The factoring of the polynomial f(x) = x^4 + 3x³ - 2x² + 48x - 288 is given as follows:
f(x) = (x - 4i)(x + 4i)(x + 6)(x - 3).
How to factor the polynomial?The polynomial is factored by it's linear factors, given as follows:
f(x) = (x - x')(x - x'')(x - x''')(x - x'''').
In which x', x'', x''' and x'''' are the roots of the polynomial f(x).
In this problem, we have that -4i is a root, hence, by the Conjugate Roots Theorem, 4i is also a root and hence:
f(x) = (x - 4i)(x + 4i)(x - x''')(x - x'''').
f(x) = (x² + 16)(x - x''')(x - x'''').
As a fourth order polynomial can be the product of two second order polynomials, hence:
x^4 + 3x³ - 2x² + 48x - 288 = (x² + 16)(ax² + bx + c)
x^4 + 3x³ - 2x² + 48x - 288 = ax^4 + bx³ + (c + 16)x² + 16bx + 16c.
Hence the coefficients are given as follows:
a = 1.b = 3.16c = -288 -> c = -18.Then the equation for the final two factors is:
x² + 3x - 18 = 0.
Which is simplified as follows:
x² + 3x - 18 = (x + 6)(x - 3).
Hence the simplified polynomial is:
f(x) = (x - 4i)(x + 4i)(x + 6)(x - 3).
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Activity No. 16
Find the roots of the following radical expressions.
1. √64
2. ⁴√ -1
3. ³√1000
Write each expression in radical form or with a rational exponent, and simplify.
1. (⁴√ 1)⁷
2. ³ √-64
3. (2⁹)¹/³
4. (11²)¹/²
The required roots of the given radical expressions are √64 = 8, ⁴√ -1 = 4i, and ³√1000 = 10. The required expressions for the given radical are ³√-64 = 12i, (2⁹)¹/³ = 8, and (11²)¹/²=11.
What are the radical expressions?Any expression that contains a radical symbol is considered a radical expression. To find the square roots, cubic roots, and higher, radical symbols are used. For example, √7, √6, and √11 are in the radial form.
The radical expressions are given in the question
As per the given question, the required solution would be as:
√64 = √(2 × 2 × 2 × 2 × 2 × 2)
√64 = 2 × 2 × 2 = 8
⁴√ -1 = 4i (apply the imaginary number rule)
³√1000 = ³√(10 × 10 × 10)
³√1000 = 10
(⁴√ 1)⁷ = (1)⁷ = 1
³√-64 = (³√2 × 2 × 2 × 2 × 2 × 2 × -1 )
³√-64 = 2 × 2(³√-1)
³√-64 = 4(3i) = 12i (apply the imaginary number rule)
(2⁹)¹/³ = [tex]2^{9/3} = 2^3 = 8[/tex]
(11²)¹/² = [tex]11^{2/2} = 11[/tex]
Thus, the required roots of the given radical expressions are √64 = 8, ⁴√ -1 = 4i, and ³√1000 = 10. The required expressions for the given radical are ³√-64 = 12i, (2⁹)¹/³ = 8, and (11²)¹/²=11.
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Which equation represents a linear function?
A. y = -x² - 4
B.-3x² + 1 = y
C.y=x²
D.x+2=y
Answer: D. x + 2 = y
Step-by-step explanation:
The only answer option that represents a linear function is option D. The other three, A, B and C, all represent quadradic functions. You can tell they're quadradic functions because the highest degree is a 2, or a square. This gives a parabola and is not linear.
With this information in mind, the answer is;
"D.x+2=y"
Choose a value of k so that the relation below is not a function.
{(-2,8),(0,3),(k,1),(-4,7)}
Responses
2
5
-1
-2
Answer:
-2
Step-by-step explanation:
it cannot be a function if there are more than one of the same x’s
Find a recursive rule for the sequence 3, 6, 12, 24, ...
Answer:
?
