Answer:
f(x) = [tex]3x^3 - 3x^2 - 6x[/tex]
Step-by-step explanation:
Which polynomial function has x intercepts -1,0, and 2 and passes through the point (1,-6)?
There are 3 distinct and real roots given in the question, which means that the function must be a third degree polynomial. The roots are -1, 0, and 2. This means that f(x) = 0 at these points. The general form of the cubic equation is given by:
f(x) = ax^3 + bx^2 + cx + d; where a, b, c, and d are arbitrary constants.
From the given data:
f(-1)=0 implies a*(-1)^3 + b*(-1)^2 + c(-1) + d = -a + b - c + d = 0. (Equation 1).
f(0)=0 implies a*(0)^3 + b*(0)^2 + c(0) + d = 0a + 0b + 0c + d = 0. (Equation 2).
f(2)=0 implies a*(2)^3 + b*(2)^2 + c(2) + d = 8a + 4b + 2c + d = 0. (Equation 3).
f(1)=0 implies a*(1)^3 + b*(1)^2 + c(1) + d = a + b + c + d = -6. (Equation 4).
Equation 2 shows that d = 0. So rest of the equations become:
-a + b - c = 0;
8a + 4b + 2c = 0; (Divide 2 on both sides of the equation to simplify).
a + b + c = -6
This system of equation can be solved using the Gaussian Elimination Method. Converting the system into the augmented matrix form:
• 1 1 1 | -6
• -1 1 -1 | 0
• 4 2 1 | 0
Adding row 1 into row 3:
• 1 1 1 | -6
• 0 2 0 | -6
• 4 2 1 | 0
Dividing row 2 with 2 and multiplying row 1 with -4 and add it into row 3:
• 1 1 1 | -6
• 0 1 0 | -3
• 0 -2 -3 | 24
Multiplying row 2 with 2 and add it into row 3:
• 1 1 1 | -6
• 0 1 0 | -3
• 0 0 -3 | 18
It can be seen that when this updated augmented matrix is converted into a system, it comes out to be:
• a + b + c = -6
• b = -3
• -3c = 18 (This implies that c = -6.)
Put c = -6 and b = -3 in equation 1:
• a + (-3) + (-6) = -6
• a = -6 + 3 + 6
• a = 3.
So f(x) = [tex]3x^3 - 3x^2 - 6x[/tex] (All conditions are being satisfied)!!!
Explain the rule for multiplying two negative integers. Use a number line or algebra tiles to illustrate three examples. Make a sketch of your work.
Multiplying two negative numbers would always give you a positive product
Example 1) -2 * -1 = 2
Example 2) -5 * -4 = 20
Example 3) -10 * -3 = 30
Factor the expression using the GCF. 44−11
[tex]\( 44 - 11 \)[/tex] factors to [tex]\( 33 \)[/tex] when using the greatest common factor method.
To factor the expression [tex]\( 44 - 11 \)[/tex] using the greatest common factor (GCF), we first need to find the GCF of the two numbers.
The numbers 44 and 11 have a common factor of 11.
Now, we can factor out 11 from both terms:
[tex]\( 44 - 11 = 11 \times (4 - 1) \)[/tex]
This simplifies to:
[tex]\( 44 - 11 = 11 \times 3 \)[/tex]
Finally, we calculate the product: [tex]\( 11 \times 3 = 33 \)[/tex]
So, the factored expression for [tex]\( 44 - 11 \)[/tex] using the GCF is [tex]\( 33 \)[/tex].
To summarize:
[tex]\( 44 - 11 = 11 \times (4 - 1) \)[/tex]
[tex]\( 44 - 11 = 11 \times 3 \)[/tex]
[tex]\( 44 - 11 = 33 \)[/tex]
Therefore, [tex]\( 44 - 11 \)[/tex] factors to [tex]\( 33 \)[/tex] when using the greatest common factor method.
Complete Question:
Factor the expression using the GCF. 44 - 11 = _____
Will mark brainliest if right
An item is regularly priced at $59. It is now priced at a discount of 55% off the regular price. What is the price now?
Answer: $32.45
Step-by-step explanation:
Because....
55% × 59 = $32.45
You could also write 55% as 0.55.
So it will look like this :
0.55 × 59 = 32.45.
It will still give you the same answer :)
* Hopefully tis helps:) Mark me the brainliest:)!!
What is the minimum y value on the graph of y = sinx - 6?
-7
-5
5
7
Answer:
The correct answer option is -7.
