Write a polynomial function of least degree with integral coefficients that has the given zeros. -1/3, 2/3, -1/4
Step-by-step explanation:
"integral" means here that the factors in the polynomial are integers.
we have 3 zeros.
the least degree of a polynomial with 3 zeros is 3.
and yes, that works also for 2 zeros (the degree must be at least 2, it must be at least a quadratic equation).
or 4 zeros (at least 4th degree).
it in general n zeros (at least nth degree).
constructing this out of the given zeros is easy.
what happens, when I multiply something by 0 ? the total result will be 0.
and so, we simply multiply 3 short terms with each other, where each term turns 0 for one of the given zeros.
what expression in x turns 0, when x = -1/3 ?
well : x + 1/3 or with integers 3x + 1 (multiplied by 3)
and for x = 2/3 ?
x - 2/3 or with integers 3x - 2
and for x = -1/4 ?
x + 1/4 or with integers 4x + 1
so, our polynomial function (of at least 3rd degree) is then
(3x + 1)(3x - 2)(4x + 1)
basically this could be already the result, depending on what your teacher wants.
for the fully extended form we need to do the multiplications :
(3x + 1)(3x - 2) = 9x² - 6x + 3x - 2 = 9x²- 3x - 2
(9x²- 3x - 2)(4x + 1) = 36x³ + 9x² - 12x² - 3x - 8x - 2 =
= 36x³ - 3x² - 11x - 2
the requested polynomial function is
f(x) = 36x³ - 3x² - 11x - 2
Please help!
Using the graph below, which of the following equations represents the line that is parallel to line FG and passes through the (8,−3) point?
my only answers can be the ones in the attached image (just had a random one ticked)
Answer:
the second option: [tex]7x+4y=44[/tex]
Step-by-step explanation:
So when a line is parallel, it means that it has the same slope and a different y-intercept, it's important that there is a different y-intercept, otherwise it would be the same line, and the "two lines" would intersect at infinite points.
Anyways by looking at the graph you have two points (-8, 5) and (-4, -2). So the run in this case was 4 and the rise was -7. This is a slope of -7/4. So we have the equation: [tex]y=-\frac{7}{4}x+b \text{ where b}\ne-9}[/tex]. Since it passes through the point (8, -3) we can plug that in as (x, y) to solve for b (the y-intercept)
Plug in (8, -3) as (x, y)
[tex]-3=-\frac{7}{4}(8)+b[/tex]
Multiply the -7/4 and 8
[tex]-3 = -14+b[/tex]
add 4 to both sides
[tex]11 = b[/tex]
So this gives us the equation:
[tex]y=-\frac{7}{4}x+11[/tex]
Since it's asking for it in standard form you move the 7/4 x to the other side
Add 7/4x to both sides
[tex]\frac{7}{4}x+y=11[/tex]
Multiply both sides by 4 to cancel out the fraction
[tex]7x+4y=44[/tex]
Mary has $6$ identical basil plants, and three different window sills she can put them on. How many ways are there for Mary to put the plants on the window sills
There are 28 ways for Mary to put the plants on the sills.
We can use the Binomial Theorem to solve this problem.
We have 6 plants and 3 sills, so we have6 + 3 − 1 = 8 total objects.
Since we have 8 objects, we can use the Binomial Theorem to expand [tex](x + y)^{8}[/tex].
The coefficient [tex]x^{6} y^{2}[/tex] will be the number of ways for Mary to put the plants on the sills. We can expand [tex](x + y)^{8}[/tex]
using the Binomial Theorem:
[tex](x + y)^{8}=(\left\ {{8}\atop {0}} \right. )x^{8}+ (\left\ {{8}\atop {1}} \right. )x^{7}y+(\left\ {{8}\atop {2}} \right. )x^{6}y^{2} +(\left\ {{8}\atop {3}} \right. )x^{5}y^{3} +(\left\ {{8}\atop {4}} \right. )x^{4}y^{4}+(\left\ {{8}\atop {5}} \right. )x^{3}y^{5} +(\left\ {{8}\atop {6}} \right. )x^{2}y^{6} +(\left\ {{8}\atop {7}} \right. )x^{1}y^{7} +(\left\ {{8}\atop {8}} \right. )y^{8}[/tex]
Since we are only interested in the coefficient of [tex]x^{6} y^{2}[/tex] , we can ignore all terms that do not have [tex]x^{6}[/tex] and [tex]y^{2}[/tex].
