In other words, theorem is a proposition that can be demonstrated or proved to be true through logical argument.
Proof:
(⇒) Suppose is odd.
Thus, for some integer , = 2 + 1.
Now + 1 = (2 + 1) + 1 = 2 + 2 = 2( + 1).
Therefore, + 1 is even.
(⇐) Suppose + 1 is even.
Then, for some integer , + 1 = 2.
Thus, for some integer , = 2 - 1 = 2( - 1) + 1 .
Since 2 | 2( + 1), 2 | ( + 1).
Therefore, is odd.
Thus, we have shown that for any integer , is odd if and only if + 1 is even.
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Work out the value of 5 cubed - 10 squared.
Give your answer as a power of 5
Answer:
5 cubed is 5 x 5 x 5 = 125.
10 squared is 10 x 10 = 100.
So, 5 cubed - 10 squared = 125 - 100 = 25.
We can also express 25 as a power of 5 by noting that 25 = 5 squared. Therefore:
5 cubed - 10 squared = 5 squared x 5 - 10 squared = 5^(2+1) - 10^(2) = 5^3 - 10^(2) = 125 - 100 = 25.
So, the answer is 5 squared or 5^2.
Step-by-step explanation:
Answer:
25
Step-by-step explanation:
5 cubed - 10 squared
[tex]5^{3} - 10^{2} \\ = 125 - 100\\= 25[/tex]
Hope this helps!
Brainliest and a like is much appreciated!
A rectangular piece of carpet has dimensions of 6 feet by 8 feet. A larger rectangular piece of carpet has dimensions that are times longer. What area, in square feet, does the larger piece of carpet cover?
Answer:
45
Step-by-step explanation:
Pls help! This is due today!
Answer:
3(a−1)
———
a+1
Step-by-step explanation:
Suppose that 20% of all copies of grade 7 Science text book fail a certain binding strength test. Let X denote the number among 15 randomly selected copies that fail the test.
b. What is the probability that at least 8 fail the test?
binomial distribution
Question: Suppose that 20% of all copies of grade 7 Science textbook fail a certain binding strength test. Let X denote the number among 15 randomly selected copies that fail the test.b. What is the probability that at least 8 fail the test?To find the probability that at least 8 fail the test, we will use the binomial probability formula.The binomial distribution is a statistical distribution that describes the probability of success or failure in an experiment or survey with two possible outcomes.Suppose a trial consists of n independent experiments, each of which has two possible outcomes: either success with probability p or failure with probability q = 1 – p. If X is the number of successes in n trials, then X is said to follow a binomial distribution with parameters n and p. The probability that X = x is given by;P(X = x) = nCxpx(1-p)n-xwhere nCx = n! / x! (n-x)! is the binomial coefficient which counts the number of ways that x successes can be chosen from n trials.Using the formula, we have;P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)Here n = 15, p = 0.2 and q = 0.8P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)= 15C8 (0.2)^8 (0.8)^7 + 15C9 (0.2)^9 (0.8)^6 + 15C10 (0.2)^10 (0.8)^5 + 15C11 (0.2)^11 (0.8)^4 + 15C12 (0.2)^12 (0.8)^3 + 15C13 (0.2)^13 (0.8)^2 + 15C14 (0.2)^14 (0.8)^1 + 15C15 (0.2)^15 (0.8)^0= 0.0094 + 0.0267 + 0.0524 + 0.0864 + 0.1174 + 0.1312 + 0.1184 + 0.0352= 0.5771Therefore the probability that at least 8 fail the test is 0.5771 (rounded to four decimal places).Answer: 0.5771
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What is the factor of the equation
Answer:
(2x - 3) (x + 3)
Step-by-step explanation:
Let's check
(2x - 3) (x + 3)
2x² + 6x - 3x - 9
2x² + 3x - 9
So, (2x - 3) (x + 3) is the correct answer.
(Please answer quickly!!!!) Given 7.05(−18.2), find the product.
−128.31
−12.83
77.55
578.10
Answer:
-128.31
First choice
Step-by-step explanation:
Using a calculator:
7.05(−18.2) = 7.05 x -18.2 = -128.31
Given the polynomial f(x) = x^3 + 3x^2 − x − 3, which of the following is true?
(x + 3) is a factor since f(3) = 0.
(x + 3) is a factor since f(−3) = 0.
(x − 3) is a factor since f(3) = 0.
(x − 3) is a factor since f(−3) = 0.
The true statement is that (x + 3) is a factor of f(x) = x³ + 3x² - x - 3 since f(-3) = 0.
How to find the factor of a polynomial ?A polynomial is an expression that consists of variables, terms, exponents and constants.
The factors are the polynomials which are multiplied to produce the original polynomial.
The factorisation of a polynomial is breaking the polynomial as a products.
Therefore,
f(x) = x³ + 3x² - x - 3
Hence, let's use the factor (x + 3)
Therefore,
f(-3) = (-3)³ + 3(-3)² - (-3) - 3
f(-3) = -27 + 27 + 3 - 3
f(-3) = 0
Therefore, the factor is (x + 3) since f(-3) = 0
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5 Exam-style Samia invests £3000 in an account for one year.