Step-by-step explanation:
We are asked to determine the minimum value of y on the graph of [tex]y = sin x - 6 [/tex].
[tex]y' = cos(x) = 0[/tex]
[tex]x = arccos(0) = \pm\frac{\pi }{2} +2k\pi[/tex] where 'k' is any integer.
So, [tex] y = sin ( \frac { \pi} { 2 } ) - 6 = 1 - 6 = -5 [/tex] (it is the absolute minimum value)
and [tex]y=sin(\frac{-\pi}{2}+2\pi )-6 = sin(\frac{3\pi}{2})-6 [/tex] = -7 (absolute minimum y value)
Answer:
The correct answer option is -7.
We are asked to determine the minimum value of y on the graph of . where 'k' is any integer.
So, (it is the absolute minimum value)and = -7 (absolute minimum y value)
DEFG is an isosceles trapezoid find the measure of E
Answer:
The last option (62 degrees)
Step-by-step explanation:
Angle F is the same measure as angle E, just like angle D is the same measure as G.
The measure of angle ∠E will be 62°. Then the correct option is D.
What is a trapezium?It is a polygon that has four sides. The sum of the internal angle is 360 degrees. In a trapezium, one pair of opposite sides are parallel.
DEFG is an isosceles trapezoid.
Then the angle ∠E and ∠F will be congruent.
∠E = ∠F
∠E = 62°
Then the correct option is D.
More about the trapezium link is given below.
https://brainly.com/question/22607187
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Someone help me out please
Answer:
Do you need work shown?
Step-by-step explanation:
(I don't know where to comment here)
69.96 divided by 132
0.53 is 69.96 divided by132
Answer:
The answer is 0.53
A phone number contains 7 digits. How many different numbers can be made using the digits 0–9 if the first digit is not 0 and all of the digits can be repeated?
The Options are:
A. 10 × 96
B. 9 × 107
C. 9 × 106
D. 107
Answer:
[tex]9 \cdot 10^6[/tex]
Step-by-step explanation:
A phone number contains 7 digits. How many different numbers can be made using the digits 0–9 if the first digit is not 0 and all of the digits can be repeated
In 7 digit phone number, the first number cannot be 0
So only 1 to 9 are used to get the first digit. 9 numbers can be used
other digits can use number from 0 to 9. 10 number can be used
So possible numbers can be made is
[tex]9 \ times \ 10 \ times \ 10 \ times \ 10 \ times \ 10 \ times \ 10 \ times \ 10[/tex]
[tex]9 \cdot 10^6[/tex]
The correct option is C.[tex]\ 9 \times 10^6[/tex]
To solve the problem of finding how many different [tex]7[/tex]-digit phone numbers can be made using the digits [tex]0–9[/tex], with the first digit not being 0 and digits allowed to repeat, we can follow these steps:
1. First Digit Choices
The first digit has 9 possible choices ([tex]1[/tex] through [tex]9[/tex]) since it cannot be 0.
2. Remaining Digits Choices
Each of the remaining [tex]6[/tex] digits can be any of the [tex]10[/tex] digits ([tex]0[/tex] through [tex]9[/tex]).
So, the total number of different [tex]7[/tex]-digit phone numbers can be calculated by multiplying the number of choices for each digit:
[tex]\[9 \text{ choices for the first digit} \times 10 \text{ choices for each of the remaining 6 digits}\][/tex]
This can be represented mathematically as:
[tex]\[9 \times 10^6\][/tex]
Calculating [tex]\(10^6\)[/tex]
[tex]\[10^6 = 1,000,000\][/tex]
So,
[tex]\[9 \times 1,000,000 = 9,000,000\][/tex]
The sum of the squares of two consecutive integers is 85. Using n to represent the smaller of the two consecutive integers, express this statement in algebraic form
The algebraic representation of the given statement is:
[tex]n^2 + (n + 1)^2 = 85[/tex]
Expanding and simplifying:
[tex]n^2 + (n^2 + 2n + 1) = 85[/tex]
[tex]2n^2 + 2n + 1 = 85[/tex]
[tex]2n^2 + 2n + 1 - 85 = 0[/tex]
[tex]2n^2 + 2n - 84 = 0[/tex]
Dividing the equation by 2 to simplify:
[tex]n^2 + n - 42 = 0[/tex]
Now, we can solve this quadratic equation using the quadratic formula:
[tex]n = [-b ± √(b^2 - 4ac)] / (2a)[/tex]
Where a = 1, b = 1, and c = -42:
n = [-(1) ± √((1)^2 - 4(1)(-42))] / (2(1))
n = [-1 ± √(1 + 168)] / 2
n = [-1 ± √169] / 2
n = [-1 ± 13] / 2
This yields two possible values for n:
n₁ = (-1 + 13) / 2 = 12 / 2 = 6
n₂ = (-1 - 13) / 2 = -14 / 2 = -7
Since n represents the smaller of the two consecutive integers, we discard the negative value.
