The criterion used to decide on the line that best fits a set of data points is called the "least squares" criterion. Specifically, the criterion states that the line that best fits the data is the one that minimizes the sum of the squared errors between the observed data points and the corresponding predicted values on the line.
The correct answer is b. least-squares.
1. The least squares criterion is a widely used method to determine the line that provides the best fit for a set of data points. It aims to minimize the overall difference between the observed data points and the values predicted by the line.
2. To achieve this, the criterion calculates the error between each observed data point and the corresponding predicted value on the line. The errors are then squared to eliminate the effect of positive and negative differences canceling each other out.
3. The squared errors are summed up, and the line that minimizes this sum of squared errors is considered the best-fitting line. The idea behind this criterion is to find the line that provides the "best compromise" in terms of overall fit to the data.
4. By minimizing the sum of squared errors, the least squares criterion provides a measure of how well the line represents the observed data points. It takes into account both the magnitude and direction of the errors, giving more weight to larger errors. The line with the smallest sum of squared errors is considered the line that best fits the data.
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Find the nth term of the arithmetic sequence (an) whose initial term a and common difference d are given below. What is the forty-fifth term?
a₁ = 2; d = 1/4
Answer:
a₄₅ = 13
Step-by-step explanation:
The n-th term of an arithmetic sequence with first term a₁ and common difference d is given by the formula ...
aₙ = a₁ +d(n -1)
SetupYou want the 45th term where a₁ = 2 and d = 1/4. Putting these values into the formula gives ...
a₄₅ = a₁ +d(n -1) = 2 +(1/4)(45 -1)
SolutionEvaluating this expression, we have ...
a₄₅ = 2 +44/4 = 2 +11
a₄₅ = 13
The 45th term of the sequence is 13.
The forty-fifth term of the sequence whose initial term a = [tex]2[/tex] and common difference d = [tex]\frac{1}{4}[/tex] is [tex]13[/tex]
How to find the nth term of the Arithmetic series?
The nth term of the arithmetic series is find by [tex]Tn = a+(n-1)d[/tex] where Tn is the nth term of the series a is called the initial number and is the common difference between two number. n is the number of term of that arithmetic series.
In the given series initial term a = [tex]2[/tex] and common difference d = [tex]\frac{1}{4}[/tex]
We have the find the forty-fifth term of the given series.
= [tex]Tn=a+(n-1)d[/tex]
= [tex]Tn = 2+(45-1)\frac{1}{4}[/tex]
= [tex]Tn=2+44\cdot\frac{1}{4}[/tex]
= [tex]Tn = 2+11[/tex]
= [tex]Tn = 13[/tex]
So, the forty-fifth term of the sequence whose initial term a = [tex]2[/tex] and common difference d = [tex]\frac{1}{4}[/tex] is [tex]13[/tex]
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Two consecutive positive integers have a product of 30. what are the integers?
The two consecutive integers are 5 and 6
How to find integers in an expression?let
the first integer = x
second integer = x + 1
Therefore,
x(x + 1) = 30
x² + x = 30
x² + x - 30 = 0
(x - 5)(x + 6) = 0
Therefore,
x = 5 and x = -6
Hence, let's use 5 because it's positive.
So, the numbers are 5 and 6
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Use the fact that 1.5 is the same as 3/2 to select all of the equations below that are equivalent toT
The equivalent expressions of T=A^1.5 are: T= (A)^3/2 and T= ((A)^1/2)^3
According to the statement
we have to show that the 1.5 is same as 3/2 are equivalent to T.
So,
Equivalent expressions are expressions with the same value.
The Given expression is
T=A^1.5
we can write it as a
T= (A)^3/2
Because we know that the
1.5 = 3/2
So, it become T= (A)^3/2.
And From
T=A^1.5
we can write it as a
T= ((A)^1/2)^3
because we know that the
3/2 = (1/2)^3.
So, The equivalent expressions of T=A^1.5 are: T= (A)^3/2 and T= ((A)^1/2)^3.
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Johnny created a chart using stick people to depict the frequency of the data. What type of chart did Johnny create
Johnny created a Tally marks chart.
