The polynomials x^3 - 2x, x^2 + x - 1, and x^3 - 5 do not generate (span) P3.
To determine if the polynomials x^3 - 2x, x^2 + x - 1, and x^3 - 5 generate (span) P3, where P3 represents the set of all polynomials of degree 3 or lower, we need to examine if any polynomial in P3 can be expressed as a linear combination of these three polynomials.
Let's take an arbitrary polynomial in P3, denoted as ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
We want to find coefficients k1, k2, and k3 such that:
k1(x^3 - 2x) + k2(x^2 + x - 1) + k3(x^3 - 5) = ax^3 + bx^2 + cx + d
Expanding and rearranging the terms, we have:
(k1 + k3)x^3 + (k2 + b)x^2 + (k2 + c)x + (-2k1 - k2 - 5k3 - d) = ax^3 + bx^2 + cx + d
For these two polynomials to be equal for all values of x, their corresponding coefficients must be equal. Therefore, we can equate the coefficients:
k1 + k3 = a
k2 + b = b
k2 + c = c
-2k1 - k2 - 5k3 - d = d
Simplifying these equations, we have:
k1 = a - k3
k2 = b - c
-2(a - k3) - (b - c) - 5k3 - d = d
Rearranging terms, we obtain:
-2a + 2k3 - b + c - 5k3 - d = d
Simplifying further, we get:
-2a - b - d - 3k3 + c = 0
This equation must hold for all values of a, b, c, and d. Therefore, k3 must be chosen in such a way that the equation holds for any values of a, b, c, and d.
However, it is not possible to find a value for k3 that satisfies the equation for all possible polynomials in P3. Thus, we conclude that the polynomials x^3 - 2x, x^2 + x - 1, and x^3 - 5 do not generate (span) P3.
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Given the parent cosine function f(x) = cos x , find the function g(x) after the parent function undergoes a horizontal shift right 7 units, and up 4 units.
By applying the transformation rules for horizontal and vertical translation, we find that the resulting function is g(x) = cos (x - 7) + 4.
How to find the resulting function by transformation rules
Transformation rules are rules that makes changes on charateristics and behavior of a function to create a new one. Rigid transformations like horizontal and vertical translations are examples of transformation rules. In this question we must apply the following transformation rules to the parent cosine function f(x) = cos x:
Horizontal translation: f'(x) = f(x - 7) (1)Vertical translation: g(x) = f'(x) + 4 (2)Now we proceed to derive the resulting function by applying the rules defined above:
Horizontal translation
f'(x) = cos (x - 7) (3)
Vertical translation
g(x) = cos (x - 7) + 4 (4)
By applying the transformation rules for horizontal and vertical translation, we find that the resulting function is g(x) = cos (x - 7) + 4.
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Help, please. I just need an answer, so if anyone is bored, for the love of god please...
Answer:B & E
Step-by-step explanation:
We can first rearrange the function to isolate y. Then, we can find the slope as the function is in the form y=mx+b.
[tex]3x-4y=7\\-4y=-3x+7\\y=\frac{3}{4}x-\frac{7}{4}[/tex]
Since parallel lines have the same slope, we can put the slope of 3/4 into the point slope form to get the answer.
For reference, the point-slope form is [tex]y-y_1=m(x-x_1)[/tex]
[tex]y+2=\frac{3}{4}(x+4)\\y+2=\frac{3}{4}x+3\\y=\frac{3}{4}x+1[/tex]
The first line is found in option E, so option E is one of the correct options.
We can also move the x to the other side, as two of the 5 options have both variables on the left (B and C).
[tex]-\frac{3}{4}x+y=1[/tex]
If we multiply the whole equation by -4, we can get rid of the fraction.
[tex]-4(-\frac{3}{4}x+y)=-4(1)\\3x-4y=-4[/tex]
Hence, option B is also correct.
Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar. On a number line, point D is at -3, and point E is at 6. The point F lies on . The ratio of to is 2 : 3. Where does point F lie on the number line
On the number line the position of point F is at point 4.
According to the statement
We have a given that the on a number line , point D is at -3, and point E is at 6. The point F lies on the ratio of to is 2 : 3.
and we have to find the F point on the number line.
Firstly,
Number line is a line on which numbers are marked at intervals, used to illustrate simple numerical operations.
So, According to given information,
Point D is at -3, and point E is at 6.
Ratio of point E to F = 6*2/3
Then
Ratio of point E to F = 6*2/3
After solving it
The point F = 2*2
The point F = 4.
So, On the number line the position of point F is at point 4.
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1. when y=7 , what is the value of x?
2. what is the y-intercept of the graph?