Therefore, the smaller integer (n) is 6.
To solve this problem algebraically, we first translate the given statement into an equation. We know that the sum of the squares of two consecutive integers can be represented as[tex]n^2 + (n + 1)^2, v[/tex]where n is the smaller integer. Setting this expression equal to 85, we get the equation [tex]n^2 + (n + 1)^2 = 85.[/tex]
We then expand and simplify this equation to get a quadratic equation in standard form: [tex]2n^2 + 2n - 84 = 0.[/tex]
Next, we use the quadratic formula to solve for n, which gives us two possible values. Since we are looking for the smaller of the two consecutive integers, we discard the negative solution.
Thus, the smaller integer is n = 6.
Complete question:
The sum of the squares of two consecutive integers is 85. Using n to represent the smaller of the two consecutive integers, express this statement in algebraic form
simplify this expression 6x/9(x+y)
Simplify 6x/9 to 2x/3
2x/3(x + y)
Simplify
= 2x(x + y)/3
12 people are entered in a race. if there are no ties, in how many ways can the first three places come out.
Answer:
1320
Step-by-step explanation:
This is a simple permutation.
Since you start with 12 people and there are 3 winners, after every winner the number pf people available to win decrease.
First, there are 12 people, then after the first win there are 11 people, and after the 2nd win, for the third win the there are 10 people.
Thus, to find the total number of combinations, you multiply:
12*11*10=1320
Thus, there are 1320 possible combinations
How fast is a train going that makes a 330 mile trip in 4 hours? need help fast please!!
ANSWER
The train is moving 82.5mi/hr
EXPLANATION
How fast the train is going is the same as the speed of the train:
The speed of the train is calculated using the formula,
[tex]Speed = \frac{distance}{time} [/tex]
[tex]Speed = \frac{330}{4} [/tex]
[tex]Speed = 82.5[/tex]
Therefore the train is moving 82.5mi/hr
Grade 4
96. Shelly uses a scoop to fill a container with
flour. The scoop holds cup of flour.
If Shelly uses 12 scoops of flour to fill the
container, how many cups of flour does she
use?
Answer:
12
Step-by-step explanation:
Answer:
12
Step-by-step explanation:
The Pythagorean identity 1+cot^2 theta=csc^2 theta can be converted to the other Pythagorean identity 1+tan^2 theta=sec^2 theta. Show the steps on how to do that.
Answer: view image. its a proof (in red)
Step-by-step explanation:
How is the percent efficiency of a machine determined?
A. (force / distance) × 100%
B. (work output / work input) × 100%
C. (work input / work output) × 100%
D. (input force / output force) × 100%
The answer is B. I just answered this question earlier!
Answer:
i need the ansrew too
Step-by-step explanation:
hehe
A photo printer can print 78 color pictures in 24 seconds. Which equation represents the relationship between t, the time in seconds, and p, the number of pictures printed
3.25 • p = t
Or
p = t/3.25
Hope this helps!
The time it takes to print a number of photos can be shown as a proportionality relationship p = (78/24) * t, with p being the number of photos printed and t being time in seconds.
Explanation:The relationship between the time (t) needed to print pictures and the number of pictures (p) printed by a photo printer can be represented by a simple proportionality relationship. This means that we can express this relationship as a rate. In this case, the printer prints 78 pictures in 24 seconds, so we can write the relationship as:
p / t =78/24
Or in other words:
p = (78/24) * t
This equation suggests that for any given unit of time t, you simply multiply t by the rate (78/24) to find the number of pictures p that can be printed in that time.
Learn more about Proportionality relationship here:https://brainly.com/question/34960188
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g(x)=9+4x
h(x)=x+21÷5
Write (h×g)(x) as an
expression in terms of x.
