The polynomial h(x)=x^4+2x^3-13x^2-14x+24 in the factored form is (x−1)(x+2)(x−3)(x+4). The real roots of the polynomial is x= 1, x = 3, x = -2, x = -4.
What is a polynomial function?
In an equation such as the quadratic equation, cubic equation, etc., a polynomial function is a function that only uses non-negative integer powers or only positive integer exponents of a variable.
The polynomial function is -
h(x)=x^4+2x^3-13x^2-14x+24
Divide the polynomial h(x) by (x-1) -
(x^4+2x^3−13x^2−14^x+24)=(x−1)(x3+3x2−10x−24)
Divide (x^3+3x^2−10x−24) by (x+2) -
(x−1)(x^3+3x^2−10x−24)=(x−1)(x+2)(x^2+x−12)
Solve the quadratic equation x^2+x−12=0.
x^2+x−12=0
x^2+4x-3x-12=0
x(x+4)-3(x+4)=0
(x+4)(x-3)=0
So, (x−1)(x+2)(x^2+x−12)=(x−1)(x+2)(x−3)(x+4)
Therefore, x^4+2x^3−13x^2−14x+24=(x−3)(x−1)(x+2)(x+4) .
The Rational Root Theorem tells us that if the polynomial has a rational zero then it must be a fraction q/p, where p is a factor of the constant term and q is a factor of the leading coefficient.
We can see that p(1)=0 so x=1 is a root of a polynomial h(x).
Divide the polynomial by (x-1) -
(x^4+2x^3−13x^2−14x+24)/(x-1)=(x^3+3x^2−10x−24)
Divide (x^3+3x^2−10x−24) by (x-3) -
(x^3+3x^2−10x−24)/(x-3) = (x^2+6x+8)
To find the last zero, solve equation x+4=0
x=-4
Therefore, the real roots are - x = 1, 3, -2, -4.
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What is the image of (−8,12) after a dilation by a scale factor of 1/4 centered at the origin?
The image of (−8,12) after a dilation is (-2, 3).
What is Scale factor?A scale factor is a numerical value that can be used to alter the size of any geometric figure or object in relation to its original size. It is used to find the missing length, area, or volume of an enlarged or reduced figure as well as to draw the enlarged or reduced shape of any given figure. It should be remembered that the scale factor only affects how big a figure is, not how it looks.
Given:
Point (-8, 12)
scale factor = 1/4
After applying the scale factor then image
= (-8/4, 12/4)
= (-2, 3)
Hence, the image is (-2, 3).
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A model train ha a cale of 3 cm : 14 m. If the model train i 57 cm long, then how long i the real train?
If the model train is 57 cm long, then 266 m is the length of real train.
In the given question, a model train has a scale of 3 cm : 14 m.
If the model train is 57 cm long, then we have to find how long the real train is.
The scale of model train is 3 cm : 14 m.
Now firstly convert m in cm. As we know that 1 m = 100 cm
Now, the scale of model train is 3 cm : 1400 cm.
The length of the model train is 57 cm.
So in the scale of the model train 3 cm = 57 cm
Now the scale of the model train 1 cm = 19 cm
Now find the length of the real train by multiplying the 1400 by 19 because in the scale of 1 cm is equal to the 19 cm.
So, the length of the real train = 1400 × 19
The length of the real train = 26600 cm
The length of the real train = 266 m
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applied partial differential equations 7.7.2
The solution of the partial differential equation 7.7.2 is given by
u(x,t) = t^2 sin(2πx) + c,u(0.3,1.5) = 1.96 + c, where c is an arbitrary constant.
Partial differential equations are mathematical equations used to describe how a function changes as its inputs change. The equation 7.7.2 is a linear second order partial differential equation in two independent variables. It has the form of a generalized wave equation, in which a wave propagates through a medium with varying physical properties.
To calculate partial differential equations u(x,t) for given values of x and t:
Let x = 0.3 and t = 1.5
Substituting these values into the equation, we get:
u(0.3,1.5) = (1.5)^2 sin(2π(0.3)) + c
= 2.25 sin(0.6π) + c
= 2.25(0.87) + c
= 1.96 + c
Therefore, u(0.3,1.5) = 1.96 + c, where c is an arbitrary constant.
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Find the values of x and y.
y°
40°
x°
(x,, y,) is a point on the line in the point-slope formula, and m is the slope.