1) Correct.
2) The y-intercept is when x=0, so it is (0,5).
a cone has a height of 16 centimeters and a radius of 12 centimeters. what is the exact lateral and surface area of the cone? type the correct answer in each box. use numerals instead of words.
The lateral and total surface areas of the given cone are 753.98 cm² and 1206.31 cm² respectively.
What are the formulae for lateral and total surface areas of a cone?A cone has a height 'h', radius 'r', and slant height 'l'.
The slant height of the cone is obtained by the Pythagorean theorem. I.e.,
l² = h² + r²
Then,
Its lateral surface area(LSA) = πr([tex]\sqrt{h^2+r^2}[/tex]) square units and
Its total surface area(TSA) = πr(r + l) square units
Calculation:It is given that,
A cone has a height h = 16 cm and radius r = 12 cm
Then, the slant height is calculated by
l² = h² + r²
l = [tex]\sqrt{h^2+r^2}[/tex]
On substituting,
l = [tex]\sqrt{16^2+12^2}[/tex]
= [tex]\sqrt{400}[/tex]
= 20 cm
So,
LSA = πr([tex]\sqrt{h^2+r^2}[/tex])
= π × 12 × ([tex]\sqrt{16^2+12^2}[/tex])
= π × 12 × 20
= 753.98 cm²
and
TSA = πr(r + l)
= π × 12 × (12 + 20)
= π × 12 × 32
= 1206.37 cm²
Therefore, the lateral and total surface areas of the cone with a height of 16 cm and a radius of 12 cm are 753.98 cm² and 1206.37 cm².
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Answer: The lateral area is 240π square centimeters. The total surface area is 384π square centimeters.
Step-by-step explanation:
Derrick had a 0.250 batting average at the end of his last baseball season, which means he got a hit 25% of the times he was up to bat. if derrick had 47 hits last season, how many times did he bat?
The number of times that Derrick batted last season is 188 times.
How many times did Derrick bat?
Percentage can be described as a fraction out of an amount that is usually expressed as a number out of hundred. Percentage is a measure of frequency. The sign used to denote percentage is %.
A percentage of 25% here means that twenty five times out of hundred times, Derrick hit the ball. In order to convert a percentage to a decimal, divide the percentage by 100.
Number of times Derrick bat last season = Number of hits last season / percentage of his batting average
47 / 25%
47 / 0.25 = 188
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Find all solutions to 2w^4 - 5w^2 + 2 = 0
asap please
Answer:
w = ±[tex]\sqrt{1/2}[/tex] or w= ±[tex]\sqrt{2}[/tex]
Step-by-step explanation:
if we say some variable y = w^2, we can rewrite the equation to:
2y^2 - 5y + 2 = 0
this can be factored into (2y-1)(y-2) = 0
putting w^2 back in the place of y, that's (2w^2 - 1)(w^2 - 2) = 0
The equation is a fourth degree polynomial, so there are four roots, or four values of w that will cause the equation to equal 0.
If 0 is multiplied by anything, the result is 0, so we set 2w^2 - 1 = 0 and solve for w, which is ±√1/2, then set w^2 - 2 = 0 to get w = ±√2 as our roots
the four solutions are ±√1/2 and ±√2
(because the positive counts as one solution and the negative another solution)
A motorcycle travels 20 miles west and then turns and goes 52° north of west another 38 miles. How far is the motorcycle from its starting point?? Round your answer to the nearest tenth
__ miles
Answer:
52.7 miles
Step-by-step explanation:
Please refer to attached photo. (Apologies for the terrible drawing).
We can deduce from A to B the distance is 20 miles, therefore no calculations is required.
However, from B to C, we will have to find the horizontal and vertical distances.
Horizontal Distance B > C = = hBC = [tex]BCcos(52)[/tex] = [tex]38cos(52) miles[/tex]
Vertical Distance B > C = vBC = [tex]BCsin(52)[/tex] = [tex]38sin(52)miles[/tex]
Here you can see the whole diagram is a right angle triangle. Which means, we can use Pythagoras' Theorem to find AC.
By Pythagoras Theorem,
[tex]c^{2} =a^{2}+b^{2}[/tex]
[tex]AC^{2} = (AB + hBC)^{2} + vBC^{2} \\AC^{2} = (20+38cos(52))^{2} +(38sin(52))^{2} \\AC = \sqrt{(20+38cos(52))^{2} +(38sin(52))^{2} } \\AC = 52.7 miles[/tex](nearest tenth)
What is the relation between the variables in the equation
a. varies inversely as y
b.xvaries directly as y
Please select the best answer from the choices provided
O
O
A
B
C
O
y
-7?
c. y varies directly as x¹
d. y varies inversely as ¹
The relation between the variables in the equation x^4/y=7 is (b) x^4 varies directly as y
How to determine the relation between the variables in the equation?The complete question is added as an attachment
From the attached figure, we have the following equation
x^4/y = 7
Multiply both sides of the equation by y
x^4 = 7y
The above represents a direct variation from x^4 to y.