A frequency distribution table displays the frequency of each data set in an organized way. It helps us to find patterns in the data and also enables us to analyze the data using measures of central tendency and variance. The first step that a mathematician does with the collected data is to organize it in the form of a frequency distribution table. All the calculations and statistical tests and analyses come later.
Tally Marks are used to keep a track of numbers in the quickest possible way. Tally marks are used for counting and are represented as a set of five lines in which there are four vertical lines (one vertical line is made for each of the first four numbers) and the fifth number is represented by a diagonal line across the previous four numbers. It can be understood through the given figure
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Myron put $5000 in a 5 year CD paying 3% interest compounded monthly. After 2 years he withdrew all his money and as an early withdrawal penalty he paid back all the interest he made during the first year. How much money was Myron left with
Answer:
$7128.80
Step-by-step explanation:
First, create an equation.
Use the formula f(t)=P(1+b)^t
Plug in the information.
Time will equal months.
f(12)=5000(1+0.03)^12
Solve.
f(12) = 7128.80
Myron was left with $7128.80.
Hope this helps!
There are 26 prize tickets in a bowl, labeled A to Z. What is the probability that a prize ticket with a vowel will be chosen, not replaced, and then another prize ticket with a vowel will be chosen? Does this represent an independent or dependent event? Explain.
The probability that a prize ticket with a vowel will be chosen, not replaced, and then another prize ticket with a vowel will be chosen is 0.031 or 3.1%. So, this represents a dependent event.
What are independent and dependent events?Independent events are the events in which the outcome of one event does not affect the outcome of the other event.Dependent events are the events in which the outcome of one event affects the outcome of the other event.Calculation:It is given that there are 26 prize tickets in a bowl, labeled A to Z.
There are 5 vowels- A, E, I, O, U
So, the probability of choosing a ticket with a vowel will be - (5/26)
Another ticket with a vowel will be chosen without replacing the first ticket. Then, its probability is (4/25)
So, the required probability is,
(5/26) × (4/25) = 2/65 = 0.031
The event of choosing the first ticket with a vowel effect the event of choosing the second ticket with a vowel and vice-versa. So, this represents dependent events.
Therefore, the probability is 3.1 %.
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A partition of a positive integer $n$ is any way of writing $n$ as a sum of one or more positive integers, in which we don't care about the order of the numbers in the sum. For example, the number 4 can be written as a sum of one or more positive integers (where we don't care about the order of the numbers in the sum) in exactly five ways: \[4,\; 3 1,\; 2 2,\; 2 1 1,\; 1 1 1 1.\] So 4 has five partitions. What is the number of partitions of the number 7
There are 15 partitions for the number 7. Where the partitions are nothing but writing an integer in different ways as a sum of two or more integers.
What is a partition?A partition of a positive integer is the way of writing the integer as a sum of one or more positive integers. Where the order of the integers in the sum is not considered.
What is the number of partitions of the number 7?There are 15 partitions. They are as follows:
71+62+53+41+1+51+2+41+3+32+2+31+1+1+41+1+2+31+2+2+21+1+1+1+31+1+1+2+21+1+1+1+1+21+1+1+1+1+1+1These are the 15 possible partitions of the number 7.
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Flies Mosquitoes The table shows the number of flies and the number of mosquitoes that the frog eats for two lunches. Based on the ratio,
complete the missing values in the table.
On Monday: 35 mosquitos.
On Tuesday: 6 flies.
How to estimate the missing values of the table?As you can notice in the diagram, the frog eats 3 flies for every 7 mosquitoes (for lunch). Then you can define this ratio as pursues:
3: 7 or 3/7.
Based on the table below:
If the frog swallows 15 flies on Monday, then the number of mosquitos that it eats can be estimated as pursues:
3/7 = 15/mosquitoes
mosquitoes [tex]$= \frac{15*7}{3}[/tex]
mosquitoes = 35
If the frog eats 14 mosquitoes on Tuesday, then the number of flies that it eats can be estimated as pursues:
3/7 = flies/14
flies [tex]$= \frac{14*3}{7}[/tex]
flies = 6
The complete question is:
A frog catches insects for his lunch. The frog likes to eat flies and mosquitoes in a certain ratio, which is shown in the diagram. The table shows the number of flies and the number of mosquitoes that the frog eats for two lunches.