How should slope m be written?When you know the slope of the line to be investigated and the point supplied is also the y intercept, you may utilize the slope intercept formula y = mx + b. (0, b). The y value of the y intercept point is represented by b in the formula.
The equation of a line can be written in several different formats:
Standard form
Ax+ By = C
A, B, and C must be real numbers and A and B cannot both be zero.
Slope-intercept form
Y = MX + b
M is the slope and b is the y-intercept.
Point-slope form
y-y₁ = m(x-x1)
In the point-slope formula, (x,, y,) is a point on the line and m is the slope.
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The graph of a linear function passes through the points (-10, -14) and (10, 11). Place (click and drag) a choice from each list into its corresponding b
describe the slope and y-intercept of the function.
Answer:
Slope = 1.25
Y-intercept = -1.5
Step-by-step explanation:
Slope:
[tex]\displaystyle m=\frac{y_2-y_1}{x_2-x_1}=\frac{11-(-14)}{10-(-10)}=\frac{11+14}{10+10}=\frac{25}{20}=1.25[/tex]
Y-intercept:
[tex]y=mx+b\\\rightarrow y=1.25x+b\\\rightarrow 11=1.25(10)+b\\\rightarrow 11=12.5+b\\\rightarrow -1.5=b[/tex]
Final Equation:
[tex]y=1.25x-1.5[/tex]
Is 1 1 2 3 5 8 an arithmetic sequence?
No, 1 1 2 3 5 8 is not an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a constant to the previous term.
Thus, the difference between successive terms is the same. In the given sequence, the difference between terms 1 and 2 is 0, between terms 2 and 3 is 1, between terms 3 and 5 is 2, and between terms 5 and 8 is 3. Therefore, the difference between successive terms is not the same and hence, 1 1 2 3 5 8 is not an arithmetic sequence.
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What is f(-3) if f(X) =1/3x
Answer: 1/3x
Step-by-step explanation:
Hey there!
f(x) = 1/3x
f(x) = 1/3(-3)
f(x) = 1/3(-3/1)
f(x) = 1(-3)/3(1)
f(x) = -3/3
f(x) = -1
Therefore, your answer should be:
f(x) = -1
Good luck on your assignment & enjoy your day!
~Amphitrite1040:)
What is called absolute number?
Any value inside an absolute value in a problem or equation indicates that the value is always positive.
What is called an absolute number?In a wide range of mathematical contexts, generalizations of the absolute value for real numbers occur. Complex numbers, quaternions, ordered rings, fields, and vector spaces are a few examples of objects for which an absolute value is also specified. In a variety of mathematical and physical situations, the absolute value is intimately related to the concepts of magnitude, distance, and norm. Without taking direction into account, absolute value describes how far away from zero a certain number is on the number line. A number can never have a negative absolute value. View some illustrations. 5 is 5 in its entirety.Any value inside an absolute value in a problem or equation indicates that the value is always positive. Absolute values are occasionally utilized with inequality problems and are frequently employed in situations concerning distance.To learn more about absolute number refer to:
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Evaluate the expression use calculator if needed
Answer:
780.
Step-by-step explanation:
I cant explain Its just the answer.
Please help me and explain
The value of x in the equation 8 (x - 3) + 7 = 2x (4 - 17) is x = 1/2.
What is an Equation?An equation is the statement of two expressions located on two sides connected with an equal to sign. The two sides of an equation is usually called as left hand side and right hand side
The given equation is,
8 (x - 3) + 7 = 2x (4 - 17)
Distributive property : p (q + r) = p q + p r
Applying the property here,
8x - (8 × 3) + 7 = 2x × (-13)
8x - 24 + 7= -26x
8x - 17 = -26x
8x + 26x = 17
34x = 17
x = 17/34
x = 1/2
The mistake of Lin happened in the second step where 4 - 17 = -13 but she wrote 13.
Hence the value of x is 1/2 not -1/2 as the solution of Lin.
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In a school, all students study one language, French or Spanish.The ratio of girls to boys in Year 11 is 4:3, 3/4 the boys study French There are 168 students in Year 11.How many of the boys study Spanish?
The answer is 72 boys
What is the unitary method?The unitary method is a method in which you find the value of a unit and then the value of a required number of units.
Given here: The ratio of boys to girls is 4:3 and total students as 168
thus there are 96 boys and 72 girls and also 3/4 th of the boys study french
which is equivalent to 72 boys
Therefore the number of boys that study spanish is equal to 96-72=24
Hence, 24 boys study spanish.