Therefore, we are left with [tex](\left\ {{8}\atop {2}} \right. )x^{6}y^{2}[/tex]
[tex](\left\ {{8}\atop {2}} \right. )=\frac{8!}{2!(8-2)!} =\frac{8!}{2!6!} =\frac{8*7*6!}{2!6!} =28\\[/tex]
Therefore, there are 28 ways for Mary to put the plants on the sills.
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Let P(x) = 3x³-ax+b. If P(1) = P(-1) and P(2)=16, find the values of a and b
P(1) = P(-1)
P(1) = 3 - a + b
P(-1) = -3 + a + b
-> 3 - a + b = -3 + a + b
-> 3 - a + b + 3 - a - b = 0
-> 6 - 2a = 0
-> a = 3.
P(2) = 24 - 2a + b -> 24 - (2a - b) = 16 -> 2a - b = 8
-> 6 - b = 8
b = -2.
So, a = 3 and b = -2
Recheck : P(1) = 3 - 3 + (-2) = -2
P(-1) = -3 + 3 + (-2) = -2 => P(1) = P(-1) (true)
P(2) = 24 - 6 + (-2) = 16.
Given that a function, g, has a domain of -20 ≤ x ≤ 5 and a range of -5 ≤ g(x) ≤ 45 and that g(0) = -2 and g(-9) = 6, select the statement that could be true for g.
Thw statement that's true for the function g is g(-13) = 20.
How to illustrate the information?From the information given, the function, g, has a domain of -20 ≤ x ≤ 5 and a range of -5 ≤ g(x) ≤ 45.
In this case, g(-13) = 20. Here, x = -13 is in the domain and 20 is also in the range. Therefore, this is true for g.
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A plane leaves town A for town B.at 0540hours.If the journey takes 6.5hours at what time does the plane reach its destination?
The plane reaches its destination at 12h10 minutes if the plane leaves town A for town B.at 0540hours
What is the time zone?For legal, commercial, and social purposes, a time zone is a region that observes a uniform standard time. Instead of strictly following longitude, time zones tend to follow the borders between countries and their subdivisions.
We have:
A plane leaves town A for town B.at 0540hours.If the journey takes 6.5hours
05h40min
Total time takes = 6 hours and 30 minutes
= 5 hours + 40 minutes + 6 hours + 30 minutes
= 11 hours + 70 minutes
= 12 hours 10 minutes
or
= 12h10 minutes
Thus, the plane reaches its destination at 12h10 minutes if the plane leaves town A for town B.at 0540hours
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Translate algebraic words to symbols. Traple the product of ab
The translation of the given algebraic words to symbols is: C. 3(ab).
How to Translate Algebraic Words to Symbols?The algebraic word, "triple" connotes "3 times'. "Product" means multiplication. Therefore, we would have the following translation:
Product of ab = ab
Triple of ab = 3(ab).
Therefore, the algebraic words translated to symbols is: 3(ab).
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7. A number m is such that when it is divided by 30, 36, and 45 the remainder is always 7, find the smallest possible value of m. (3 marks)
Answer:
187
Step-by-step explanation:
Least common multiple:
To find the required number,
Find the LCM of 30 , 36 and 45Then add 7 to the LCM30 = 2 * 3 * 5
36 = 2 * 2 * 3 * 3 = 2² * 3²
45 = 2 * 3 * 3 = 2 *3²
LCM = 2² * 3² * 5
= 4 * 9 * 5
= 180
Required number = 180 + 7
= 187
The Nuthouse offers a mixture of soy nuts and almonds, Almonds
sell for $7 per pound and soy nuts sell for $5.50 per pound. The
Nuthouse will make 20 pounds of the mixed nuts and sell the
mixture for $5.95 per pound. The following equations represent
the problem of keeping the cost of the mixture consistent with
the ingredients.
x + y = 20.