Based on the ratio, complete the missing values in the table.
Day Flies Mosquitoes
Monday 151515
Tuesday
141414
Disregard Above
Use Picture
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A carpenter is making doors that are 2058 millimeters tall. If the doors are too long they must be trimmed, and if they are too short they cannot be used. A sample of 17 doors is taken, and it is found that they have a mean of 2069 millimeters with a standard deviation of 24. Assume the population is normally distributed. Is there evidence at the 0.02 level that the doors are either too long or too short
There is not enough evidence at the 0.02 level that the doors are either too long or too short. So, the null hypothesis is accepted.
How to decide whether the null hypothesis is rejected or accepted?Using the p-value approach for the hypothesis test, whether the null hypothesis is rejected or accepted.
If the p-value is greater than the significance level then the null hypothesis is accepted.If the p-value is less than the significance level then the null hypothesis is rejected.Calculation:It is given that,
Sample size n = 17
Sample mean X = 2069
Standard deviation σ = 24
Population mean μ = 2058
Significance level α = 0.02
Hypothesis:
The null hypothesis: H0: μ = 2058
The alternative hypothesis: Ha: μ ≠ 2058
So, the z-test score for the given distribution is,
z-score = (X - μ)/(σ/√n)
On substituting,
z-score = (2069 - 2058)/(24/√17)
= 11/5.82
= 1.89
Thus, the p-value for the z-score of 1.89 is 0.0587 ≅ 0.06.
Since 0.06 > 0.02(significance level), the null hypothesis is accepted. So, there is no sufficient evidence that the doors are either too long or too short.
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equation of a line that passes through the points (4, 5) and (7, -7).
Answer:
y=-4x+21
Step-by-step explanation:
1. Draw these points on a graph.
2. Point A is (4,5) and Point B is (7,-7).
3. You find the slope (gradient) by the Change in Height/ Change in Horizontal distance.
4. Now put that slope and one point into the "Point-Slope Formula"
5. Simplify Algebraically.
Answer:
y = - 4x + 21
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
calculate m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (4, 5 ) and (x₂, y₂ ) = (7, - 7 )
m = [tex]\frac{-7-5}{7-4}[/tex] = [tex]\frac{-12}{3}[/tex] = - 4 , then
y = - 4x + c ← is the partial equation
to find c substitute either of the 2 points into the partial equation
using (4, 5 )
5 = - 16 + c ⇒ c = 5 + 16 = 21
y = - 4x + 21 ← equation of line
Which algebraic expression has a term with a coefficient of 9?
• A. 6 + X- 9
• B. 6x - 9
C. 6(x + 5)
• D. 9x ÷ 6
Answer:
D
Step-by-step explanation:
Coefficient is the number next to the variable, and D is the only one where that number is 9. All the others are 6.
I keep getting (10,5) or (0,5) someone pls help
Answer:
(4,-3)
Step-by-step explanation:
Multiplying the bottom equation by -6, we get -6x+12y=-60.
Adding this to the first equation gives 15y=-45, and thus y=-3.
So, it follows that 6x+3(-3)=15, and thus x=4.
Using the quadratic formula, solve the
equation below to find the two possible
values of t.
6x^2-35=-11x
Give each value as a fraction in its
simplest form.
Answer:
x= 19/12, -41/12
Step-by-step explanation:
I'm guessing x is "t".
The quadratic formula is x=(-b+-sqrtb^2-4ac)/2a I know that this is hard to read sorry!
Plug in a, b, and c.
