By cancelative property, the roots of quadratic equation x² + 4 · x - 20 are: Exact: - 2 + 2√6, - 2 - 2√6, Approximate: 2.899, -6.899.
How to find the roots of a polynomial by cancelative property
In this problem we must factorize a quadratic equation and use an algebraic property known as cancelative property, defined as:
If a · b = 0, then a = 0 or b = 0.
In addition, we can apply the following factorization rule:
x² - (r₁ + r₂) · x + r₁ · r₂ = (x - r₁) · (x - r₂)
First, factorize the quadratic equation:
x² - (- 2 + 2√6 - 2 - 2√6) · x + (- 2 + 2√6) · (- 2 - 2√6)
(x + 2 - 2√6) · (x + 2 + 2√6)
Second, apply cancelative property:
x + 2 - 2√6 = 0
x = - 2 + 2√6
x + 2 + 2√6 = 0
x = - 2 - 2√6
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(9a^4-31a^3-21a^2+6a-13)/(a-4) Synthetic Division
The quotient from the synthetic division (9a^4-31a^3-21a^2+6a-13)/(a-4) is 9a⁴ - 22a³ - a² + 5a - 8/a -4
What is synthetic division?Synthetic division can be defined as a mathematical method of dividing a polynomial by a binomial x - c
Such that, c is a constant.
The steps to take in synthetic division;
Arrange the set up for synthetic divisionWrite down the leading coefficient in the bottom rowMultiply the value of c by the value written on the bottom rowAdd the column that was created in step 3Repeat until the process till the last coefficientGiven the polynomial;
(9a^4-31a^3-21a^2+6a-13)/(a-4)
[tex]\sqrt[a -4]{9a^4 -31a^3 +6a -13}[/tex]
c of divisor x-c
4 coefficients of dividend
9 -31 -21 6 -13
_____9__-22__-1___ 5_______
9 -22 -1 5 -8
The quotient is the;
9a⁴ - 22a³ - a² + 5a - 8/a-4
Hence, the expression is 9a⁴ - 22a³ - a² + 5a - 8/a-4
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Find the surface area of this rectangular prism. Be sure to include the correct unit in your answer.
The surface area of the rectangular prism is 182m²
How to determine the surface areaIt is important to note that the surface area of a rectangular prism is expressed as;
Surface area = 2(wl + hw + hl)
Where;
w is the width of the rectangular prismh is the height of the rectangular prisml is the length of the rectangular prismNow, from the diagram show, we have the values as;
Length = 3
Height = 7
Width = 7
Substitute the values into the formula
Surface area = 2(7×3 + 7×3 + 7 ×7)
multiply the values
Surface area = 2(21 + 21 + 49)
expand the bracket
Surface = 182m²
Thus, the value is 182m²
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!!! asap
Bryce graphs an arithmetic sequence and draws a line through the points. The line has a negative slope and a positive y-intercept. Write an arithmetic sequence that could represent Bryce's sequence
The possible equation of the sequence is f(x) = -2x + 5
How to determine the possible sequenceFrom the question, we have the following parameters that can be used in our computation:
Sequence type = arithmetic
Also, we have the following features
slope = negative
y-intercept = positive
A linear equation (or an arithmetic sequence) can be represented as
y = mx + c
Where
slope = m
y-intercept = c
Using the given features, we can use the following assumed values
m = -2
c = 5
So, the sequence is
f(x) = -2x + 5
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A fresh fish retailer receives a shipment of sardines in sealed packs. Together, there were 3 pounds in 12 packs. If the retailer received 48 packs of sardines, how many pounds were in the shipment?
Answer:
So, the question is asking how many pounds are there in 48 shipments if ever 12 shipments are 3 pounds, so what you want to do is so how many times 12 will go into 48 and then multiply that by 3!
48/12 = 4
4 * 3 = 12
The total 48 shipments weigh 12 pounds!
Step-by-step explanation:
Hope it helps! =D
Answer:
The weight of one pack of sardines is 3 pounds / 12 packs = 0.25 pounds/pack
So the weight of 48 packs of sardines is 0.25 pounds/pack * 48 packs = 12 pounds.