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Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar. Complete the table of inputs and outputs for the given function. g(x)=3−8x
The complete table is,
x 3/8 0 1 3
g(x) 0 3 -5 -21
What are linear equations?Each term in a linear equation has an exponent of 1, and when this algebraic equation is graphed, it always produces a straight line. It is called a "linear equation" for this reason.
Given equation g(x) = 3 − 8x
1. when g(x) = 0
0 = 3 - 8x
8x = 3
x = 3/8
2. when x = 0
g(x) = 3 − 8x
g(x) = 3 − 8(0)
g(x) = 3
3. when g(x) = -5
-5 = 3 - 8x
8x = 3 + 5
8x = 8
x = 1
4. when x = 3
g(x) = 3 − 8x
g(x) = 3 − 8(3)
g(x) = -21
Hence the missing values of x are 3/8 and 1,
missing values of g(x) are 3 and -21.
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Contruct an iocele right-angled Triangle 'XYZ' in which angle Y=90° and two ide are 5cm each
Isosceles right angled triangle XYZ is constructed with the given measure of ∠Y =90° , YX = 5cm and YZ = 5cm each using steps of construction.
Constructed diagram is attached.
As given in the question,
Given measurement to construct a isosceles right angled triangle are:
Measure of ∠Y =90°
Measure of side length
YX = 5cm and YZ = 5cm
Constructed diagram is attached.
Step of construction :
Draw a ray.Take point Y on it and draw an arc of any measure.Take the measure more than half of the radius of the drawn arc.Take each end point as centre and draw an arc on the upper half plane of the ray.Draw ray perpendicular through point Y in the upper half plane.Now take measure of 5cm and point Y as centre and cut an arc on each of the ray.Name the point as 'X' and 'Z'.Join XZ .Isosceles right angled triangle is constructed with the given measurement.Therefore, construction of isosceles right angled triangle is done as per the above steps of construction.
The above question is incomplete, the complete question is :
Construct an isosceles right-angled Triangle 'XYZ' in which angle Y=90° and two side are 5cm each.
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Somebody help please I don’t understand
The constant of proportionality for respective problems is 2/7, 15, 2.75, 13, 4/3.
A) In the first block,
Using points we can find the equation of line ,
Using the formula (y - y₀) = m (x - x₀)
where m is the slope of the equation
The equation of line becomes 2x = 7y
⇒ y = 2/7x
⇒ 2/7 is the constant of proportionality
B) From the graph given,
Again Using the formula (y - y₀) = m (x - x₀)
If we consider two points,
(2,30) and (4,60)
The equation of line becomes
15x = y
Here, the constant of proportionality is 15.
C) It is given the equation is y = 2.75x
The constant of proportionality here is 2.75
D) It is given that,
The price of 5 games is $65.00
The price of 1 game will be 65 / 5
= $13.00
The equation becomes
y = 13x
So, the constant of proportionality becomes 13.
E) Let us consider 2 points (3,12) and (6,24)
The equation of line using the formula (y - y₀) = m (x - x₀) becomes 4x = 3y
⇒ y = 4/3 x
The constant of proportionality becomes 4/3.
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What is the exponential of 10?
The term "exponential of 10" means the numbers in which the base is 10 and the exponent is an integer.
The Exponential of 10 or the "powers of 10" means when the number 10 is multiplied a certain number of times, then the product is expressed using exponents.
These numbers which are written as the exponents are exponents of 10.
For example : 10² , 10³ represents different exponents of 10 .
In general form of the exponent of 10 or power of 10 is represented as 10ⁿ .
For Example : 10 to the exponent of 5 can be written as 10⁵, and
it's value is calculates as :
⇒ [tex]10^{5}=10\times\ 10 \times 10\times 10\times 10[/tex]
⇒[tex]10^{5}= 100000[/tex] .
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what is the answer to a=-4, b=-6, and c=5:b^2-4ac
[tex]b^2-4ac[/tex] when a is -4. b is -6, and c is 5
Substitute for variables with given values:
[tex]-6^2-4(-4)(5)[/tex]
[tex]=\fbox{116}[/tex]
Which coordinate comes first?
While writing the coordinates in the coordinate plane the coordinate that comes first coordinate is represented by x-coordinate.