Step-by-step explanation:
c.r.=6/3=12/6=2
[tex]c.r=\frac{a_{n}}{a_{n-1}} \\\frac{a_{n}}{a_{n-1}} =2\\a_{n}=2a_{n-1}[/tex]
Three of the following statements are true. Which one is NOT true?
|-12| > 1
Ice skaters are skating on a solidly frozen lake. Which of the following could possibly be the temperature of the lake
Answer:
Below 30⁰F or 0⁰C
Step-by-step explanation:
Water freezes at those temperatures
Question 7 of 10
A construction worker needs to put a rectangular window in the side of a
building. He knows from measuring that the top and bottom of the window
have a width of 8 feet and the sides have a length of 15 feet. He also
measured one diagonal to be 17 feet. What is the length of the other
diagonal?
A. 8 feet
OB. 17 feet
O C. 15 feet
OD. 23 feet
The length of the other diagonal is 17 feet.
What is diagonal?
When two polygonal or polyhedral vertices are not on the same edge, a diagonal is a line segment connecting them.
Main Body:
Given:
A construction worker needs to put a rectangular window on the side of a building.
He knows from measuring that the top and bottom of the window have a width of 8 feet and the sides have a length of 15 feet.
He also measured one diagonal to be 17 feet.
Solution:
As we know that the rectangle is a particular type of parallelogram.
So, the rectangle has diagonals that have equal lengths.
Therefore, one diagonal length is given as 17 feet.
Hence, the other diagonal length will be equal to the given diagonal length.
So, the length of the other diagonal is 17 feet.
Hence, Option B is the correct answer.
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Which is the graph of f(x) = 100(0.7)x?
On a coordinate plane, a curve approaches y = 0 in quadrant 1 and curves up into quadrant 2. It goes through (2, 49) and (0, 100).
On a coordinate plane, a curve approaches y = 0 in quadrant 1 and curves up into quadrant 2. It goes through (2, 9) and (0, 100).
On a coordinate plane, a curve approaches y = 0 in quadrant 1 and curves up into quadrant 2. It goes through (2, 70) and (0, 100).
On a coordinate plane, a curve approaches y = 0 in quadrant 1 and curves up into quadrant 2. It goes through (2, 30) and (0, 100).
The graph of f(x) = 100(0.7)^x is represented by On a coordinate plane, a curve approaches y = 0 in quadrant 1 and curves up into quadrant 2. It goes through (2, 49) and (0, 100)
How to find the pointsUsing the exponential function f(x) = 100(0.7)^x the graph is plotted and attached
Other information that can be abstracted from the equation directly
f(x) = 100(0.7)^x
at x = 2
f(x) = 100(0.7)^2
f(x) = 100(0.49)
f(x) = 49
hence the ordered pair is written as (2, 49)
f(x) = 100(0.7)^x
at x = 0
f(x) = 100(0.7)^0
f(x) = 100(1)
f(x) = 100
hence the ordered pair is written as (0, 100)
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The answer is A. its correct on EDGE 2022
can you help me please
Answer:
[tex]-2 > x[/tex]
Step-by-step explanation:
We can solve the given inequality like we'd solve a regular equation. The only problem we should be looking out for is division/multiplication by a negative number - doing so would result in flipping the direction the inequality sign is facing (i.e. dividing by -1 would change < to > and vice versa.)
[tex]3x + 7 > 4x + 9 \text{ // Subtract 3x}\\3x + 7 - 3x > 4x + 9 - 3x\\\to 7 > x + 9 \text{ // Subtract 9}\\7 - 9 > x + 9 - 9\\\to -2 > x[/tex]
3x + 7 > 4 x + 9
STEPS/SOLUTION3x + 7 > 4 x + 9
1ST STEP (SUBTRACT 7 BOTH SIDES)3x + 7 - 7 > 4 x + 9- 73x > 4x + 22ND STEP (SUBTRACT 4X BOTH SIDES)3x - 4x > 4x + 2 - 4x-x > 23RD STEP (DIVIDE BOTH SIDE BY THE SAME FACTOR, AND FLIP ALSO THE SIGN AS IT IS NEGATIVE)-x/-1 < 2/-1-1 ÷ -1 and 2 ÷ -1ANSWERx < -2Ascending order 2 1/3 2.3 23.5% 25/9
Ascending order - least to greatest
Answer: 2.35%, 2.3, [tex]2\frac{1}{3}[/tex], [tex]\frac{25}{9}[/tex]
Explanation:
0.235 (23.5%) is the least
2.3 (the actual decimal in the entire group) is the second least
2.33 ([tex]2\frac{1}{3} =\frac{7}{9}[/tex] as an improper fraction and [tex]\frac{7}{9} = 2.33[/tex]is the third least
2.78 ([tex]\frac{25}{9}[/tex] is already an improper fraction. In decimal version, it is 2.78) is the greatest.