30 POINTS!
A eyeglasses store is checking their inventory. The table shows the relationship between the number of days since the opening of the eyeglasses store and the total amount of sunglasses in their inventory.
Sunglasses Inventory
Number of Days Total Amount of Sunglasses
2 52
5 46
7 42
10 36
Which graph shows the relationship given in the table?
graph with the x axis labeled number of days and the y axis labeled total amount of sunglasses and a line going from the point 0 comma 52 through the point 1 comma 50
graph with the x axis labeled number of days and the y axis labeled total amount of sunglasses and a line going from the point 0 comma 56 through the point 1 comma 54
graph with the x axis labeled number of days and the y axis labeled total amount of sunglasses and a line going from the point 0 comma 52 through the point 1 comma 49
graph with the x axis labeled number of days and the y axis labeled total amount of sunglasses and a line going from the point 0 comma 56 through the point 1 comma 53
The images are also the answers
A graph which shows the relationship given in the table is: B. graph with the x axis labeled number of days and the y axis labeled total amount of sunglasses and a line going from the point 0 comma 56 through the point 1 comma 54.
How to construct and plot the data on a graph?In this scenario, the number of days would be plotted on the x-axis (x-coordinate) of the scatter plot while the total amount of sunglasses would be plotted on the y-axis (y-coordinate) of the scatter plot through the use of Microsoft Excel.
From the scatter plot (see attachment) which models the relationship between the number of days and total amount of sunglasses, a linear equation for the line of best fit is given by:
y = -2x + 56
In conclusion, we can reasonably infer and logically deduce that the y-intercept is equal to (0, 56).
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Answer:
YOUR WELCOME THE ANSWER IS
Step-by-step explanation:
B (-:{
An item on sale costs 30% of the original price. If the original price was $90 what is the sale price
Please can somebody help me do this further maths gcse question in ‘transformations of graphs and functions’
Answer:
y=f(x)-6 is a vertical translation down by 6 units so draw the same graph but 6 units down with the point at which it crosses the y-axis as (0,-4)
(−4x + 3)(−2x2 − 8x + 2)
On solving the expression, the standard form is obtained as 8x³ + 26x² - 32x + 6.
What is an expression?
Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.
The given expression is - (−4x + 3)(−2x² − 8x + 2)
To solve this expression first expand the equation using arithmetic operation of multiplication.
(−4x + 3)(−2x² − 8x + 2)
-4x(−2x² − 8x + 2) + 3(−2x² − 8x + 2)
8x³ + 32x² - 8x - 6x² - 24x + 6
Collect the like terms together -
8x³ + 32x² - 6x² - 8x - 24x + 6
Perform the arithmetic operation of addition and subtraction -
8x³ + 26x² - 32x + 6
Therefore, the standard form is 8x³ + 26x² - 32x + 6.
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Line segment SU is dilated to create S'U' using the dilation rule DQ,2. 5. What is the distance, x, between points U' and U?
4 units
4. 8 units
6 units
10 units
If Line segment SU is dilated to create S'U' using the dilation rule DQ₂.₅ , then the distance "x" between points U' and U is (c) 6 units .
The Scale Factor is written as ratio of the distance from center to image of a point and distance from center to corresponding point.
the scale factor from the figure can be written as : QU'/QU ;
From the figure , we can see that QU = 4 units and scale factor is = 2.5 ;
⇒ 2.5 = QU'/4 ;
On simplifying ,
we get ;
⇒ QU' = 2.5 × 4 = 10 ;
From the figure : UU' = QU' - QU ;
⇒ UU' = 10 - 4 = 6 .
Therefore , the distance between points U and U' denoted by "x" is 6 units .
The given question is incomplete , the complete question is
Line segment SU is dilated to create S'U' using the dilation rule DQ₂.₅ , What is the distance, x, between points U' and U?
