Answer:
Step-by-step explanation:
The basic concept for both is the same, except with grouping you have more than 3 terms and after factoring out what is common in each group, you are left with identical terms in a set of parenthesis that can then also be factored out. For example, this would be factoring by grouping (then I will show you a factoring by GCF to better illustrate the idea):
If our polynomial is
[tex]x^3-x^2+3x-3=0[/tex]
we could group those terms into groups of 2 (without changing their order at all) to get:
[tex](x^3-x^2)+(3x-3)=0[/tex]
and we factor out what's common in each group to get:
[tex]x^2(x-1)+3(x-1)=0[/tex]
You can see that the (x - 1) is common now and can be factored out, leaving us with
[tex](x-1)(x^2+3)[/tex] (the secod term now can be factored if you need the complete linear factorization, but it will lead to imaginary numbers).
An example of factoring out the GCF is to factor out what's common in all the terms. It HAS TO BE COMMON IN ALL THE TERMS to do it this way. For example, in the polynomial
[tex]6x^3-4x^2+2x-8=0[/tex] the GCF for all the terms is a 2, so that's all we can factor out. It has to factor out of every term every time, no exceptions:
[tex]2(3x^3-2x^2+x-4)=0[/tex]
The method of GCF factoring often just serves to simplify the polynomial down a bit, but sometimes will still require factoring.
Final answer:
Factoring by grouping is used for polynomials with four or more terms, arranging them into groups with common factors, while factoring by GCF involves finding the largest common factor for all terms in an expression.
Explanation:
Factoring by grouping and factoring by the Greatest Common Factor (GCF) are two methods used in algebra to simplify expressions or solve equations.
Factoring by GCF involves finding the largest factor that is common to all terms in an expression.
For example, in the expression 4x + 8, the GCF is 4, so we can factor it as 4(x + 2).
Factoring by grouping, on the other hand, is used when an expression has four or more terms.
The method involves grouping terms so that each group has a common factor that can be factored out.
For instance, with the polynomial x^3 + 3x^2 + 4x + 12, we group the terms into (x^3 + 3x^2) + (4x + 12).
We then factor out the GCF from each group, yielding x^2(x + 3) + 4(x + 3).
Recognizing that (x + 3) is common between the groups, we can then factor further to get (x^2 + 4)(x + 3).
whats the odd number?
Answer:
An odd number is an integer which is not a multiple of two; A number that when divided by two, leaves a remainder. If it is divided by two the result is a fraction. One is the first odd positive number. ... Some sequential odd numbers are: {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35}, and so on.
Step-by-step explanation:
Answer:
I cant see the numbers, but I will tell you the definiton of an odd number:
An odd number is a number that leaves a remainder or a fraction when divided by 2.
Rita is hiking along a trail that is 13.8 miles long. So far she has hiked along one-tenth of the trail.
How far has Rita hiked?
Rita has hiked
miles.
Rita has hiked 1.38 miles so far.
Step-by-step explanation:
Given,
Total length of the hiking trail = 13.8 miles
Distance covered by Rita so far = [tex]\frac{1}{10}\ of\ the\ trail[/tex]
Distance covered in miles = [tex]\frac{1}{10}*13.8[/tex]
Distance covered in miles = 1.38 miles
Rita has hiked 1.38 miles so far.
Keywords: fraction, division
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A researcher conducted a study investigating the effect of computer screen time on a person’s amount of REM sleep. The researcher randomly assigned 100 people into four groups. The first group was given 2 hours of computer screen time each day, the second group was given 4 hours each day, the third group was given 6 hours each day, and the fourth group was given 8 hours each day. The study was conducted for one week, and the amount of REM sleep for each person was measured. What is the explanatory variable in the study?
Answer:
Amount of screen time
Step-by-step explanation:
Explanatory variables are those variables that are not totally independent and are used to predict or explain differences in the response variable.
Having pointed what an explanatory variable is, we'll list the explanatory variable(s) in the question, using the definition above.
The Variable used to predict results here is the amount of screen time.
In that case, it is the explanatory variable.
And it's of numerical type.
In the described study, the explanatory or independent variable is the amount of computer screen time each day, as it is the variable that is being manipulated by the researcher in order to examine its effect on REM sleep.
Explanation:In this particular research study, the explanatory variable, or the independent variable, is the amount of computer screen time each day. This is because the explanatory variable is the factor manipulated by the researcher to observe any potential effects on the outcome variable, which in this case, is the amount of REM sleep.