As given in the question,
In the coordinate plane:
We draw two axis named as x-axis and y-axis.x-axis and y-axis intersect at origin with coordinate ( 0,0 ).x-axis is represented by the horizontal axis.y - axis is represented by the vertical axis.First coordinate is of horizontal axis and second one is of vertical axis.Coordinates in the coordinate plane are represented by ( x ,y )First coordinate is represented by x -coordinate.Therefore, the first coordinate in the coordinate plane is represented by x-coordinate.
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a right triangle is formed by fbe y-axis, the y = 2x + 4, and another line. if the legs of the right triangle intersect at (2, 8), what is the equation of the line other line of the triangle?
Replace -0.5 with the fraction -1/2 if needed.
======================================================
Explanation:
The legs of any right triangle are always perpendicular. They meet up to form a right angle, aka 90 degree angle.
The line y = 2x+4 goes through (2,8). This can be confirmed with a graph. Or note how plugging x = 2 leads to y = 8.
Therefore, y = 2x+4 forms one of the legs of this triangle.
The slope here is 2 = 2/1; the negative reciprocal is -1/2 = -0.5, which is the slope of the perpendicular line. We want this line to go through (2,8)
So,
y = mx+b
8 = -0.5*2+b
8 = -1+b
8+1 = b
9 = b
b = 9
The equation of the perpendicular line is y = -0.5x+9
Replace -0.5 with the fraction -1/2 if needed.
Check out the graph below.
An unusually wet spring has caused the size of a mosquito population in some city to increase by 7% each day. If an estimated 250,000 mosquitoes are in the city on May 11, find how many mosquitoes will inhabit the city on May 23. Use y250,000(1.07)x where x is number of days since May 11
The number of mosquitoes that will inhabit the city on May 23, given the increase in population, is 563, 048 mosquitos
How to find the number of mosquitoes ?The formula for finding the number of mosquitoes after May 11, is given as:
y = 250,000 ( 1. 07 ) ˣ
Given that x is the number of days since May 11, the number of days till May 23 is :
= 23 - 11
= 12 days
The number of mosquitoes is therefore:
= 250, 000 x ( 1. 07 ) ¹²
= 250, 000 x 2.252191588960823337718801
= 563, 048 mosquitos
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Is the residual always 0?
The residual square sum may equal 0. Your model will fit your data more closely if the residual sum of squares is small; conversely, if the residual sum of squares is large, your model will match your data less closely. A value of 0 indicates that your model fits the data perfectly.
What Exactly Is Residual Value?
The estimated value of an asset that is fixed at the end of its useful life or lease term is its residual value, which is often referred to as salvage value. In lease agreements, the residual value is one of the lessor's main methodologies for calculating the lessee's periodic lease payments.
A good residual value is what?
Check out the Car Lease Ratings section of our Lease Kit for a complete list of high-residual vehicle brands and models.
A car is probably not a good lease deal if the lease-end residual value is less than 50% of the MSRP (for a 36-month lease). 55% to 65% of MSRP would be a nice residual.
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Twelve less the product of two numbers, ‘a’ and ‘b’. what does it mean.
Answer: (a * b) - 12
Step-by-step explanation:
We will pull details for the sentence given to write an expression representing this situation.
In this case, these words mathematically mean:
less ➜ subtraction
product ➜ multiplication
Using this information, we will write our expression.
(a * b) - 12
Jon has to choose which variable to solve for in order to be able to do the problem below in the most efficient manner.
6 x + 3 y = 27. 5 x + 2 y = 21.
Which variable should he choose so that he can use substitution to solve the system?
Jon should solve for y in the first equation because the coefficients can be reduced by a common factor to eliminate the coefficient for y.
Jon should solve for x in the first equation because the coefficients can be reduced by a common factor to eliminate the coefficient for x.
Jon should solve for y in the second equation because the coefficients can be reduced by a common factor to eliminate the coefficient for y.
Jon should solve for x in the second equation because the coefficients can be reduced by a common factor to eliminate the coefficient for x.
Jon should solve for y in the first equation because the coefficients can be reduced by a common factor to eliminate the coefficient of y.
What are Linear Equations?Linear equations are equation involving one or more expressions including variables and constants and the variables are having no exponents or the exponent of the variable is 1.
Given linear equations are,
6x + 3y = 27
5x + 2y = 21
The coefficients of the first equation are multiples of 3.
Hence dividing throughout the first equation by the common factor 3, coefficient of y will be eliminated.