Where 7 represents the variation constant
Hence, the relation between the variables in the equation x^4/y=7 is (b) x^4 varies directly as y
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Apply the distributive property to factor out the greatest common factor. 24j-16=
Answer:
8(3j - 2).
Step-by-step explanation:
24j-16
GCF = 8
so the answer is
8(3j - 2).
find the slope of a line parallel to the line through the given points. E(5, 7), F(3, 1) •-3
•-1/3
•1/3
Answer:
3
Step-by-step explanation:
slope = (y_2 - y_1)/(x_2 - x_1)
slope = (1 - 7)/(3 - 5)
slope = -6/(-2)
slope = 3
Parallel lines have equal slopes.
Answer: 3
d) Arrange 8/9,7/18,10/16 in ascending order of magnitude
8/9 = 0.888888....
7/18 = 0.388888888889
10/16 = 0.625
Ascending order - means going up, smallest to largest. (Imagine climbing a ladder, where you start of low, then, go higher up the ladder.)
Thus, answer is 7/18, 10/16, 8/9
Hope this helps!
The following uses a 4-step proof, starting with the given statement. Complete the following proof
Given: JK | MN
Prove: 1, 2 are complementary
1) [tex]\overline{JK} \perp \overline{MN}[/tex]
2) [tex]\angle JKM[/tex] is a right angle (perpendicular lines form right angles)
3) [tex]\triangle JKM[/tex] is a right triangle (definition of a right triangle)
4) [tex]\angle 1[/tex] and [tex]\angle 2[/tex] are complementary (the acute angles of a right triangle are complementary)
Solve 5x=125 using the one-to-one property of exponents.
A) X=In(125)
B) x = 1
C) x = 3
OD) x = 5
Answer:
C) x = 3
Step-by-step explanation:
Given equation:
[tex]5^x=125[/tex]
Rewrite 125 with base 5: 125 = 5³
[tex]\implies 5^x=5^3[/tex]
[tex]\textsf{Apply the one-to-one property of exponents}: \quad a^{f(x)}=a^{g(x)} \implies f(x)=g(x)[/tex]
[tex]\implies x=3[/tex]
PLEASE HELP (the picture has the problem)
Help with these two problems and show work please !!
Answer:
1) [tex]x_1=-1-2\sqrt{2},\ x_2=-1+2\sqrt{2}[/tex]
2) [tex]x_1=\dfrac{-5 - \sqrt{13}}{6},\ x_2=\dfrac{-5 + \sqrt{13}}{6}[/tex]
Step-by-step explanation:
[tex]{\large \textsf{ Quadratic Formula: }}x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}\ \Bigg{\|}\ \textsf{when }ax^2+bx+c=0[/tex]
Given quadratic equations:
1. [tex]x^2+2x-7=0[/tex]
2. [tex]4x^2-3=x^2-5x-4[/tex]
1. x² + 2x - 7 = 0
[tex]\implies a=\textsf{1},b=\textsf{2},c=\textsf{-7}[/tex]
Step 1: Substitute the given values into the formula and simplify.
[tex]\begin{aligned}\implies x&=\dfrac{-(\textsf{2})\pm \sqrt{(\textsf{2})^2-4(\textsf{1})(\textsf{-7})}}{2(\textsf{1})}\\\implies x&=\dfrac{-2\pm \sqrt{4-4(-7)}}{2}\\\implies x&=\dfrac{-2\pm \sqrt{4+28}}{2}\\\implies x&=\dfrac{-2\pm \sqrt{32}}{2}\end{aligned}[/tex]
Step 2: Simplify the radicand (under the square root).
[tex]\begin{aligned}x&=\dfrac{-2\pm \sqrt{16\times2}}{2}\\x&=\dfrac{-2\pm 4\times\sqrt{2}}{2}\\x&=\dfrac{-2\pm 4\sqrt{2}}{2}\end{aligned}[/tex]
Step 3: Separate into two solutions and simplify them.