7x + 5.5y = 119.
x = pounds of almonds and y = pounds of soy nuts.
How many pounds of soy nuts should be used in the mixture?
URGENT!!!
Answer:
14 pounds
Step-by-step explanation:
The given equations can be solved for y by substituting for x. The first equation is convenient for writing x in terms of y.
Solutionx = 20 -y . . . . . . . subtract y from the first equation
7(20 -y) +5.5y = 119 . . . . . substitute for x in the second equation
140 -1.5y = 119 . . . . . . . . simplify
21 = 1.5y . . . . . . . . . . . add 1.5y -119 to both sides
14 = y . . . . . . . . . . . .divide by 1.5
14 pounds of soy nuts should be used in the mixture.
__
Additional comment
There are many ways to solve a system of two linear equations. The attachments shows a matrix solution using a suitable calculator. It tells us that x=6 and y=14, as we found above.
which number produced a rational number when added to 1/5
The number that produced a rational number when added to 1/5 is D. -2/3.
What is a rational number?A rational number is a number that can be written as a ratio (or fraction) of two integers.
For example, when 1/5 is added to -2/3, it produces a rational number, -7/15, that remains as a fractional value.
Thus, the number that produced a rational number when added to 1/5 is D. -2/3.
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Given that f(x)=x2−4 and g(x)=x+3 , what are all the values of x for which f(g(x))=0 ?
Answer:
the answer is (x )
math way helps
Step-by-step explanation:
The requried function of function f(g(x)) is given as x² + 6x +5.
What are functions?Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.
here,
Given functions,
f(x)=x²−4 and g(x)=x+3
In order to form a function of function we have to function g(x) in function f(x)
Now,
f(g(x)) = f(x)=(g(x))²−4
f(g(x)) = f(x)=(x+3)²−4
f(g(x)) = x² + 6x + 9 -4
f(g(x)) = x² + 6x +5
Thus, the requried function of function f(g(x)) is given as x² + 6x +5.
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Rhombus wxyz is graphed on a coordinate plane. what is the area of the rhombus? 24 square units 28 square units 32 square units 48 square units
The area of the rhombus WXYZ is 48 unit².
From the diagram:
Diagonal WY = 8 units, Diagonal XZ = 6 units, hence:
Area = 8 * 6 = 48 unit².
A rhombus is a special case of a parallelogram. In a diamond, the opposite sides are parallel and the opposite sides are equal in angle. In addition, all sides of the rhombus are the same length and the diagonal bisects at right angles. Diamonds are also known as diamonds.
The diagonals of the rhombus intersect at right angles to form a non-uniform triangle. The opposite angles are equal. However, if all angles of the diamond are 90 degrees, the diamond is said to be square.
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Determine whether the geometric series 27 + 18 + 12 + 8 + ... converges or diverges, and identify the sum if it exists.
The given geometric series as shown in the question is seen to; Be converging with its' sum as 81
How to identify a converging or diverging series?We are given the geometric series;
27 + 18 + 12 + 8 + ...
Now, we see that;
First term; a₀ = 27
Second Term; a₁ = 2(27/3)
Third term; a₂ = 2²(27/3²)
Fourth term; a₃ = 2³(27/3³)
Thus, the formula is;
2ⁿ(27/3ⁿ)
Applying limits at infinity gives;
2^(∞) * (27/3^(∞)) = 0
Since the terms of the series tend to zero, we can affirm that the series converges.
The sum of an infinite converging series is:
S_n = a/(1 - r)
S_n = 27/(1 - (2/3)
S_n = 81
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Triangle A C B is cut by line segment E D. Line segment E D goes from side B C to side A C. The length of B A is 4 x minus 6 and the length of E D is x + 2. The length of B E is x. Sides B E and E C are congruent. Sides A D and D C are congruent.
What is the length of BC?
From the markings on the diagram, we can tell E is the midpoint of BC and
is the midpoint of AC
We can apply the
theorem: ED = One-halfBA.
Substituting in the expressions for the lengths and solving for x, we get x =
.
Now, since BE = x, then BC =
.