------------------------------------------------------------------------
Have a great day and God bless! :D
The Rogers family and the Walker family each used their sprinklers last summer. The water output rate for the Rogers family's sprinkler was 30 L per hour. The
water output rate for the Walker family's sprinkler was 20 L per hour. The families used their sprinklers for a combined total of 45 hours, resulting in a total
water output of 1150 L. How long was each sprinkler used?
Rogers family's sprinkler:
hours
Walker family's sprinkler: hours
The Rogers family's sprinkler worked for 25 hours and Walker family's sprinkler worked for 20 hours
The water output rate of Roger's family's sprinkler = 30 liter per hour
The water output rate of Walker family's sprinkler = 20liter per hour
Total number of hours combined = 45 hours
Consider the number of hours that Rogers family's sprinkler work = x
The number of hours that Rogers family's sprinkler work = y
Then the equation will be
x + y = 45
x = 45 -y
Total water output = 1150 liters
30x + 20y = 1150
Here we have to use substitution method
30(45-y) + 20y = 1150
1350 - 30y + 20y = 1150
1350 - 10y = 1150
10y = 1350 - 1150
10y = 200
y = 200/10
y = 20 hours
Then,
x = 45 - y
x = 45 - 20
x = 25 hours
Hence, the Rogers family's sprinkler worked for 25 hours and Walker family's sprinkler worked for 20 hours
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The following data were collected from a random sample of people, who identified their favorite type of juice. The results are shown in the following two-way table.
What proportion of the children identified orange as their favorite type of juice?
A 4001,0004001,000
B 400700400700
C 4002,0004002,000
D 6001,3006001,300
E 1,0002,000
The proportion of the children identified orange as their favorite type of juice is 400 / 1000.
What is proportion?In general, the term "proportion" refers to a part, share, or amount that is compared to a total.
According to the concept of proportion, two ratios are in proportion when they are equal.
A mathematical comparison of two numbers is called a proportion. According to proportion, two sets of provided numbers are said to be directly proportional to one another if they increase or decrease in the same ratio. "::" or "=" are symbols used to indicate proportions.
Given:
From the table
Adults who like oranges= 600
and, Children who like oranges = 400
So, proportion of the children identified orange as their favorite type of juice
= 400/ 1000
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PLEASE HELP find the measure of each marked angle
Through the use of a protractor, angles are measured in degrees (°). In order to calculate or draw angles in terms of degrees, a protractor is a measuring tool that is utilized.
Does a linear pair have a 180 degree angle?
Two lines converge at a single point to generate a pair of linear angles. If, following the junction of the two lines, the angles are next to one another, they are said to be linear. A linear pair always has an angle total of 180 degrees.Angles next to each other that add up to two right angles, or 180 degrees, are said to be in a linear pair. This indicates that the non-shared sides of them are aligned in a straight line.When the total of two angles equals 180 degrees, the angles are said to be supplementary. It is referred to as a linear pair if the two additional angles are close to one another.How to find the measure of the angle?
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IXL Please Help Fast !
Answer:
x=9
Step-by-step explanation:
9x-7=5x+29
9x=5x+36
4x=36
x=9
Hopes this helps please mark brainliest
caroline bought 3 eye shadows and 4 lipsticks and paid a total of 39.53 write an equation to represent this situation assuming each eye shadow cost 4.79 how much did caroline pay for each lipstick
ANSWER:
The equation used to represent the situation is 3x + 4y = 39.53, and Caroline spent 6.29 per lipstick.
EXPLANATION:
We will use x for eyeshadow and y for lipstick.