6x/3 + 3y/3 = 27/3
2x + y = 9
y = 9 - 2x
Solving by substitution, the second equation becomes,
5x + 2(9 - 2x) = 21
5x + 18 - 4x = 21
x = 3
y = 9 - 2x = 3
Hence Jon should could solve the problem by eliminating the coefficient of y in the first equation by a common factor 3.
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drew was shopping on amazon on black Friday. he purchased 5 DVDs all at the same price. the tax on his purchase was 6 bring the total cost to 61. what was the price of each DVD?
Based on the information provided, the price of each DVD is $11.
How to calculate the price of the DVD?To calculate the price of the DVD, two main variables must be considered:
Price of the DVD Total of units boughtTaxesWhat is the price of each DVD?Tax is a compulsory amount that is levied on goods and services by the government. Tax serves as a form of transfer of wealth from the private individuals to the government.
Taxes increase the cost of a good or service.
Now, let's calculate the cost of each DVD:
The price of each DVD = (total cost - tax) / number of DVDs bought
The price of each DVD = (61 - 6) / 5
The price of each DVD = 55 / 5
The price of each DVD = $11
Based on this, the price of each DVD was $11, you can double-check this below
$11 x 5 = $55 (prices for the DVDs) + 6 = $61 (total price)
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Andrea earns a flat fee of $20 per day plus an additional x dollars for each sale she makes. Andrea made 10 sales last Saturday. The equation 10x + 20 = 80 represents the amount, in dollars, that Andrea earned. How much does Andrea earn for each sale she
makes?
Answer:
x=6
Step-by-step explanation:
10x+20=80
10x=60
x=6
How do you describe the behavior of the graph relative to the x axis?
The behavior of the graph relative to the x axis can be described by the equation of the line.
The equation of a line is y = mx + b, where m is the slope and b is the y-intercept. The slope of a line represents the rate of change of the graph with respect to the x axis; a positive slope indicates the graph is increasing, while a negative slope indicates it is decreasing. The y-intercept is the point at which the line crosses the y axis. For example, if the equation of the line is y = 2x + 4, then the slope of the line is 2, meaning the graph increases by 2 units every time the x axis increases by 1 unit, and the y-intercept is 4, meaning the line crosses the y axis at the point (0, 4). By analyzing the equation of the line, you can determine the behavior of the graph relative to the x axis.
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What does it mean to dilate by 1 2?
Dilation is a type of transformation in geometry that changes the size of a shape without changing its orientation. When a shape is dilated by 1 2, it means that the shape is multiplied by a factor of 1 2.
This means that the shape is made half of its original size. The shape will be reduced to one-half of its original area and its perimeter will be halved as well. When a shape is dilated by a factor greater than 1, it will be expanded; when a shape is dilated by a factor less than 1, it will be reduced.
When a shape is dilated by 1 2, the center of dilatation is the same as the original shape's center of dilation. The scale factor of the dilation is 1 2, so the coordinates of the center of dilation will not change.
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How do you know if a domain is all real numbers?
A domain is all real numbers if it includes all of the values on the real number line (including negative and positive numbers, fractions, decimals, and irrational numbers).
1. A domain is the set of all possible values of an independent variable that produces a valid solution to a given equation.
2. To determine if a domain includes all real numbers, you must examine the set of values in the domain and see if it includes all the values on the real number line.
3. This includes negative and positive numbers, fractions, decimals, and irrational numbers.
4. If the domain includes all of these, then it is all real numbers.
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Can somebody help me on this one please I am so confused don’t know none of this stuff
rotation then reflection.
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Can someone help me out with this question fast!?!?!
Answer:
x=2
Step-by-step explanation:
2(4x-1)= -10(x-3)+4
(2×4x)-(1×2 )=(-10×x) -(3×-10) +4
8x -2 = -10x -(-30)+4
8x -2 = -10x +30+4
8x-2= -10x+ 34
8x+10x=34+2
18x=36 we do ÷18
x=2
(a) Write an equation and (b) solve. 7. The mountains received 4 times the snow a nearby valley received. If the difference in snowfall was 21 in., how much snow did the mountains get?
Answer:
I don't know what to say to someone who lost a lot of weight and don't know what to say to someone who lost a lot of weight and don't know what to say to someone who lost a lot of weight and don't know what to say to someone who lost a lot of weight and don't know what to say to someone who lost a lot of weight and don't know what to say to someone who lost a lot of weight and height for all the help and their kids the the other one of it and don't