From the markings on the diagram, we can tell E is the midpoint of BC and D is the midpoint of AC. We can apply the triangle midsegment theorem: ED = ½BA. Substituting in the expressions for the lengths and solving for x, we get x = 5. Now, since BE = x, then BC = 10.
What is triangle midpoint theorem?Triangle midpoint theorem states that the line segment which joins the midpoints of two (2) sides of a triangle is parallel to the third side, and it's congruent to one-half of the third side.
By applying the triangle midpoint theorem, we can find the value of x:
ED = ½BA
x + 2 = ½(4x - 6)
2x + 4 = 4x - 6
4x - 2x = 6 + 4
2x = 10
x = 10/2
x = 5.
BC = x + x
BC = 5 + 5
BC = 10.
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The angle by which turns clockwise about point b to coincide with is 63.9 degrees. if from point b, a point e is drawn directly opposite point c so that b, e, and c are on the same straight line, the angle by which turns counterclockwise to coincide with is 243.9 degrees.
AB turns clockwise to coincide with BC.
To find the angle by which turns counterclockwise to coincide with is 243.9 degrees:
An angle -
An angle is a figure in Euclidean geometry created by two rays, called the sides of the angle, that share a common termination, called the vertex of the angle. Angles formed by two rays are located in the plane containing the rays. Angles are also generated when two planes intersect. These are known as dihedral angles.So,
= ABD + DBC
= 33.3° + 30.6°
= 63.9°
b) If E is drawn directly opposite C.
EBC is a straight line, so the sum of angles =180°
ABE + ABD + DBC = 180°
ABE + 33.3° + 30.6° = 180°
ABE + 63.9° = 180°
ABE = 180 - 63.1
ABE = 116.9
Therefore, AB turns clockwise to coincide with BC.
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3y > 2x + 12
2x + y ≤ -5
The solution for the inequalities 3y>2x+12, 2x+y<=-5 is the value of x<-27/8 and y<7/4.
Given 3y>2x+12 and 2x+y<=-5.
We are given two inequalities 3y>2x+12 and 2x+y<=-5. Inequality are like equations but in greater than ,less than or in combination with equal to. To solve them we need to first write them properly.
2x-3y<-12
2x+y<=-5
Now assume these are equalities
2x-3y=-12
2x+y=-5
now subtract 2 from 1
2x-3y-2x-y=-12+5
-4y=-7
y=7/4
put the value of y in 2x-3y=-12
2x-3(7/4)=-12
2x-21/4=-12
2x=-12+21/4
2x=(-48+21)/4
2x=-27/4
x=-27/8
Now put the signs between the values
x<-27/8
y<7/4.
Hence the solution of the inequalities 3y > 2x + 12
2x + y ≤ -5 is the value of x<-27/8 and y<7/4.
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-x+4y=9
pls find ordered pair
We can write the ordered pairs that meet this condition as:
(x, y) = (x, (9 + x)/4 )
How to find the ordered pair?An ordered pair on the X-Y plane is written as (x, y).
Here we also have the relation:
-x + 4y = 9
If we isolate y, we get:
4y = 9 + x
y = (9 + x)/4
Now, we can write the ordered pairs that meet this condition as:
(x, y) = (x, (9 + x)/4 )
To get the particular ones you can replace the value of x, for example, if x = 0, then:
(x, (9 + x)/4 ) = (0, 9/4)
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The table gives the scores of 6 students from a class of 25 in a competitive exam. the point estimate of the mean score for the students is
The mean of the scores if 6 students in a competitive exam is 25 marks.
Given scores of 6 students are: 10,30,50,40,20 in a competitive exam.
We have to find out the mean of the mark of 6 students in a competitive exam.
Mean is the sum of all the values in a set of data, such as numbers, measurements, divided by the number of values. It is also known as average.
Mean=∑X/n
∑X=10+30+50+40+20
=150
N=6
Now to calculate the mean we have to divide 150 by number of students which is 6.
Mean=150/6
=25 marks.
Hence the mean of the scores of 6 students is 25 marks.
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Question is incomplete as it should include :
Marks of students =10,30,50,40,20.