The equation would be: 3x + 4y = 39.53
3(4.79) + 4y = 39.53
14.37 + 4y = 39.53
14.37 - 14.37 + 4y = 39.53 - 14.37
4y = 25.16
4y / 4 = 25.16 / 4
y = 6.29
Check.
3x + 4y = 39.53
3(4.79) + 4(6.29) = 39.53
39.53 = 39.53
I hope this helps! :D
The perimeter of the triangle is 10x^2-6x+1 and the side lengths are 5x+7 and 8x^2-4x.What is the length of the missing side?
Answer:
2x^2-7x-6
Step-by-step explanation:
P=A+B+C=10x^2-6x+1
A=5x+7
B=8x^2-4x
C=?
C=P-(A+B)= 10x^2-6x+1-(5x+7+8x^2-4x)
=10x^2-6x+1-(8x^2-6x+7)
=2x^2-7x-6
The given quadrilateral ABCD is an isosceles trapezoid. Which choice is NOT true about ABCD?
A) 2x + 10 = 4x - 30
B) angle A+ angle B+ angle C+ angle D=180°
C) angle A+ angle B+ angle C+ angle D=360°
D) AB = CD
Solve for angle DAB.
A) 50°
B) 20°
C) 30°
D) 65°
Answer:
B) angle A+ angle B+ angle C+ angle D=180°A) 50°========================
GivenABCD is an isosceles trapezoidSolutionPart 1Choices:
A) 2x + 10 = 4x - 30
TRUE, as A and D are congruent anglesB) angle A+ angle B+ angle C+ angle D=180°
NOT true, as the sum of interior angles is 360° not 180°.C) angle A+ angle B+ angle C+ angle D=360°
TRUED) AB = CD
TRUE, as marked same.Part 2Find the value of x:
2x + 10 = 4x - 304x - 2x = 10 + 302x = 40x = 20Find the measure of angle DAB:
m∠DAB = 2*20 + 10 = 50°Correct choice is A.
Answer:
b) A + B + C + D = 180° a) 50°Step-by-step explanation:
1) Finding the incorrect option?
The incorrect statement is,
→ angle A+ angle B+ angle C+ angle D=180°
(Because, A + B + C + D = 360°)
Hence, option (b) is the answer.
2) Solving for angle DAB?
Forming the equation,
→ 2x + 10 = 4x - 30
Now the value of x will be,
→ 2x + 10 = 4x - 30
→ 2x - 4x = -30 - 10
→ -2x = -40
→ x = -40/-2
→ x = 40/2
→ [ x = 20° ]
Then value of angle DAB is,
→ A = 2x + 10
→ A = 2(20) + 10
→ A = 40 + 10
→ [ A = 50° ]
Hence, option (a) is correct.
4. Mr. Smith makes lemonade. His recipe calls for 1½ cups of water for every cup of concentrate. If Mr. Smith 2 pours all 5 cups of concentrate into a pitcher to serve at a party, how much water should he add?
He should add 7.5 or 7 1/2 cups of water.
Cup of concentrate = 1
Cup of water = 1 1/2
= 3/2
= 1.5
Cups of concentrate = 5
Cups of water = 5 * 1.5
= 7.5
= 15/2
= 7 1/2
Hence, He should add 7.5 or 7 1/2 cups of water.
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Question 5 of 7
Question 1-5
Submit Test
22-23 Algebra 1 Unit 2 EUA
A hairstylist charges $15 for an adult haircut and $9 for a child haircut. She wants to earn at least $360 dollars and cut a maximum of 30
haircuts this week. The graphs represent the hairstylist's constraints.
List two points that could represent the numbers of adult and child haircuts that meet the hairstylist's goals. Enter your response in the
space provided. You may also use the drawing tool to help explain or support your answer.
The point that could represent the numbers of adults and child haircuts that meet the stylist goals exists 25, 0.
What is meant by mathematical operation?A function that converts zero or more input values into a clearly defined output value is referred to in mathematics as an operation. The operation's complexity is measured by the number of operands.
A value is computed using operands and a math operator as part of the mathematical "operation." In order to be applied to the provided operands or numbers, the math operator's symbol has predetermined rules.