(a) 4 units
(b) 4.8 units
(c) 6 units
(d) 10 units
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Your water hose leaking at a rate of 0.25 cups per hour. How many gallons did your hose leak in a full 24 hour day ?
well, let's first find out how many cups was that in 24 hours anyhow.
so in 1hr the hose leaks 0.25 cups, so in 24 hrs that'd be (24)(0.25) = 6 cups.
Since a gallon has 16 cups, that'd be 6/16 gallons or 3/8 = 0.375 of a gallon.
please help me with my homework!
Answer:
The machine makes 45 bottles per minute.
Step-by-step explanation:
We can first find the slope through this equation: [tex]\frac{y2-y1}{x2-x1}[/tex]
Since we are given two points, (2,90) and (7,315), we can plug those two points into the formula given above:
[tex]\frac{315-90}{7-2}[/tex]
= 45
Therefore, m = 45.
Now, we can pick one of the two points given--in this case, I'll use (2,90)--to find the equation of the line: [tex]y-y1 = m(x-x1)[/tex]
For (2,90), y1 will be 90, and x1 will be 2. Let's also plug in m:
y - (90) = 45(x - 2)
We can solve for y now to find what y=mx+b is:
y = 45(x-2) + 90
y = 45x
Since the problem asks how many bottles the machine can make per minute, we can plug x = 1: 45(1) = 45
The machine makes 45 bottles per minute.
Please help GIVING 50 POINTS
Answer:
¿Qué quieres comer?
Step-by-step explanation:
Combine like terms to create an equivalent expression.
3.4 2.8d +2.8d-1.3
Answer:
Step-by-step explanation:
If the problem was supposed to be "3.4 + 2.8d + 2.8d -1.3". Then the answer would be 5.6d + 2.1. you can combine the "d" terms because they have the same variable and you can combine the number terms because they are like. This result in 2.8d+2.8d + 3.4 - 1.3. This simplifies to the correct answer given above.
Buddy walks 5000 feet due North, then turns and walks 4000 feet due East. If he walked in a straight line from his starting position to his final position, how many miles would he walk?
Answer: 1.2 miles
You can find the answer by using the pythagorean theorem.
a^2 + b^2 = c^2.
(5000)^2 + (4000)^2 = c^2
25 million +16 million = c^2
41 million = c^2
Take the square root of both sides in order to get the value of C.
C = about 6,403 feet.
1 mile = 5280 feet
6403/5280 = 1.2
Buddy would walk about 1.2 miles
i need a help with graphing
A graph of the line that passes through the two points (-3/2, -3/2) and (3/2, 3/2) is shown in the image attached below.
How to calculate the slope of any straight line?In Mathematics and Geometry, the slope of any straight line can be calculated by using this mathematical equation (expression);
Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
From the information provided above, we have the following data points located on the line:
Points on the x-coordinate = (-3/2, 3/2).
Points on the y-coordinate = (-3/2, 3/2).
Substituting the given data points into the slope formula, we have the following;
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope (m) = (3/2 + 3/2)/(3/2 + 3/2)
Slope (m) = 3/3 = 1
At point (-3/2, -3/2), an equation of this line can be calculated by using the point-slope form as follows:
y - y₁ = m(x - x₁)
Where:
m represents the slope.x and y are the data points.c represents the intercept.y - (-3/2) = 1(x - (-3/2))
y + 1.5 = x + 1.5
y = x
Next, we would use an online graphing calculator to plot the linear equation as shown in the graph attached below.
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The prism below is made of cubes which measure 1/4 of a foot on one side. What is the volume? A. 13/4 cubic feet B. 80 cubic feet C. 20 cubic feet D.5/4 cubic feet
The volume of the prism in this problem is given as follows:
C. 9/16 cubic feet.
How to obtain the volume of the rectangular prism?The volume of a rectangular prism, with dimensions length, width and height, is given by the multiplication of these dimensions, as follows:
Volume = length x width x height.
For this problem, two of the dimensions are formed by three cubes, hence their measures are:
3 x 1/4 = 3/4 foot.
The remaining dimensions is composed by four cubes, hence the measures are:
4 x 1/4 = 1 foot.