The participants were divided into four different groups, each exposed to a different amount of screen time: 2 hours, 4 hours, 6 hours, and 8 hours. These varying hours of exposure were purposely manipulated by the researcher to observe and measure any differences in the REM sleep duration. Therefore, the computer screen time is the explanatory variable or cause, and the observed alterations to the REM sleep will be the response, or the dependent variable.
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how to use the rational root theorem
Here are the steps:
1. Arrange the polynomial in descending order.
2. Write down all the factors of the constant term. These are all the possible values of p.
3. Write down all the factors of the leading coefficient. ...
4. Write down all the possible values of . ...
5. Use synthetic division to determine the values of for which P( ) = 0.
Answer:
Step-by-step explanation:
Arrange the polynomial in descending order.
Write down all the factors of the constant term. These are all the possible values of p.
Write down all the factors of the leading coefficient.
PLEASE HELP ASAP
Which pairs of rectangles are similar polygons?
Select Similar or Not Similar for each pair of rectangles.
Answer:
A & B Not Similar B & C Not Similar A & C Similar
Step-by-step explanation:
i just took the test and these are the correct answers that they gave on review of test, Youre welcome :)
Find the vertex of the graph of the function.
f(x) = 2x2 - 8x + 9
Answer:
(2, 1).
Step-by-step explanation:
f(x)=2x^2-8x+9
Find -b/2a first, it's -(-8)/2(2)=8/4=2.
Then find the y-coordinate of the vertex.
2(2)^2-8(2)+9=2(4)-16+9=8-16+9=-8+9=1
If 3(d+1)=4(d–2), then what is the value of 5(d–2)?
Answer:
Ans => 45
Step-by-step explanation:
3(d+1) = 4(d-2)
Expanding the brackets we have:
3d + 3 = 4d - 8
our aim now is to solve for d
3d - 4d = -8 - 3
-d = -11
=> d = 11
we can now get 5(d-2)
substituting d = 11
5(11-2)
5(9)
= 45
The value of 5(d-2) is 45.
3(d+1)=4(d–2)
3d + 3 = 4d - 8.
3d - 4d = -8 - 3.
-d = -11
d = 11
Substitute d = 11 into 5(d–2):
5(d-2)=5(11-2)
= 5(9)
= 45.
Mr Logan needs to buy snacks for his daughter soccer team. He decides to buy crackers and granola bars at a local store. The price of the crackers is $9 for 3 boxes, and the price of the granola bars is $6 for 3 boxes. Mr Logan ends up buying $20 worth of crackers and granola bars. Which equation represents the relationship between the number of boxes of crackers, x, and the number of boxes of granola bars, y, that Mr Logan bought at the grocery store.
Answer:
20 = 3x+2y
Step-by-step explanation:
one box of crackers is $3 (3x)
one box of granola bars is $2 (2y)
A radio dealer makes a radio with the price 20% above the cost price and allows 10% discount on it. Find his gain percent
Answer:
10%
Step-by-step explanation:
Good morning,
Answer:
8%Step-by-step explanation:
Let x be the cost price and
y the final sell price
p gain percent
the gain precent p = (20/100)-(10/100)-[(20×10)/(100×100)]
= (20-10)/100 - 2/100
= 10/100 - 2/100
= 8/100
example : x=20
x(1+20/100)(1-10/100) = 20(1+20/100)(1-10/100) = 20(120/100×90/100)
= 20(10 800÷10 000)
= 21.6
20+20p% =21.6 then 20p%=1.6 then p%=1.6/20=16/200=8/100 then p=8.
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You have budgeted $100 to improve your swimming over the summer. At your local pool, it costs $50 to join the swim association and $5 for each swim class. Write and solve an inequality to find the possible numbers of swim classes you can attend with
Answer:
[tex]50+5x\le 100[/tex]
[tex]x\le 10[/tex]
You can visit at most 10 swim classes.
Step-by-step explanation:
Let x be the number of swim classes you can attend with.
At your local pool, it costs $50 to join the swim association and $5 for each swim class. then x classes cost $5x and the total cost is $(50 + 5x).
You have budgeted $100 to improve your swimming over the summer. Hence,
[tex]50+5x\le 100[/tex]
Solve this inequality:
[tex]50+5x-50\le 100-50\\ \\5x\le 50\\ \\x\le 10[/tex]
You can visit at most 10 swim classes.
The student can join the swim association and attend up to 10 swim classes within the $100 budget.