Oh yeah, first convert it to the form Ax^2+bx+c
so 6x^2-35=-11x
6x^2+11x-35=0
(-11+-sqrt11^2-(-4*6*35))/2*6
Simply: (-11+-sqrt 121+840)/12
(-11+-sqrt961)/12
(-11+-31)/12
-11+30=19
-11-30=-41
x= 19/12, -41/12
6.6x 10⁵ in standard notation
Answer:
660000
Step-by-step explanation:
Find the surface area of a rectangular prism with a height of 16 feet, a width of 10 feet, and a length of 13 feet.
The surface area of a rectangular prism with a height of 16 feet, a width of 10 feet, and a length of 13 feet is 996 ft².
A = Surface area
l = Length
w = Width
h = Height
A = 2 ( w l + h l + h w )
= 2 · ( 10 · 13 + 16 · 13 + 16 · 10 )
= 996 ft²
The gap occupied by using a dimensional flat surface is referred to as the region. it's far measured in rectangular gadgets. The location occupied by using a 3-dimensional object by means of its outer floor is referred to as the surface location.
The surface vicinity of a strong object is a degree of the total vicinity that the surface of the item occupies.
The floor location is the overall location included via all of the faces of a 3-d item. As an example, if we need to discover the amount of paint required to paint a cube, then the surface on which the paint will be implemented is its floor place. it is continually measured in square gadgets.
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Which statement can you use to conclude that quadrilateral xyzw is a parallelogram
Answer: Option (4)
Step-by-step explanation:
A quadrilateral with two pairs of opposite congruent sides must be a parallelogram.
A fair coin is tossed three times and is tails each time. What is the probability that the next toss will be
heads
The probability that the next toss will be heads is 1/8.
What is probability?The likelihood of an event occurring is described by probability. We frequently have to make forecasts about the future in real life. We may or may not be aware of the outcome of an event. When this happens, we declare that there is a chance the event will take place.
Using the probability formula, one can determine the likelihood of an event by dividing the favorable number of possibilities by the total number of options. Since the favorable number of outcomes can never be greater than the entire number of outcomes, the probability of an event happening can range from 0 to 1.
Probability of getting two tails and next heads in three tosses is,
=1/2*1/2*1/2
=1/8
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Which expression is equivalent to 4x^2sqrt 5x^4 times 3sqrt 5x^8 , if x ≠ 0?
The expression which is equivalent to 4x²√5x⁴ × 3√5x^8 is; 60x^8.
Which expression is equivalent to the given expression?The expression given in the task content is; 4x²√5x⁴ × 3√5x^8.
Hence, upon evaluation of the product; we have;
= 12x²√25x¹².
= 60x^8.
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The prism shown in the diagram has a volume of 39 units^3. What is the volume of the pyramid? Explain your reasoning. I WILL GIVE BRAINLIEST ASAP FOR A QUICK ANSWER
The volume of the pyramid is 13 units³.
What is the volume of the pyramid?
A pyramid is a 3-dimensional shape that is made up of a rectangular base and 4 triangular faces. The volume of a pyramid is one-third that of a prism.
Volume of a pyramid = 1/3 x prism
1/3 x 39 = 13
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HI! Ok so I specifically am having an issue solving this problem? I keep coming back to 2.32 when im solving it myself and when i've asked for help its also 2.32, 2.3 rounded to be exact but the issue is my program wont accept that?
Am I solving it right or is the program im using just bugging out on me.
maybe try using the 2 and dropping the decimal
Type the correct answer in the box. Use numerals instead of words.
What is the length of the diagonal shown in the rectangular prism? Round your answer to the nearest tenth.
7 ft
ft
4 ft
5 ft
The length of diagonal is 9.5 ft
What is Pythagoras theorem?
Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.
We will find the length of diagonal as shown below:
l= 7 ft
h= 5 ft
w= 4 ft
Let diagonal be d.
Let v be the diagonal of the base.