Since the hairstylist change $15 for an adult haircut and $9 for a child haircut, then utilizing 25, 0 will be:
= 15(25) + 9(0)
= 375 + 0
= $375
She has earned at least 360 dollars and cut about 25 haircuts this week.
Therefore, the correct answer is option A. 25, 0.
The complete question is:
A hairstylist change $15 for an adult haircut and $9 for a child haircut. She wants to earn at least 360 dollars and cut a maximum of 30 haircuts this week. The graphs represent the hairstylist constants.
Which point could represent the numbers of adults and child haircuts the meet the stylist goals.
A. 25, 0
B. 15, 20
C. 10,15
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Find the remainder when f(x) is divided by x-k
f(x)=2x^4+x^3-4x^2-4x+4
k=-2
The remainder when the given function is divided by x +2 is 36
The given function is
f(x) = 2x⁴+x³-4x²-4x+4
Remainder theorem states that if any polynomial is divided by another factor in the form of x-a , then the remainder will be given by f(a)
Hence remainder = f(-2)
f(-2) = 2(-2)⁴ + (-2)³ - 4(- 2)²- 4 (-2) + 4 = 32 - 8 + 8 + 4 = 36
The Remainder Theorem can be used to tackle polynomial division according to Euclidean geometry. The Remainder Theorem is also known as Bezout's Theorem regarding Approaching Polynomials of Euclidian's Division.
The remainder theorem says that when a polynomial f(x) is divided by the factor (x - a), even though the factor isn't always an element of the polynomial, a smaller polynomial and a remnant are created. Because the result is actually the value of the polynomial f(x), where x = a, this can only occur if f(a) = 0. The remainder theorem can be used to factor polynomials quickly.
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find the perimeter 24 4
Answer:
56
Step-by-step explanation:
24x2+4x2
Simplify
48+8
Solve
56
need the answer. ............................
Answer:
The value of cos 0 is 1.Here, we will discuss the value for cos 0 degrees and how the values are derived using the quadrants of a unit circle. The trigonometric functions are also known as an angle function that relates the angles of a triangle to the length of the triangle sides.
Trigonometric functions are one of the most important topics which are used in the study of periodic phenomena like sound and light waves, the study of harmonic oscillators and finding the average temperature variations.
The three basic trigonometric ratios are sine function, cosine function, and tangent function. It is commonly defined for the angles less than a right angle, trigonometric functions are stated as the ratio of two sides of a right triangle containing the angle in which the values can be found in the length of various line segments around a unit circle. The angles of a triangle are calculated with respect to sin, cos and tan functions. Usually, the degrees are represented as 0°, 30°, 45°, 60°, 90°, 180°, 270° and 360°.Cos 0 Degree ValueTo define the cosine function of an acute angle, start with the right-angled triangle ABC with the angle of interest and the sides of a triangle. The three sides of the triangle are defined as follows:The opposite side is defined as the side opposite the angle of interest.The hypotenuse side is the side opposite the right angle and it is the longest side of a right triangleThe adjacent side is the remaining side where it forms a side of both the angle of interest and the right angleThe cosine function of an angle is equal to the length of the adjacent side divided by the length of the hypotenuse side and the formula is given by:Cos θ = Adjacent Side / Hypotenuse SideValue of Cos 0 Using Unit CircleAssume a unit circle with the center at the origin of the coordinate axes. Consider that P (a, b) be any point on the circle which forms an angle AOP = x radian. This means that the length of the arc AP is equal to x. So we define that cos x = a and sin x = b
Step-by-step explanation:
lots to read but answer is 1
PLS HELP ME RIGHT NOW WILL GIVE BRAINLEST
The required equation of the line is y = -(0.5)x + 225 expressed in slope-intercept form.
Let x represent the number of months Jason's gift card has been inactive after 1 year and let y represent Jason's card balance.
We take the two points (0, 225) and (1, 224.5) in the given table.
The required equation of the line is
⇒ y - y₁ = (y₂ - y₁)/(x₂ -x₁ )[x -x₁]
x₁ = 0, y₁ = 225
x₂ = 1, y₂ = 224.5
Substitute values in the equation, we get
y - 225 = (224.5-225 /1-0)(x - 0)
y - 225 = -(0.5)x
y = -(0.5)x + 225
Therefore, the required equation of the line is y = -(0.5)x + 225.