Explanation:This question is asking for the creation and solution of an inequality based on a budgeting scenario. The student has budgeted $100 and needs to spend $50 to join the swim association, leaving $50 for swim classes at $5 each. The problem can be represented by the inequality 5c ≤ 50, where c is the number of swim classes.
To solve the inequality, you would divide both sides by 5. The solution of this inequality is c ≤ 10. So, the student can attend up to 10 swim classes within the budget.
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Please help, giving brainliest!
Continue the sequence:
1, 3, 7, 15, 31, 63, __
Answer:
127, 255 are the next 2 terms.
Step-by-step explanation:
The difference in terms are 2,4,8,16,32 and so on.
Expand.
Your answer should be a polynomial in standard form.
(9h+3)(-h-1)
Answer:
Step-by-step explanation:
47/14 in decimal rounded to the nearest hundredth
47 / 14 ≈ 3.36
Firstly, let's covert the fraction 47 / 14 to a decimal. The decimal format of 47 / 14 will be
47 / 14 = 3.35714285714Finally, let's approximate the decimal to the nearest hundredth
3.35714285714 ≈ 3.36Always remember to use the approximate symbol when you approximate an answer.
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Final answer:
To round 47/14 to the nearest hundredth as a decimal, divide 47 by 14 to get 3.3571..., and then round this to 3.36 since the third decimal place is 7.
Explanation:
To convert the fraction 47/14 to a decimal rounded to the nearest hundredth, you first divide 47 by 14. This gives you the decimal form. After obtaining the decimal, you look at the third decimal place. If it's 5 or more, you round the second decimal place up. If it's less than 5, you leave the second decimal place as is.
Dividing 47 by 14 gives us 3.3571... To round it to the nearest hundredth, we look at the third decimal place, which is a 7, so we round up the second decimal place from 5 to 6, thus the answer is 3.36.
Find the zero(s) of the following functions algebraically: f(x)=3x-21
Answer:
x=7
Step-by-step explanation:
The function is:
f(x)=3x-21.
The zero(s) is/are any value(s) that make(s) the function equal(s) to zero.
This implies that, to find the zero(s), we equate the function to 0 and solve for x
This implies that:
3x-21=0
3x=21
Dividing through by 3, we obtain
x=7
One angle of a right triangle measures 20°. What is the measure of the other acute angle?
Answer:
Step-by-step explanation:
Sum all the angles of triangle = 180
One angle is 90 as it is a right angle.Take the unknown angle as X.
90 + 20 + x = 180
110 + x = 180
x = 180 - 110
x = 70
what is .41 as a fraction
Answer: 41/100
Step-by-step explanation: First, we can change .41 to 0.41 by putting a zero in front of the decimal point.
Using the place value chart, we can see that 0.41 means 0 units, 4 tenths a 1 hundreth. Think of 4 tenths and 1 hundredth as 41/100 so we have 41/100.
Since the fraction 41/100 is in lowest terms, this is our final answer.
y = -1/4x + 2 and 2y = 1/2x - 4
how do these lines compare?
Answer:
the line 1 has negative slope of -1/4 and line 2 has positive slope of 1/4.
Step-by-step explanation:
let the line 1 be y = -1/4x + 2 and line 2 be 2y = 1/2x - 4
these lines are in the form of y = mx + c
where m is the slope of the line and c is the intercept made with y axis.
dividing the line 2 by 2 we get [tex]y = \frac{1}{4}x -2[/tex]
line 1 meets x axis when y = 0 applying y = 0 in equation 1 we get x = 8
line 2 meets x axis when y=0 applying y = 0 in equation 2 we get x = 8
line 1 meets y axis when x = 0 applying x= 0in equation 1 we get y = 2
line 2 meets y axis when x = 0 applying x= 0in equation 2 we get y = -2
therefore line 1 passes through (8,0)and (0,2) with negative slope of -1/4
and line 2 passes through (8,0) and (0,-2) with positive slope of 1/4.
1 What is the length of carpet for carpeting the
floor of 45 m long square room with 9 m width carpet?if
cost of the carpet per meter is Rs. 225, find the total cost
of carpeting the room
Answer:
The length of the carpet = l = 5 meter
The cost of carpeting the room is Rs 6300 .
Step-by-step explanation:
Given as :
The area of the room = 45 m²
The width of the carpet = w = 9 m
Let The length of the carpet = l meter
The cost of the carpet per meter = Rs 225
Let The cost of carpeting the room = Rs x
Now, According to question
The Area of room = length × breadth
Or, 45 m² = l m × b m
Or, 45 m² = l m × 9 m
∴ l = [tex]\dfrac{45}{9}[/tex]
i.e l = 5 m
So, The length of the carpet = l = 5 meter
Again
Perimeter of carpet = 2 × length + 2 × width
Or, Perimeter of carpet = 2 × 5 m + 2 × 9 m
Or, Perimeter of carpet = 28 meters
∵ The cost of carpet per meter = Rs 225
∴ The cost of carpeting the room = Rs 225 × 28
i.e The cost of carpeting the room = Rs 6300
Hence, The length of the carpet = l = 5 meter and The cost of carpeting the room is Rs 6300 . Answer
A history Club sold rolls of wrapping paper as a fundraiser. The wrapping paper was sold in small rolls and large rolls. The club earned $3 for every small roll sold. The club earned $4.50 for every large roll sold. The club sold 10 more large rolls than small rolls. The club collected $135 more from sales of large rolls than from sales of small rolls. The equation 3 S + 135 = 4.5( s + 10) can be used to represent this situation, where s represents the number of small rose the club sold. How many small roles did the history Club sell? Of wrapping paper as a fundraiser. The wrapping paper was sold and small rose and large Rose. The club earn $3 for every small role sold. The club sold 10 more large rolls than small rolls. The club collected $135 a more from sales of large rolls than from sales of small rolls. The equation 3x + 135 = 4.5(s + 10) can be used to represent this situation, where s represents the number of small rolls the club sold. How many small rolls did the history Club sell?
Answer:
hence small rolls 60 and large rolls 70
Step-by-step explanation:
let club sold s small rolls
hence club sold s+10 large rolls
earning from small roll 3s
earning from large rolls 4.5(s+10)
hence
4.5(s+10) - 3s = 135
or 4.5(s+10) = 3s +135
3s +135 = 4.5(s+10)
3s +135 = 4.5s +45
4.5s +45 = 3s +135
4.5s -3s = 135-45
1.5s = 90
s = 90/1.5
s= 60
s+10 = 70
hence small rolls 60 and large rolls 70
The history club sold 60 small rolls of wrapping paper according to the given equation.
Explanation:The subject here is Mathematics, specifically algebra. The equation given by the problem, 3S + 135 = 4.5(S + 10), can be solved to find 'S', which represents the number of small rolls sold by the history club. Start by distributing the 4.5 across the (S + 10) on the right side of the equation. This gives 3S + 135 = 4.5S + 45. Subtract 3S from both sides of the equation to get 135 = 1.5S + 45. Subtracting 45 from both sides then gives 90 = 1.5S. Finally, divide each side of the equation by 1.5 to get S = 60. So, the history club sold 60 small rolls.
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Selena is standing on a rock cliff that is 52 feet high. She tosses a pebble upward over the edge, where it hits the top of a 12 foot high boulder. The quadratic equation that models the path of the pebble is given below. How long did it take for the pebble to hit the top of the boulder? p(t)=-16t^2+12t+52
Answer: At [tex]t=\dfrac{3}{8}[/tex], the pebble hit the top of the boulder.
Step-by-step explanation:
Since we have given that
Distance at which Selena is standing on a rock cliff = 52 feet
Height of boulder above a rock cliff = 12 foot
So, the given quadratic equation would be
[tex]p(t)=-16t^2+12t+52[/tex]
First we derivative it w.r.t t,
[tex]p'(t)=-32t+12[/tex]
Now, we will find the critical value,
[tex]-32t+12=0\\\\-32t=-12\\\\t=\dfrac{12}{32}=\dfrac{6}{16}=\dfrac{3}{8}[/tex]
Now, we will check whether it is maximum or not.
[tex]p''(t)=-32<0[/tex]
So, we get that it is maximum.
Hence, at [tex]t=\dfrac{3}{8}[/tex], the pebble hit the top of the boulder.
Answer:
The answer is t=3/8
if x+3/3 = y+2/2, then x/3 = ?
A. Y + 1
B. Y / 3
C. Y / 2
D. Y - 1
Answer:
C
Step-by-step explanation:
(x+3)/3 =(y+2)/2
x/3 +3/3 = y/2 +2/2
x/3 +1 = y/2 +1
x/3 = y/2
2
2
+
0
.
5
0
x
=
3
7
+
0
.
3
5
x
2
2
+
0
.
5
0
x
=
3
7
+
0
.
3
5
x
Solving for x...