Using Pythagoras theorem
[tex]v=\sqrt{7^2+4^2}[/tex]
[tex]=\sqrt{49+16}[/tex]
[tex]v=\sqrt{65}[/tex]
For finding the value of d again using Pythagoras theorem
[tex]d=\sqrt{v^2+h^2}[/tex]
[tex]=\sqrt{(\sqrt{65})^2+5^2 }[/tex]
[tex]=\sqrt{65+25}[/tex]
[tex]=\sqrt{90}[/tex]
=9.4868
Rounding to nearest tenth
=9.5
Hence, the length of diagonal is 9.5 ft
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I NEED HELP NOW PLSSS!!! Daredevil Robbie Knievel is building a ramp to attempt to jump his motorcycle over 35 school buses. The surface of the ramp will be 40 feet long and the ramp must stand 11 feet tall in order to clear the buses. What angle will the ramp form with the ground?
A. 7°
B. 15°
C. 16°
D. 74°
The angle that the ramp makes with the ground is 16 degrees.
What angle does the ramp make with ground?We can see that the ramp is and its support now constitutes a right angled triangle.
Thus, the hypotenuse of the triangle is 40feet while the opposite of the angle is 11 feet.
The angle made with the ground is;
sin x = 11/40
x = sin-1(11/40)
x = 16 degrees
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The table above represents what type of function?
A.
linear and nonlinear
B.
linear
C.
neither linear nor nonlinear
D.
nonlinear
B because it’s a linear graph.
Shawn graphed the points listed below.
(0, 0) (1, -1) (3, 0) (4, 4) (4, 5) (5, 4) (6, -2)
Which pair of points could be used to show that the points Shawn graphed do not represent a function?
A. (4, 4) and (4, 5)
B. (4, 4) and (5, 4)
C. (0, 0) and (3, 0)
D. (1, -1) and (6, -2)
Answer: A. (4, 4) and (4, 5)
Step-by-step explanation:
The same x value maps onto more than one y value.
A cylinder has a height of 17 feet and a radius of 6 feet. What is its volume? Use ≈ 3.14
and round your answer to the nearest hundredth.
Answer:
1921.68 [tex]ft^{3}[/tex]
Step-by-step explanation:
V = pi*r*r*h
= 3.14*6*6*17
= 1921.68
100 POINTS OKK!!!!
1. If y varies directly as x and y=5, when x=3, find the equation that relates x and y.
2. If v varies directly as w and v=21, when w=8. find the equation that relates v and w.
3. If p varies inversely with q and p=5 when q=6, find the equation that relates p and q.
4. If y varies inversely with x and y=11 when x=3 find the equation that relates x and y.
5. The force needed to break a board varies inversely with its length. If Tom uses 20 pounds of pressure to break a 1.5-foot-long board, how many pounds of pressure would he need to use to break a 6-foot-long board?
Variation is a topic that describes the relationship between or among given variables. So that the expected answers to the questions are;
y = [tex]\frac{5}{3}[/tex] xv = [tex]\frac{21}{8}[/tex]wp = [tex]\frac{30}{q}[/tex]y = [tex]\frac{33}{x}[/tex]F = 5 poundsVariation is a topic that can be used to show the relationship between or among a given number of variables. The types are direct variation, inverse variation, constant variation, and joint variation.
The required answers to the given questions are:
1. y [tex]\alpha[/tex] x
⇒ y = kx
where k is the constant of proportionality.
Given that, y = 5 and x =3, then;
5 = 3x
x = [tex]\frac{5}{3}[/tex]
Thus,
y = [tex]\frac{5}{3}[/tex] x
2. v [tex]\alpha[/tex] w
v = kw
Given: v = 21, w = 8
Then,
21 = 8k
k = [tex]\frac{21}{8}[/tex]
Thus,
v = [tex]\frac{21}{8}[/tex]w
3. q [tex]\alpha[/tex] [tex]\frac{1}{q}[/tex]
y = [tex]\frac{k}{q}[/tex]
Given: q = 5, q = 6
Then,
5 = [tex]\frac{k}{6}[/tex]
k = 5 x 6
k = 30
Thus,
p = [tex]\frac{30}{q}[/tex]
4. y [tex]\alpha[/tex] [tex]\frac{1}{x}[/tex]
y = [tex]\frac{k}{x}[/tex]
Given: y = 11, x = 3
Then,
11 = [tex]\frac{k}{3}[/tex]
k = 11 x 3
k = 33
Thus,
y = [tex]\frac{33}{x}[/tex]
5. F [tex]\alpha[/tex] [tex]\frac{1}{l}[/tex]
F = [tex]\frac{k}{l}[/tex]
Given; F = 20 pounds, l = 1.5 ft
20 = [tex]\frac{k}{1.5}[/tex]
k = 20 x 1.5
k = 30
F = [tex]\frac{30}{l}[/tex]
When l = 6 ft, then;
F = [tex]\frac{30}{6}[/tex]
F = 5 pounds
The amount of force required is 5 pounds.
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Please Help and Explain
Answer:
Step-by-step explanation:
add 4 more rows like in the pattern and you will get your answer
Answer:
36 squares.
Step-by-step explanation:
Number of squares in each step is triangular number. This form a triangular series.
1 , 3 , 6 , 10........
Number of squares in each step is given by
[tex]\sf \dfrac{n*(n+1)}{2}[/tex] here , n is the step or term.
n = 8
[tex]\sf Number \ of \ squares \ in \ 8^{th} \ step = \dfrac{8 * (8+1)}{2}[/tex]
[tex]\sf = \dfrac{8*9}{2}\\\\ = 4*9\\\\=36 \ squares[/tex]
Rewrite the function by completing the square.
Answer:
f(x) = (x -6)² +14
Step-by-step explanation:
Completing the square involves writing part of the function as a perfect square trinomial.
Perfect square trinomialThe square of a binomial results in a perfect square trinomial:
(x -h)² = x² -2hx +h²
The constant term (h²) in this trinomial is the square of half the coefficient of the linear term: h² = ((-2h)/2)².
Completing the squareOne way to "complete the square" is to add and subtract the constant necessary to make a perfect square trinomial from the variable terms.
Here, we recognize the coefficient of the linear term is -12, so the necessary constant is (-12/2)² = 36. Adding and subtracting this, we have ...
f(x) = x² -12x +36 +50 -36
Rearranging into the desired form, this is ...
f(x) = (x -6)² +14
__
Additional comment
Another way to achieve the same effect is to split the given constant into two parts, one of which is the constant necessary to complete the square.
f(x) = x² -12x +(36 +14)
f(x) = (x² -12x +36) +14
f(x) = (x -6)² +14
The graph of f(x) = 1/3
is reflected vertically, compressed horizontally by a factor of 5, shifted
right 2 units and down 6 units. Write the equation of the new graph.
The transformed function is defined by:
g(x) = -f(x/5 - 2) - 6
Where f(x) is the given function.
How to get the equation of the transformed function?Here we start with the function:
f(x) = (1/3)*x
(I don't know if this is the actual function, but that does not matter, I will solve it in a general way).
If first, we reflect it vertically, then the new function is:
g(x) = -f(x).
If now we compress it horizontally by a factor of 5, then we get:
g(x) = - f(x/5)
Now we shift it 2 units to the right, then:
g(x) = -f(x/5 - 2)
Finally, we shift it down 6 units, then we get:
g(x) = -f(x/5 - 2) - 6
That is the general form of the function g(x).
Replacing f(x) by what is defined above, we get:
g(x) = -(1/3)*(x/5 - 2) - 6
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Consider the expansion of (x^2+px-3)(3x-5),where p is a constant.If the coefficient of x of the expansion is -44,find the value of p
Answer:
p = 7
Step-by-step explanation:
(x² + px - 3)(3x - 5)
each term in the second factor is multiplied by each term in the first factor, that is
x²(3x - 5) + px(3x - 5) - 3(3x - 5) ← distribute parenthesis
= 3x³ - 5x² + 3px² - 5px - 9x + 15 ← collect like terms
= 3x³ +x²(3p - 5) - x(5p + 9) + 15
given the coefficient of the x- term is - 44 , then
-(5p + 9) = - 44 ← - (5p + 9) is th coefficient of x in the expansion
- 5p - 9 = - 44 ( add 9 to both sides )
- 5p = - 35 ( divide both sides by - 5 )
p = 7