A support wire that is attached from the top of an observation tower to 20 meters away on the ground. if the support wire and the ground form an angle of 46 degrees, what is the length of the support wire?
Trigonometric ratios are defined as the ratio of different sides of a right angle triangle with respect to one angle of the right angled triangle.
Length of the wire = Hypotenuse of the triangle = X
Base of the triangle = Distance from the bottom of the tower to the point where wire is attached on ground = 20 m
Angle b/w Hypotenuse and Base of triangle = 46°
cos(θ) = Base / Hypotenuse
cos(46) = 20 / X
X = 20 / cos(46)
X = 28.79 m
Hence, the length of the wire is 28.79 m.
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Answer this question correctly and you will be marked the Brainiest! Please don’t answer for points.
Answer:
x=4+√5 or x=4−√5
Step-by-step explanation:
Multiply both sides by x-4.
x2−8x+16=5
x2−8x+16−5=5−5(Subtract 5 from both sides)
x2−8x+11=0
For this equation: a=1, b=-8, c=11
1x2+−8x+11=0
x=
−b±√b2−4ac
2a
(Use quadratic formula with a=1, b=-8, c=11)
x=
−(−8)±√(−8)2−4(1)(11)
2(1)
x=
8±√20
2
x=4+√5 or x=4−√5
Norman makes $15 commission for selling $200 in merchandise. he makes $37.50 commission for selling $500 in merchandise. assuming that the variables are directly related and norman wants to make $150 in commission, what is the total value of merchandise he needs to sell?
So he needs to sell $2,000 in merchandise.
what is the total value of merchandise he needs to sell?
We can assume that we have a proportional relation:
y = k*x
Where y is the commission and x is what Norman sells, then:
$15 = k*$200
k = $200/$15 = 0.075
$37.50 = k*$500
k = $500/$37.50 = 0.075
Then the relation is:
y = 0.075*x
If he wants a commission of $150, then:
$150 = 0.075*x
$150/0.075 = x = $2,000
So he needs to sell $2,000 in merchandise.
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According to the rational root theorem, the numbers below are some of the potential roots of f(x) = 10x3 29x2 – 66x 27. select all that are actual roots.
The actual roots of the function [tex]f(x)=10x^{3}+29x^{2} -66x+27[/tex] are -9/2, 3/5 and 1.
Given function [tex]f(x)=10x^{3}+29x^{2} -66x+27[/tex].
Function is a relationship between two or more variables expressed in equal to form.
The roots of a polynomial function are the zeroes of the polynomial function. A polynomial function is a function that involves only non negative integer powers in an equation.
The polynomial function is given as:
[tex]f(x)=10x^{3}+29x^{2} -66x+27[/tex]
factorize the above function
[tex]f(x)=(2x+9)(5x-3)(x-1)[/tex]
Now put the function f(x) equal to zero.
f(x)=(2x+9)(5x-3)(x-1)
split the function means put all the expressions equal to zero as under:
(2x+9)(5x-3)(x-1)=0
solve each for the value of x
x=-9/2,x=3/5,x=1
Hence the roots of the function [tex]f(x)=10x^{3} +29x^{2} -66x+27[/tex] are which are also the values of x are -9/2,3/5,1.
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Answer:
Step-by-step explanation:
A company produces candy bags that each hold about 528 cubic inches of candy. Each bag is filled with any mixture of lollipop candies and gummy bear candies. When a bag contains only lollipop candies, then it has about 361 candies. When a bag contains only gummy bear candies, then it has about 697 candies. Given any candy bag produced by this company, which of the following equations could relate the approximate number of lollipop candies, 7, in the bag and the approximate number of gummy bear candies, g, in the bag? A company produces candy bags that each hold about 528 cubic inches of candy . Each bag is filled with any mixture of lollipop candies and gummy bear candies . When a bag contains only lollipop candies , then it has about 361 candies . When a bag contains only gummy bear candies , then it has about 697 candies . Given any candy bag produced by this company , which of the following equations could relate the approximate number of lollipop candies , 7 , in the bag and the approximate number of gummy bear candies , g , in the bag ?