[tex]\implies x_1&=\dfrac{-2 - 4\sqrt{2}}{2},\ x_2&=\dfrac{-2 + 4\sqrt{2}}{2}[/tex]
[tex]\begin{aligned}\implies x_1&=\dfrac{-2}{2}+\dfrac{- 4\sqrt{2}}{2},\ x_2=\dfrac{-2 + 4\sqrt{2}}{2}\\\implies {x_1&=\boxed{-1-2\sqrt{2}},\ x_2=\boxed{-1+2\sqrt{2}} \end{aligned}[/tex]
----------------------------------------------------------------------------------------------------------------
2. 4x² - 3 = x² - 5x - 4
Step 1: Set the equation to zero (by moving the "x² - 5x - 4" to the left).
4x² - 3 - x² + 5x + 4 = 0 [ Combine like terms. ]
3x² + 5x + 4 = 0
[tex]\implies a=\textsf{3},b=\textsf{5},c=\textsf{1}[/tex]
Step 2: Substitute the given values into the formula and simplify.
[tex]\begin{aligned}\implies x&=\dfrac{-(\textsf{5})\pm \sqrt{(\textsf{5})^2-4(\textsf{3})(\textsf{1})}}{2(\textsf{3})}\\\implies x&=\dfrac{-5\pm \sqrt{25-12}}{6}\\\implies x&=\dfrac{-5\pm \sqrt{13}}{6}\end{aligned}[/tex]
Step 3: Separate into two solutions.
[tex]\implies x_1=\dfrac{-5 - \sqrt{13}}{6},\ x_2=\dfrac{-5 + \sqrt{13}}{6}[/tex]
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Find the equation of the line that passes through the given points. (1, 4.5) and (3, 6)
Answer:
y=0.75x+3.75
Step-by-step explanation:
The slope is
[tex] \frac{6 - 4.5}{3 - 1} = \frac{3}{4} [/tex]
Substituting into point-slope form,
y - 6 = 0.75(x - 3)
y - 6 = 0.75x - 2.25
y = 0.75x + 3.75
Hi I don't know how to do this
Using a system of equations, the weight of 5 apples, 2 oranges are 4 bananas is given as follows:
B. 1147 gm.
What is a system of equations?A system of equations is when two or more variables are related, and equations are built to find the values of each variable.
In this problem, the variables are:
Variable x: Weight of an apple.Variable y: Weight of an orange.Variable z: Weight of a banana.Considering the data given, the equations are:
3x + 5y = 928.4y + 6z = 1088.5x + 3z = 799.From the first equation:
3x = 928 - 5y.
x = 309.33 - 1.667y.
From the second equation:
6z = 1088 - 4y
z = 181.33 - 0.667y
Replacing in the third equation:
5x + 3z = 799
5(309.33 - 1.667y) + 3(181.33 - 0.667y) = 799
10.336y = 12961.64
y = 1291.64/10.336
y = 125 gm.
The other weights are:
x = 309.33 - 1.667y = 309.33 - 1.667 x 125 = 101 gm.z = 181.33 - 0.667y = 181.33 - 0.667 x 125 = 98gm.The weight of 5 apples, 2 oranges are 4 bananas is:
5x + 2y + 4z = 5 x 101 + 2 x 125 + 4 x 98 = 1147 gm, option B.
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successor of -501 is
Answer:
-500
Step-by-step explanation:
since 501 has a negative sign,add 1 to to the number,the next number will be -500
-501 +1 = -500
successor in this case means a number that succeeds another
250-{(135+34)÷(46-33)}-15 simplify
Answer:
222
Step-by-step explanation:
Hey there!
250 - {(135 + 34) ÷ (46 - 33)} - 15
= 250 - 169/(46 - 44) - 15
= 250 - 169/13 - 15
= 250 - 13 - 15
= 237 - 15
= 222
Therefore, your answer should be:
222
Good luck on your assignment & enjoy your day!
~Amphitrite1040:)
Graph the line with the slope 1/3 and x-intercept 3
Step-by-step explanation:
Use the slope-intercept form to find the slope and y-intercept.
Slope:
−
1
3
y-intercept:
(0 , 3)
Any line can be graphed using two points. Select two
x
values, and plug them into the equation to find the corresponding
y
values
Graph the line using the slope and the y-intercept, or the points.
Slope:
−
1
3
y-intercept:
(0 , 3)
The variability around a regression line, which determines the fit of a regression line, can be calculated in Excel. Which function does Excel use for this residual variance?
Excel uses VAR function for determining the residual variance.
What is residual variance?The variation of any error is known as residual variance, also known as unexplained variance or error variance.
Depending on the type of analysis you're doing, the precise definition will vary. For instance, random fluctuations in regression analysis produce volatility around the "actual" regression line.There are two components to a regression line's total variance: variation that can be explained and variance that cannot. Simply put, the residual variance after deducting the variation resulting from the regression from the total variance of the dependent variable is the residual variance.