22+0.50x=37+0.35x
22+0.15x=37
0.15x=15
x=100
William runs at a constant speed of 8 miles per hour. Running at this constant speed, it takes William 75 seconds to run from his house to school. How many miles did William run?
The number of miles William run from his house to school is 0.167 miles
How to find distance?Number of miles run per hour = 8 mileslet
x = number of miles run in 75 seconds
Ratio of miles to time
8 miles : 3600 seconds = x miles : 75 seconds
8/3600 = x/75
8 × 75 = 3600 × x
600 = 3600x
x = 600/3600
x = 0.167 milesLearn more about distance:
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How many ways can we choose a 5 digit code out of 10 digits (includes ) if no numbers can be repeated?
Express the ratio below in its simplest form.
1/3:1/4
how do you sinplify fractions in ratios
Answer:
4/3
Step-by-step explanation:
(1/3)/(1/4)
(1/3)(4/1)
4/3
The ratio 1/3:1/4 is simplified to 4:3 by finding a common denominator. In the given proportion with a scale factor, cross-multiplication is used to find the actual dimension which results in x=8 when the scale dimension is 4.
To express the ratio 1/3:1/4 in its simplest form, find a common denominator for the two fractions, which in this case is 12. The fraction 1/3 is equivalent to 4/12, and 1/4 is equivalent to 3/12. Therefore, the ratio 1/3:1/4 becomes 4/12:3/12. Since both denominators are the same, they can be canceled out, leaving us with the simplified ratio 4:3.
Now, let's apply this to a problem with a scale factor. For the proportion 1:2=4:x, to find the value of x, you can cross-multiply. This gives us 1*x = 2*4 which simplifies to x=8. Therefore, if the scale dimension is 4, the actual dimension is 8.
the Strauss family is deciding between two lawn care services Green Lawn start charges a $49 startup fee plus $29 per month grass team charges a $25 startup fee plus that $37 per month and how many months will both line care services cost the same what will be the cost
Final answer:
Both lawn care services, Green Lawn and Grass Team, will cost the same after 3 months, with each service costing a total of $136 at that point.
Explanation:
The student asked at how many months will both lawn care services, Green Lawn and Grass Team, cost the same and what will be that cost. To solve this, we set up an equation with Green Lawn's cost model: $49 (startup fee) + $29 per month (monthly fee) and Grass Team's cost model: $25 (startup fee) + $37 per month (monthly fee). Let m represent the number of months.
The equation will look like this:
49 + 29m = 25 + 37m
To find the number of months where the costs are the same, we will solve for m:
Subtract 29m from both sides: 49 = 25 + 8mSubtract 25 from both sides: 24 = 8mDivide both sides by 8: m = 3Therefore, both services will cost the same after 3 months. The total cost at this point will be calculated by substituting m back into either cost model (we will use Green Lawn's model):
Total Cost = 49 (startup fee) + (29 × 3) (3 months of service)Total Cost = 49 + 87Total Cost = $136
Jacob deposited $4,010 in his bank account at an interest rate of 7%. The following function represents his balance after t years.
How do you solve 1/3x20
Answer:
20/3
Step-by-step explanation:
Triangles and angle bisectors? Can somebody please help me find x & y ?
Answer:
x = 12
y = 5
Step-by-step explanation:
a triangle's side is divided by an angle bisector that bisects the opposite angle. Their relative lengths equals to the relative lengths of the other two sides of the triangle
x / 8 = 9 / 6
x = 9/6 * 8 = 12
(y-3) / 3 = 4 / 6
y - 3 = 4/6 * 3 = 2
y = 5
2x=(360÷4) solve for x.
Answer:
Step-by-step explanation:
2x=(360÷4)
2x = 90
Dividing by 2
x = 90/2
x = 45
By following the steps of simplifying, isolating, and solving for x, we find that x = 45 is the value that satisfies the given equation.
To solve the equation 2x = (360 ÷ 4) for x, we need to isolate x on one side of the equation. Follow these steps to find the value of x:
Step 1: Simplify the expression on the right side of the equation.
360 ÷ 4 = 90
Step 2: Replace the right side of the equation with its simplified value.
2x = 90
Step 3: To isolate x, divide both sides of the equation by 2.
(2x) ÷ 2 = 90 ÷ 2
This simplifies to:
x = 45
The solution to the equation is x = 45. To verify, you can substitute x = 45 back into the original equation:
2(45) = 90
90 = 90
Since both sides of the equation are equal, x = 45 is indeed the correct solution.
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