361 lollipop candies or 697 gummy bear candies fills the bag of volume 528 in.³, which gives the following possible equation;
1.46•l + 0.76•g = 528How can the correct equation be found?Volume of candy bag = 528 in.³
Number of lollipop candies the bag can hold = 361 candies
Number of gummy bear candies the bag can hold = 697 candies
Therefore;
[tex]1 \: lollipop = \frac{528}{361} = 1.46[/tex]
[tex]1 \: gummy \: bear = \frac{528}{697} = 0.76[/tex]
Which gives the following equation;
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Answer:
l/361 + g/697= 1
Khan Academy Sat Practice
Find the distance between the two points in simplest radical form.
(−5,−4) and (−2,−6)
Answer: [tex]\sqrt{13}[/tex] units
Work Shown:
[tex](x_1,y_1) = (-5,-4) \text{ and } (x_2, y_2) = (-2,-6)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(-5-(-2))^2 + (-4-(-6))^2}\\\\d = \sqrt{(-5+2)^2 + (-4+6)^2}\\\\d = \sqrt{(-3)^2 + (2)^2}\\\\d = \sqrt{9 + 4}\\\\d = \sqrt{13}\\\\d \approx 3.6056\\\\[/tex]
I used the distance formula.
A slightly alternate method is to form a right triangle and use the pythagorean theorem. The hypotenuse will have the endpoints (-5,-4) and (-2,-6).
What is the exact value
of cos
7pi/8
The exact value of cos 7pi/8 is 0. 9988
How to find the value
It is important to note that the value for pi = 3. 142
= cos 7π
Substitute value of π
= cos(7× 3. 142)
= cos 21. 994
Divide 21. 994 by 8
= 2. 740
Let's find the vale of [tex]cos \frac{21. 994}{8}[/tex]
= [tex]cos (2. 749)[/tex]
= [tex]0. 9988[/tex]
Thus, the exact value of cos 7pi/8 is 0. 9988
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Based on the form of the equation, what is the best way to graph the line 3x+3y=15? Select the correct answer below: Recognize the equation as that of a vertical line passing through the x-axis at 15. Recognize the equation as that of a horizontal line passing through the y-axis at 15. Identify the slope and y-intercept, and then graph. Find the x- and y-intercepts, and then graph.
Tayloe travels 35 miles every 2 days to work. how many days would it take him to travel 100 miles . answer in fraction
The number of days Tayloe will take to travel 100 miles if he travels 35 miles every 2 days is 40/7 or 5 5/7 in fraction form.
A fraction is a representation of a part of the whole.
Written as a/b, read as "a by b", it functions as a number of parts of a total of b number of parts.
In the question, we are given that Tayloe travels 35 miles every 2 days to work.
Therefore, daily traveling done by Tayloe can be shown as a fraction, 35/2 miles.
The total traveling given to us is 100 miles.
Thus, the number of days Tayloe will take to travel 100 miles can be shown in fractional form as 100/(35/2) = (100*2)/35 = 40/7 or 5 5/7.
Thus, the number of days Tayloe will take to travel 100 miles if he travels 35 miles every 2 days is 40/7 or 5 5/7 in fraction form.
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D
Distance
ST
Speed
Time
A bus is moving at an average speed of 70 mph on the
motorway.
The journey takes 3 hours 30 minutes.
How far did the bus travel?
miles
The distance would be 231 miles.
What is speed?Speed can be calculated as the ratio of distance traveled to the time taken
Given information;
Time taken = 3 hours 30 minutes
Speed = 70 mph
We know that,
Distance = speed x time
Distance = 70 x 3.30
Distance = 231 miles
Thus, the distance would be 231 miles.
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A single card is drawn from a standard 52 deck of cards.work out the probability of choosing a heart
I honestly have no idea how to slove this question
For real,I don't know. OK
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In the figure, POG is a diameter of the circle with centre O.
Find x° + y°.
The value of x° + y° is 216°
How to determine the angles
Note that the shape is a regular pentagon
Sum of angle in a pentagon = 540
Angle on each side = 108°
x = 108°
y = 108°
x° + y° = 108 + 108
x° + y° = 216°
Thus, the value of x° + y° is 216°
Learn more about a pentagon here:
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