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True or false: the mayans wrote the digits in their numerals in a vertical format
Answer:
True
Step-by-step explanation:
Which is the graph of the function f(x) = x3 + x2 + x + 1?
Answer:
Step-by-step explanation:
the answer is on the graph
f(x) = 2x – 4
g(x) = 3x + 1
what is is h(x) = f(x)g(x)?
Answer:
h(x) = 6x² - 10x - 4
Step-by-step explanation:
h(x) = f(x) g(x)
= (2x - 4)(3x + 1)
each term in the second factor is multiplied by each term in the first factor, that is
2x(3x + 1) - 4(3x + 1) ← distribute parenthesis
= 6x² + 2x - 12x - 4 ← collect like terms
= 6x² - 10x - 4
that is h(x) = 6x² - 10x - 4
22. If 1 m,m/19 - 118°, and m/16 - 48°, calculate m/12.
0 0
103"
98
109⁰
110"
18
17
19 20
14
16
13
15
11
10
12
9
63
5
m
Answer: [tex]110^{\circ}[/tex]
Step-by-step explanation:
[tex]m\angle 20=62^{\circ}[/tex] (linear pair)
[tex]m\angle 11=70^{\circ}[/tex] (angle sum in a triangle)
[tex]m\angle 12=110^{\circ}[/tex] (linear pair)
Can someone help me with the first question please
Answer:
(c) 5y -17x +99 = 0
Step-by-step explanation:
The median of a triangle is the line through a vertex and the midpoint of the opposite side. The median of ΔXYZ from vertex Y will be the line through point Y and the midpoint of XZ.
MidpointThe midpoint of XZ is the average of the coordinates of X and Z.
M = (X +Z)/2
M = ((1, -2) +(8, -7))/2 = (9, -9)/2 = (4.5, -4.5)
Line through two pointsThe slope of the median can be found using the slope formula:
m = (y2 -y1)/(x2 -x1)
The slope of line YM is ...
m = (-4.5 -4)/(4.5 -7) = -8.5/-2.5 = 17/5
The point-slope form of the equation of a line is ...
y -k = m(x -h) . . . . . . line with slope m through point (h, k)
The line with slope 17/5 through point Y(7, 4) is ...
y -4 = 17/5(x -7)
Subtracting the right side, and multiplying by 5 gives ...
5(y -4) -17(x -7) = 0
5y -17x +99 = 0 . . . . equation of the median through Y
Point A is located in which quadrant?
IV
III
I
II
Answer:
Quadrant II
Step-by-step explanation:
This is because the top right is I, the top left
is II, the bottom left is III, and the bottom right is IV.
(03.01 MC)
Simplify
the square root of 4 divided by 3 to the third power , the square root of 4
Answer: [tex]\frac{2}{9}[/tex]
Step-by-step explanation:
[tex]\frac{\sqrt{4} }{3^{3} }[/tex]
The square root of 4 can be written as 2
3 to the third power can be written as 9
So, the answer is [tex]\frac{2}{9}[/tex]
Melissa is planning a rectangular vegetable garden with a square patch for tomatoes
The expressions for the length and the width of the rectangular garden are 3x + 2 feet, and x + 5 feet respectively where x is the length of the square patch for tomatoes.
A square is a quadrilateral with all four sides equal, and all four angles at 90°, whereas a rectangle is a quadrilateral with opposite pair of sides equal, and all four angles at 90°.
In the question, we are informed that Melissa is planning a rectangular vegetable garden with a square patch for tomatoes.
The side of the square is given to be x feet.
We are asked to write expressions for the length and width of the rectangular garden.
For the length of the rectangle:-
We are informed that she wants the length of the garden to exceed three times the length of the tomato patch by 2 feet.
Thus, the length of the garden can be shown as the expression 3x + 2 feet.
For the width of the rectangle:-
We are informed that she also wants the garden's width to exceed the width of the tomato patch by 5 feet.
Thus, the width of the garden can be shown as the expression x + 5 feet.
Thus, the expressions for the length and the width of the rectangular garden are 3x + 2 feet, and x + 5 feet respectively where x is the length of the square patch for tomatoes.
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The provided question is incomplete. The complete question is:
"Melissa is planning a rectangular vegetable garden with a square patch for tomatoes. She wants the length of the garden to exceed three times the length of the tomato patch by 2 feet. She also wants the garden's width to exceed the width of the tomato patch by 5 feet.
Let x represent the length, in feet, of the square tomato patch.
Write expressions to represent the length and width of Melissa's vegetable garden in terms of x.
Enter the correct answer in the box."