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help please real answers *10 points*!!!!!
Answer: B SSS
Step-by-step explanation:
Which expression is equivalent to
Help me figure this out please
Two polygons are similar if their respective sides have a constant ratio and their corresponding angles are congruent (in other words, if they are proportional).
What is polygons?Two polygons are similar if their respective sides have a constant ratio and their corresponding angles are congruent (in other words, if they are proportional).Two polygons that share the same shape but not the same size are said to be similar polygons. Congruent angles and proportional sides are present in similar polygons, as well as similar angles in dissimilar polygons.Rectangles have a lot in common. Rhombuses have a lot in common. Squares resemble each other. Similarity exists among all congruent polygons.Triangles are said to be comparable to one another if any two of their respective angles are identical to any two of their respective angles in another triangle.To learn more about Polygons refer to:
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A sports utility vehicle ( SUV ) was purchased for $ 23,000 . According to the automobile sales association , the buyer can expect the SUV to lose about 11 % of its value annually In 10 years , approximately for how much could an owner expect to sell the SUV ?
The amount that the owner can expect to sell the SUV in 10 years, given the loss in value annually is $7, 171 .80
How to find the selling price of the car?Assuming the selling price of the SUV will be the value of the SUV in 10 years, then the selling price can be found by the formula:
= Purchase price of SUV x ( 1 - percentage loss in value) ^ number of years
Purchase price of SUV = $23, 000
Percentage loss in value = 11%
Number of years = 10 years
The selling price of the SUV is therefore:
= 23, 000 x ( 1 - 11%) ¹⁰
= $7, 171 .80
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Use the tree diagram to see if the customer buys at least 1 of the 2 customers buys a banana icecream give your answer in a simple form
The probability that at least one of the two customers buy a banana ice cream is of:
55/64.
How to calculate the probability?The probabilities are calculated from the tree diagram given by the image at the end of the answer.
The nodes that end up with none customers buying a banana ice cream are given as follows:
Cookie dough -> Cookie dough.
The probability of each node is given by:
3/8.
Hence the probability that none of the two customers buy a banana ice cream is of:
3/8 x 3/8 = 9/64.
None and at least one customer buying a banana ice cream are complementary events, hence the probability is obtained as follows:
1 - 9/64 = 64/64 - 9/64 = 55/64.
Missing InformationThe tree diagram that models the situation is given by the image presented at the end of teh answer.
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Which sequence shows the numbers in order from least to greatest?
A.
B.
C.
D.
Answer: B
Step-by-step explanation:
Because 7 and 2/3 is almost the same as III -7.82 III and [tex]\sqrt65[/tex] is 8.0622577483 so first is 7 2/3 then III -7.82 III and then the root of [tex]\sqrt65[/tex]
PLEASE GIVE BRAINLIEST THANK YOU!!!
Directions: Select all the correct answers.
Identify the products that are less than the value represented by the model. The entire grid represents one whole.
2.7×0,2
0.23 x 0.4
1.5 x 0.6
O 1.85 x 0.3
O
0.75 × 1.1
O
O
The products that are less than the value represented by the model are
0.23 x 0.4 , 1.5 x 0.6 , 1.85 x 0.3.
From the question, we have
the model has a grid of size 10 *10 = 100
Out of 100, 75 are covered.
75 grid = 75/100 =0.75
The products that are less than the 0.75 are
0.23 x 0.4 , 1.5 x 0.6 , 1.85 x 0.3.
Divide:
Repetitive subtraction is the process of division. It is the multiplication operation's opposite. It is described as the process of creating equitable groups. When dividing numbers, we divide a larger number down into smaller ones such that the larger number obtained will be equal to the multiplication of the smaller numbers. One of the four fundamental mathematical operations, along with addition, subtraction, and multiplication, is division. Division is the process of dividing a larger group into smaller groups so that each group contains an equal number of things. It is a mathematical operation used for equal distribution and equal grouping.
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