At the end of the year interest is added to her account.
Samia pays tax on the interest at a rate of 20%.
She pays £7.80 tax.
Work out the percentage interest rate for the account.
The percentage interest rate added to the account at the end of the year is 1.3%
What percentage interest rate for the account?Tax is the compulsory levy paid by citizens to the government for buying goods and services.
Amount Samia invest = £3,000
Amount of interest = x
Rate of tax = 20%
Amount of tax = £7.80
So,
20% of x = £7.80
0.2x = £7.80
x = £7.80/0.2
x = £39
Percentage of interest rate:
x% of £3000 = £39
x% × 3000 = 39
x% = 39/3000
x% = 0.013
x% = 1.3%
Hence, 1.3% is the percentage of Samia investment added to the account.
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A house painter charges a fee for supplies and an hourly fee for the time spent painting. A paint job
that takes 3 h costs $140. A paint job that takes 5 h costs $200.
Answer:
y=30x+50
Step-by-step explanation:
3x30+50=140
5x30+50=200
3n-6=-21 ieuhfuewigdhfu43w
Answer:
-5
Step-by-step explanation:
3n-6=-21
+6 +6
3n=-15 --> divide both sides by 3
n=-5
Carmen's swimming pool has the shape of a rectangular prism
with a length of 20 feet, a width of 10 feet, and a depth of 5
feet. The pool will be filled with water until the water level is 1 1/2 feet from the top of the pool. Which volume of water will be filled into the pool?
A. 1000 cubic feet
B. 700 cubic feet
C. 500 cubic feet
D. 300 cubic feet
[tex]1\dfrac{1}{2} = \dfrac{3}{2} = 1.5[/tex]
[tex]20 (10) (5-1.5) = 200(3.5) = 700 \ cubic \ feet[/tex]
Answer:
(B) 700 cubic feet
Step-by-step explanation:
The height of water in the pool is 5 - 1 1/2 = 3 1/2 feet.
The volume of water needed to fill the pool up to 3 1/2 feet is:
20 feet x 10 feet x 3 1/2 feet = 700 cubic feet
Therefore, the answer is (B) 700 cubic feet.
Which inequality describes the graph?
Answer:
Step-by-step explanation:
I think y ≤ 3 - 3x
A tractor dealer puts a markup of 21% on cost on a part for which it paid $420. Find (a) the selling price as a percent of cost, (b) the selling price, and (c) the markup
the selling price as a percent of cost is 120.52%, b- the selling price is $508.20 and the markup is 20.95% of cost.
(a) To find the selling price as a percent of cost, we need to first find the selling price and then divide it by the cost and multiply by 100%.
The markup on cost is 21%, which means the dealer sells the part for:
420 + 0.21*420 = $508.20
So the selling price is $508.20 and the cost is $420.
The selling price as a percent of cost is:
508.20/420 * 100% = 120.52%
So the selling price is 120.52% of the cost.
(b) The selling price is $508.20.
(c) The markup is the difference between the selling price and the cost, expressed as a percentage of the cost:
Markup = (Selling price - Cost)/Cost * 100%
= (508.20 - 420)/420 * 100%
= 20.95%
So the markup is 20.95% of the cost.
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To find the area of the blue parallelogram, you can move the red triangle to the green triangle to make a rectangle. After doing this, what can you conclude about the formula for the area of a parallelogram?
The area of a parallelogram can be obtained from the formula for a rectangle by using the same base and height, leading to the formula Area of parallelogram = base x height.
By moving the red triangle to the green triangle, we create a rectangle with the same base and height as the original parallelogram. Therefore, the area of the original parallelogram is equal to the area of the rectangle, which is given by the formula:
Area of rectangle = base x height
We can conclude that the formula for the area of a parallelogram is also given by:
Area of parallelogram = base x height
This is because a parallelogram can be divided into two congruent triangles, and the area of each triangle is half the area of the parallelogram. Therefore, the area of a parallelogram is equal to the base times the height, which is the same as the formula for the area of a rectangle.
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5. Explain the properties of logarithms that you
would need to use to solve the following:
log₂2x + log₂4 = 5
Answer:
x = 4
Step-by-step explanation:
log a + log b = log ab
log(base b) x = y, then b^y = x
log₂2x + log₂4 = 5
log₂ (2x × 4) = 5
log₂ 8x = 5
8x = 2^5
8x = 32
x = 4
Help me right one gets brainiest
Answer:
7x = 7
with solution of x = 1
This is the first choice
Step-by-step explanation:
Choice 1
7x = 7 => x = 7/7 = 1
Choice 2
7x = 7x provides no additional info and therefore there are an infinite number of solutions; any value of will satisfy
Choice 3
x + 1 = x + 1; infinite solutions for reasons cited under choice 2
Choice 4
x + 1 = x + 2
Eliminating x from both sides gives 1 = 2
Such an equation will have no solution; no value of x can satisfy the equation
Correct answer: Choice 1: 7x = 7
Function: f (x) = x + 2
Step-by-step explanation:
let f(x) be y
y=x+2
make x the subject of the formula
x=2-y
Answer:
Step-by-step explanation:
Find the interest on 20,000 at 6% interest for 3 years