(h×g)(x) =
For this case we have the following functions:
[tex]g (x) = 9 + 4x\\h (x) = \frac {x + 21} {5}[/tex]
We must find [tex](h * g) (x)[/tex]. By definition of composite functions we have to:
[tex](h * g) (x) = h (x) * g (x)[/tex]
So:
[tex](h * g) (x) = 9 + 4x * \frac {x + 21} {5}[/tex]
We apply distributive property:
[tex](h * g) (x) = \frac {9x + 9 * 21 + 4x ^ 2 + 4x * 21} {5}\\(h * g) (x) = \frac {9x + 189 + 4x ^ 2 + 84x} {5}\\(h * g) (x) = \frac {4x ^ 2 + 93x + 189} {5}\\[/tex]
Answer:
[tex](h * g) (x) = \frac {4x ^ 2 + 93x + 189} {5}[/tex]
A radius is _____ the diameter.
The answer is 1/2.
2r = d.
Hope this helps!
A Radius Of The Circle Is Always Half The Diameter. This Means That The Radius Is Half Way Across A Circle.
I need help ASAP 6b+30
What exactly does the question consist of
6(b+5) would be the factorization
What would be the answer to this?
Answer:
D
Step-by-step explanation:
Distribute the negative 1 in the second group.
[tex]5k^4*-1=-5k^4 \\ 5k^3*-1=-5k^3 \\ -k*-1=k[/tex]
Now add by combining like terms.
[tex]3k^4-5k^4=-2k^4 \\ -2k^3-5k^3=-7k^3 \\ k+k=2k[/tex]
[tex]-2k^4-7k^3+2k[/tex]
What is the range of possible sizes for side x?
Answer:
all real numbers
Step-by-step explanation:
How do I solve this?
Answer:
[tex]\large\boxed{x=\log_\frac{5}{3}5}[/tex]
Step-by-step explanation:
[tex]3^x=5^{x-1}\qquad\text{use}\ a^{n-m}=\dfrac{a^n}{a^m}\\\\3^x=\dfrac{5^x}{5^1}\qquad\text{multiply both sides by 5}\\\\5\cdot3^x=5^x\qquad\text{divide both sides by}\ 3^x\\\\5=\dfrac{5^x}{3^x}\qquad\text{use}\ \left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}\\\\5=\left(\dfrac{5}{3}\right)^x\qquad\text{logarithm both sides}\ \log_\frac{5}{3}\\\\\log_\frac{5}{3}5=\log_\frac{5}{3}\left(\dfrac{5}{3}\right)^x\qquad\text{use}\ \log_ab^n=n\log_ab\\\\\log_\frac{5}{3}5=x\log_\frac{5}{3}\dfrac{5}{3}\qquad\text{use}\ \log_aa=1\\\\\log_\frac{5}{3}5=x[/tex]
transform each polar equation to an equation in rectangular coordinates and identify its shape.
(a) r=6
(b) r= 2 cos theta
please show ur work
Answer:
(a) [tex]x ^ 2 + y ^ 2 = 6 ^ 2[/tex] circle centered on the point (0,0) and with a constant radius [tex]r = 6[/tex].
(b) [tex](x-1) ^ 2 + y ^ 2 = 1[/tex] circle centered on the point (1, 0) and with radio [tex]r=1[/tex]
Step-by-step explanation:
Remember that to convert from polar to rectangular coordinates you must use the relationship:
[tex]x = rcos(\theta)[/tex]
[tex]y = rsin(\theta)[/tex]
[tex]x ^ 2 + y ^ 2 = r ^ 2[/tex]
In this case we have the following equations in polar coordinates.
(a) [tex]r = 6[/tex].
Note that in this equation the radius is constant, it does not depend on [tex]\theta[/tex].
As
[tex]r ^ 2 = x ^ 2 + y ^ 2[/tex]
Then we replace the value of the radius in the equation and we have to::
[tex]x ^ 2 + y ^ 2 = 6 ^ 2[/tex]
Then [tex]r = 6[/tex] in rectangular coordinates is a circle centered on the point (0,0) and with a constant radius [tex]r = 6[/tex].
(b) [tex]r = 2cos(\theta)[/tex]
The radius is not constant, the radius depends on [tex]\theta[/tex].
To convert this equation to rectangular coordinates we write
[tex]r = 2cos(\theta)[/tex] Multiply both sides of the equality by r.
[tex]r ^ 2 = 2 *rcos(\theta)[/tex] remember that [tex]x = rcos(\theta)[/tex], then:
[tex]r ^ 2 = 2x[/tex] remember that [tex]x ^ 2 + y ^ 2 = r ^ 2[/tex], then:
[tex]x ^ 2 + y ^ 2 = 2x[/tex] Simplify the expression.
[tex]x ^ 2 -2x + y ^ 2 = 0[/tex] Complete the square.