Hence the volume is given as follows:
V = 1 x 3/4 x 3/4
V = 9/16 cubic feet.
Missing InformationThe problem is given by the image presented at the end of the answer.
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find the distance from the given point to each of the following. (3, 9, -7) (a) the xy-plane 7 correct: your answer is correct. (b) the yz-plane 3 correct: your answer is correct. (c) the xz-plane 9 correct: your answer is correct. (d) the x-axis correct: your answer is correct. (e) the y-axis correct: your answer is correct. (f) the z-axis correct: your answer is correct.
Distance from point (3,9,-7) to xy-plane is 7, yz-plane is 3, xz-plane is 9, x-axis, y-axis and z-axis is 0.
What is axis ?
An axis is a line about which a body rotates or a system oscillates. In mathematics and physics, an axis is often represented as a straight line that serves as a reference point for the measurement of coordinates.
The correct answers are :
(a) The distance from the point (3, 9, -7) to the xy-plane is the distance from the point to the z-axis, which is 7, since the xy-plane is at z = 0
(b) The distance from the point (3, 9, -7) to the yz-plane is the distance from the point to the x-axis, which is 3, since the yz-plane is at x = 0
(c) The distance from the point (3, 9, -7) to the xz-plane is the distance from the point to the y-axis, which is 9, since the xz-plane is at y = 0
(d) The distance from the point (3, 9, -7) to the x-axis is the distance from the point to the x-axis, which is 0, since the point is already on the x-axis
(e) The distance from the point (3, 9, -7) to the y-axis is the distance from the point to the y-axis, which is 0, since the point is already on the y-axis
(f) The distance from the point (3, 9, -7) to the z-axis is the distance from the point to the z-axis, which is 0, since the point is already on the z-axis
Distance from point (3,9,-7) to xy-plane is 7, yz-plane is 3, xz-plane is 9, x-axis, y-axis and z-axis is 0.
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What is the equation of the line that passes through the point (-3,-8) and has a slope of 0
Answer:
The equation of a line in the form of y = mx + b, where m is the slope, x is the variable and b is the y-intercept.
Given a point and the slope, we can find the equation of the line that passes through that point. We know that the slope of the line is 0, so we can write the equation of the line as:
y = 0x + b
We also know that the line passes through the point (-3,-8), so we can substitute these values into the equation and solve for b, the y-intercept:
-8 = 0(-3) + b
-8 = 0 + b
b = -8
So the equation of the line that passes through the point (-3,-8) and has a slope of 0 is:
y = 0x - 8
This line is a horizontal line with y-intercept -8, which means that the line passes through the point (0,-8)
I read a total of 42 books over 7 months. If Craig has read 48 books so far, how many months has he been with his book club? Solve using unit rates.
Answer:
8
Step-by-step explanation:
42/7 =
6..
Since +6 more to 42 = 48 =
42+6=
48.
Just add 1 more month =
7+1=
8
Need more explanation I am happy to help :)
note: enter your answer and show all the steps that you use to solve this problem in the space provided. use the circle graph shown below to answer the question. the image is a color pie chart divided into 5 portions, and is titled pets owned by sixth graders.
starting from the 12 o'clock position at the top and going clockwise toward the right is dark blue portion labeled dogs, 45 percent. next portion if 30 sixth graders own birds, how many sixth graders were surveyed?
Total number sixth graders were surveyed is 600.
"percent" is Latin for "out of 100," so 5 % is literally 5 out of 100
Now--if it were 5 kids with birds, you would know that there were 100 total kids. But it's 30 kids, which is way more than 5; so your total number of kids is going to be way more than 100.
So what you need to do here is figure that "30 sixth graders own birds
i.e. 30 is 5% of some number."
How many times bigger than 5 is 30? (6 times, because 5 x 6 = 30, or you can think of it as 30 divided by 6 = 5)
So if you had to do 5 TIMES 6 to get to 30, you have to do 100 TIMES 6 to get the total number sixth graders were surveyed.
What's 100 x 6?
= 600.
So your total number sixth graders were surveyed is 600.
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(A)Find the length of the arc.
(B)Find the area of the sector.
r = 8 cm, Theta = 45°
a. The length of the arc is 6.28cm
b. The area of the sector is 25.12cm²
What are arc and sector?The arc of a circle is defined as the part or segment of the circumference of a circle. A sector is the space bounded by the arc and the radii of the circle. The smaller area is known as the minor sector and the bigger area is known as the major sector.
The length of an arc = tetha /360 ×2πr
= 45/360 × 2×3.14× 8
= 2×3.142 = 6.28cm
The area of a sector = tetha / 360 × πr²
= 45/360 × 3.14 × 8× 8
= 3.14× 8
= 25.12 cm²
Therefore, the length of the arc is 6.28cm and the area of the sector is 25.12cm²
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Think about what you know about breaking apart numbers with addition. Fill in each box. Use words, numbers. and picture. Show as many ideas as you can.
The breaking of number using addition is discussed above.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is the breaking apart of numbers with addition.
We can breaking apart the numbers with addition as follows. Assume the number to be 50. We can break the number 50 as follows -
50 = 25 + 25
50 = 15 + 10 + 15 + 10
50 = 10 + 10 + 10 + 10 + 10
50 = 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5
Therefore, the breaking of number using addition is discussed above.
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I Need help with this I’ve tried stuff and stuff but it’s not working
Answer:
y= -2x + 0
Step-by-step explanation:
to get to y you need to multiply by -2 and add 0 to that to get the final y answer
Find the selling price.
Cost to store: $65
Markup: 80%
The selling price for the article if it is equal to marked price will be $117
using profit and loss formulas.
What is profit and loss formula?
Profit and Loss formula is used in mathematics to determine the price of a commodity in the market and understand how profitable a business is. Every product has a cost price and a selling price. Based on the values of these prices, we can calculate the profit gained or the loss incurred for a particular product. The important terms covered under this are cost price, fixed, variable and semi-variable cost, selling price, marked price, list price, margin, etc.
For example, for a shopkeeper, if the value of the selling price is more than the cost price of a commodity, then it is a profit and if the cost price is more than the selling price, it becomes a loss.
Now,
Given Cost price=$65
Marked up price = 180%*$65
As
Marked up price = selling price
Therefore
selling price= 180%*$65
selling price =$117
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A mathematics teacher wanted to see the correlation between test scores and homework. The homework grade (x) and test grade (y) are given in the accompanying table. Write the linear regression equation that represents this set of data, rounding all coefficients to the nearest tenth. Using this equation, find the projected test grade, to the nearest integer, for a student with a homework grade of 37.
Regression equation:
Final Answer:
You can substitute a homework grade of 37 into the equation in place of x, to find the projected test grade.
What is linear regression?Linear regression is a statistical method that is used to study the relationship between two continuous variables, often referred to as the independent variable (x) and the dependent variable (y). The goal of linear regression is to find the best-fitting line that describes the relationship between these two variables. The best-fitting line is known as the regression line, and it is represented by the equation y = mx + b, where m is the slope of the line, and b is the y-intercept.
To find the linear regression equation, we need to use the least squares method to find the line of best fit that describes the relationship between the homework grade (x) and the test grade (y). The equation of the line is in the form of y = mx + b, where m is the slope and b is the y-intercept.
To find the slope (m) and y-intercept (b) we can use the following formulas:
m = (n(Σxy) - (Σx)(Σy)) / (n(Σx²) - (Σx)²)
b = (Σy - m(Σx)) / n
where n is the number of data points, Σx and Σy are the sum of x and y values, Σxy is the sum of the product of x and y values, and Σx² is the sum of the squares of x values.
Given the data points, we can use this information to calculate the slope and y-intercept, and write the linear regression equation.
Given the linear regression equation of the form y = mx + b
You can substitute a homework grade of 37 into the equation in place of x, to find the projected test grade.
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e
5 Louay wants a cap with less than white
stars. Which cap could Louay buy?
Cap 1
Cap 1 only
Cap 2 only
C
Cap 2 or 3
D Cap 1 or 3
A
B
Cap 2
Cap 3
The correct option regarding which cap Louay can buy is given as follows:
D. Cap 1 or 3.
How to obtain the fraction of white stars?The fraction of white stars is obtained using a proportion, dividing the number of white stars by the total number of stars.
Hence the fractions for each cap are given as follows:
Cap 1: 1/3 = 0.33 < 1/2 = 0.5.Cap 2: 2/4 = 0.5 = 0.5.Cap 3: 2/5 = 0.4 < 1/2 = 0.5.Meaning that Cap 1 and Cap 3 have the desired ratio of white stars relative to total stars, and thus the correct option is given by option D.
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I believe A and C are part of the answer but I am unsure of the rest
Please find attached the image of the enlargement of the shape X by a scale factor of -1 created using MS Word, which is a drawing equivalent to option B. The correct option is therefore;
B.
What is an enlargement transformation?An enlargement transformation is one in which the pri-image object size is scaled about a center of enlargement, using a scale factor, with the sign of the scale factor indicating the direction of the enlargement, which for a +ve sign is on the same side, and for a -ve sign is on the other side of the center of enlargement.
An enlargement by a scale factor of -1, can be obtained by producing the image on the alternate side of the center of enlargement, which with a scale factor of -1, is the same size as the size of the image.
Therefore, the dilation by a scale factor of -1, with the center at the origin, produces an image which is equivalent to a reflection of the original object (pre-image), across the y-axis followed by a reflection across the y-axis.
The image of the object following the dilation will therefore be the mirror image of the mirror image of the object reflected across the y-axis, which takes the shape of a vertically flipped, left flipped F, which is option B
Please find attached a drawing of the shape of the image of the enlargement of shape X by a scale factor of -1, created with MS Word
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If 3+x<5 and 8+x>5,What range of values of x satisfies both inequalities?
The range of values of x which satisifies both of the inequalities is:
(-3, 2)
What range of values of x satisfies both inequalities?Here we have two inequalities:
3 + x < 5
8 + x > 5
And we want to see which range of values of x satisfie both of these inequalities.
If we isolate x on both inequalities, we will get:
3 + x < 5
x < 5 - 3
x < 2
And the other one is:
8 + x > 5
x > 5 - 8
x > -3
Then we havethe compound inequality:
x < 2
x > -3
Then the range of values that satisfies both inequalities is (-3, 2)
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Terra argues that is not a function because two different inputs produce the same output, as shown in the table.
Is Terra correct in claiming that is not a function?
Input Output
-1 2
0 0
1 2
Terra is
correct/incorrect
. In order for a relationship to be a function, each input value must result in
only one input/ multiple outputs
. In this equation, each input value does result in only one output value. It is
allowed/not allowed
for more than one input to have the same out. This does not mean that the relationship is not a function.
Terra is correct. In order for a relationship to be a function, each input value must result in only one input.
What is a function?A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a connection between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain. The usual way to refer to a function is as f(x), where x is the input. A function is often represented as y = f. (x).
We know that for a function each input value must result in a unique output value. Terra claims this statement, and hence she is correct.
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Given p(x) = x^3+4x^2-3x+2 and p(x)/x-1= x^2+5x+2 R4 Which statement is true? Select all that apply. A.p(1)= 4 B.p(x)= (x - 1)(x^2 + 5x + 2) +4/x-1 C.p(x)= (x-1)-(x^2+5x+2)+4 D.p(-1)= 4
The correct statement relative to the polynomial function is given as follows:
A. p(1)= 4
B. p(x)= (x^2 + 5x + 2) +4/x-1
How to obtain the correct statement?The polynomial function for this problem is given as follows:
p(x) = x^3+4x^2-3x+2
Dividing the polynomial function by x - 1, the quotient and the remainder are given as follows:
Quotient: x² + 5x + 2Remainder: 4.Meaning that:
The function can be written as: p(x)= (x^2 + 5x + 2) +4/x-1.By the Remainder Theorem, p(1) = 4.Hence the correct options are given by option A and option B.
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