[tex]x ^ 2 -2x + 1 + y ^ 2 = 1[/tex]
[tex](x-1) ^ 2 + y ^ 2 = 1[/tex] It is a circle centered on the point (1, 0) and with radio [tex]r=1[/tex]
What is the equation (4, 5) m=-1/4 solved in point slope form?
Answer:
y - 5 = -1/4(x - 4)Step-by-step explanation:
The point-slope form of an equation of a line:
[tex]y-y_1=m(x-x_1)[/tex]
We have m = -1/4 and the point (4, 5). Substitute:
[tex]y-5=-\dfrac{1}{4}(x-4)[/tex]
chris bought 4 4/5 pounds of raisins. he shared the raisins equally between himself and five friends. how many raisins did each person get? i already know the answer but i just need to show the work. please answer quickly!!
first off let's convert the mixed fraction to improper fraction, and then do the division, since it was divided among all 6, he and 5 friends.
[tex]\bf \stackrel{mixed}{4\frac{4}{5}}\implies \cfrac{4\cdot 5+4}{5}\implies \stackrel{improper}{\cfrac{24}{5}} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{24}{5}\div 6\implies \cfrac{24}{5}\div \cfrac{6}{1}\implies \cfrac{24}{5}\cdot \cfrac{1}{6}\implies \cfrac{24}{6}\cdot \cfrac{1}{5}\implies \cfrac{4}{1}\cdot \cfrac{1}{5}\implies \cfrac{4}{5}[/tex]
Answer:
0.8 or 4/5
Step-by-step explanation:
4/5=.8 4.8/6=.8
pls HURRY 12 Points!!!
what is the area of this trapezoid?
A:96 in²
B:132 in²
C:168 in²
D:1344 in²
Answer:
(B) 132 in²
Step-by-step explanation:
Top Length = 3 + 8 + 3 = 14 in
Bottom Length = 8 in
Area of the trapezoid
= 1/2 (14 + 8) x 12
= 1/2 (22) (12)
= 132 in²
Answer:
B
Step-by-step explanation:
The area (A) of a trapezoid is calculated using the formula
A = [tex]\frac{1}{2}[/tex] h (a + b)
where h is the perpendicular height and a, b the parallel bases.
here h = 12, a = AB = 8 and b = DC = 3+ 8 + 3 = 14
A = [tex]\frac{1}{2}[/tex] × 12 × (8 + 14) = 6 × 22 = 132 in² → B
Suppose you roll a regular 6-faced die. What is the probability of rolling: a 6?, a 2?, and a 4?
3/6 because there are 6 sides and there are 3 numbers that you want to roll. They are even numbers so if you want to roll half of the numbers but not the other half well you have 3/6
The area of a parallelogram is found using the formula bh, where b represents the length of the base and h represents the length of the height. What is the area of a parallelogram that has a base of 10 centimeters and a height of 12 centimeters?
A 1,012 square centimeters
B 120 square centimeters
C 22 square centimeters
D 1,200 square centimeters
Answer:
120 square centimeters
Step-by-step explanation:
The question says the formula already.
Area = bh
b = base length b = 10
h = height length h = 12
Area = 10 · 12
Area = 120 square centimeters
Answer: B. 120 square centimeters
Step-by-step explanation:
Given : The area of a parallelogram is found using the formula [tex]bh[/tex], where b represents the length of the base and h represents the length of the height.
Then , the area of a parallelogram that has a base of 10 centimeters and a height of 12 centimeters will be:-
[tex]\text{Area}=10\times12\\\\\Rightarrow\text{Area}=120\text{ square centimeters}[/tex]
Therefore , the area of the parallelogram = 120 square centimeters
The slope intercept form of the equation of a line that passes through point (-2, -13) is y = 5x - 3. What is the point slope form of the equation for this line?
A. y - 13 = 5(x - 2)
B. y + 13 = 5(x + 2)
C. y - 2 = 5(x - 13)
D. y + 2 = 5(x + 13)
Answer:
B. y+13=5(x+2)
Step-by-step explanation:
I personally do y=_+_(x-_) when solving these equations
Plug in -2 the x coordinate for (x- (-2) to be (x+3)
Plug in -13 for the y intercept.
5 from 5x is the slope.
Thus y=-13+5(x+3)
Then just move 13 to the other side
Divide. Write your answer in simplest form.
5/6 ÷ 8
Answer:
5/48
Step-by-step explanation:
Answer:
5/48
Step-by-step explanation: