Use the remainder theorem to find P (1) for P(x) = -2x⁴+3x³ + 6x² -7. Specifically, give the quotient and the remainder for the associated division and the value of P (1).
Answers
Step 1
Given;
[tex]P(x)=-2x^4+3x^3+6x^2-7[/tex]Required; To find P(1)
Step 2
Set x=1
[tex]\begin{gathered} P(1)=-2(1)^4+3(1)^3+6(1)^2-7 \\ =-2\cdot \:1+3\left(1\right)^3+6\left(1\right)^2-7 \\ =-2\cdot \:1+3\cdot \:1+6\cdot \:1-7 \\ =-2+3\cdot \:1+6\cdot \:1-7 \\ =-2+3+6\cdot \:1-7 \\ =-2+3+6-7 \\ =0 \end{gathered}[/tex]Using the factor remainder theorem;
Hence;
[tex]\begin{gathered} Quotient=-2x^3+x^2+7x+7 \\ Remainder=0 \\ P(1)=0 \end{gathered}[/tex]
Related Questions
Help!!! What types of symmetry does this figure have?? I need an explanation plss
Answers
Answer: C
Rotational symmetry means that the object looks the same if you rotate it by a certain amount of degrees. This object demonstrates rotational symmetry because if you rotate it around, it will look the same.
Reflectional symmetry is when you put a line down the middle of the shape and it looks the same on both sides. This object also demonstrates reflectional symmetry because if you place a line in the middle from left to right, up to down, or diagonal, all sides look the same.
Therefore, the object demonstrates both rotational and reflectional symmetry.
Determine whether the following sequence is arithmetic, geometric, or neither.1, 4, 9, 16, ...neithergeometricarithmetic
Answers
Answer : neither
Explanation:
There is no common ratio or common difference between the values.
Common difference
16-9 = 7
9-4= 5
4-1= 3
Common ratio
16/9 =16/9
9/4= 2.25
4/1 = 4
Solve the linear equation 5x-10y=-5Make sure you provide me with the solution.
Answers
Answer
y-intercept = 1/2
x -intercept = -1
The solutions are x = -1 and y = 1/2
Step-by-step explanation:
Given the following linear equation
5x - 10y = -5
To find x and y-intercepts
Step 1: isolate x to find y
let x = 0, to find y
5(0) - 10y = -5
0 - 10y = -5
-10y = -5
Divide both sides by -10
-10y/-10 = -5/-10
y = 1/2
(0, 1/2)
5x - 10y = -5
To find x, let y = 0
5x - 10(0) = -5
5x - 0 = -5
5x = -5
Divide both sides by 5
5x/-5 = -5/5
x = -1
(-1, 0)
Therefore, y-intercept = 1/2 and x -intercept = -1
The slope-intercept form of the equation is given as
y = mx + c
Given 5x - 10y = -5
Step 1 : re-write the equation
-10y = -5 - 5x
10y = -(-5 - 5x)
10y = 5x + 5
Divide through by 10
10y/10 = 5x/10 + 5/10
y = 1/2x + 1/2
5x
10y = -(-5 - 5x)
10y = 5x + 5
8 ft 3 ft The shaded squares in this diagram show Kyle's plan for tiling a floor. How many square feet of tile will be use? 4 ft . 3 ft 8 ft
Answers
SOLUTION
The square feet of tile needed is the area fo the shape given
Consider the image below
The area of the shape is the sum of the individual arear of the of the two Rectangle and the Area of the square
[tex]\begin{gathered} area\text{ of the rectangel is } \\ l\times b \\ \text{where } \\ b=3,l=8 \\ \end{gathered}[/tex]Then
[tex]\text{Area of rectangle =3x8=24 square feet}[/tex]Since the other rectangle at the base have the same dimension, The are is
[tex]\text{Area of rectangle at the base =24square f}eet\text{ }[/tex]Then, we obtain the Area of the Square
[tex]\begin{gathered} \text{Area os Square=l}^2 \\ \text{Where l=4 ft} \\ \text{then} \\ \text{Area of square =4}^2=4\times4=16ft^2 \end{gathered}[/tex]Hence, the area of the Shape is
[tex]\begin{gathered} \text{Area of Shape=2(24)+16} \\ 48+16=64ft^2 \end{gathered}[/tex]The Square feet tile that will be used is 64 square feet
hello, there can help me with this question.A grove of redwood tree has an estimated 3,400,000 board feet of lumber at thestart of 2022. If redwoods grow at an exponential rate of 9% each year, then whatwould be the growth formula? Within which year will the grove reach one billion boardfeet of lumber?
Answers
Answer:
Explanation:
a) Firstly, we want to write the growth formula
We have the general form as:
[tex]L\text{ = I\lparen1 + r\rparen}^t[/tex]where:
L is the estimated board feet of lumber
l is the is the initial estimated board feet of lumber (the initial estimated board feet of lumber in 2022)
r is the percentage rate of increase which is 9% (9/100 = 0.09)
t is the number of years to reach the estimated board feet of lumber
With respect to the question given, we have the formula as:
[tex]\begin{gathered} L\text{ = 3,400,00\lparen1 + 0.09\rparen}^t \\ L\text{ =3,400,000\lparen1.09\rparen}^t \end{gathered}[/tex]b) We want to get the value of t when L is 1 billion
Substituting the values, we have it that:
[tex]\begin{gathered} 1000000000\text{ = 3400000\lparen1.09\rparen}^t \\ divide\text{ both sides by 3,400,000} \\ 294.12\text{ = 1.09}^t \\ ln\text{ 294.12 = tln 1.09t } \\ t\text{ = }\frac{ln\text{ 249.12}}{ln\text{ 1.09}}\text{ = 66 years} \end{gathered}[/tex]In 66 years (2088) , the groove will reach one billion board feet of lumber
Hi, can you help me to solve this problem please!
Answers
Given,
The function is,
[tex]g(x)=x^2+10x+6[/tex]Taking x = a - 7 then,
[tex]\begin{gathered} g(a-7)=(a-7)^2+10(a-7)+6 \\ =(a^2-14a+49)+10a-70+6 \\ =a^2-14a+10a+49-70+6 \\ =a^2-4a-15 \end{gathered}[/tex]Hence, the function is f(a-7) = a^2-4a-15.
Simplify:(2x - 3)²A. 4x²-9OB.4x² +9OC. 4x² - 6x-9D. 4x² - 12x +9
Answers
Solution
Simplify the expression
[tex](2x-3)^2[/tex]Step 1: Open the bracket
[tex]\begin{gathered} (2x-3)^2 \\ (2x-3)(2x-3) \\ 4x^2-6x-6x+9 \end{gathered}[/tex]Step 2: Collect like terms
[tex]\begin{gathered} 4x^2-6x-6x+9 \\ 4x^2-12x+9 \end{gathered}[/tex]Therefore the correct answer is Option D
Tammy, Michael, and Aldo sent a total of 119 text messages during the weekend. Tammy sent 9 more messages than Aldo. Michael sent 3 times as many messages as Aldo. How many messages did they each send?
Answers
Tammy = x
Michael = y
Aldo = z
x + y + z = 119 Eq 1
x = z +9 Eq 2
y = 3z Eq 3
We need to solve the equation system
Substitute eq 2 in 1
z + 9 + y + z = 119
y + 2z = 110 eq 4
Then we multiply eq 3 and 4
y + 2z = 110
-y + 3z = 0
5z = 110, z =22
then eq 3 will be rewritten as
y = 3x22 = 66
Finally we do the same with eq 2.
x = 22 + 9 = 31
So
Tammy sent 31
Michael 66
Aldo 22
The credit remaining on a phone card (in dollars) is a linear function of the total calling time made with the card (in minutes). The remaining credit after 24 minutes of calls is $21.64, and the remaining credit after 55 minutes of calls is $17.30 . What is the remaining credit after 79 minutes of calls?
Answers
Given data:
The credit that remains after 24 minutes is $21.64.
The credit that remains after 55 minutes is $17.30.
The expression for the equation is,
[tex]\begin{gathered} y-21.64=(\frac{17.30-21.64}{55-24}(x-24) \\ y-21.64=-0.14x+3.36 \\ y=-0.14x+25 \end{gathered}[/tex]Substitute 79 for x in the above expression.
[tex]\begin{gathered} y=-0.14(79)+25 \\ =13.94 \end{gathered}[/tex]Thus, the credit that remains after 79 minutes is $13.94.
Round all answers to the nearest hundredth.Solve for the radius. r= Determine which value to use for the height.h= Solve for the Volume. V=
Answers
Given:
A figure of a cone
Required:
1. Find the radius and round it to the nearest hundredth
2. Determine which value to use for the height.
3. Volume of the cone.
Explanation:
Using Pythagoras Theorem
[tex]undefined[/tex]Write each fraction in simplest form. Write simplified if the fraction is already in simplest form. 18/27 a. 6/9b. 2/3c. 1/3d. simplified
Answers
Answer:
b. 2/3
Explanation:
Given the fraction:
[tex]\frac{18}{27}[/tex]We can write each of the numbers as a product below:
[tex]\frac{18}{27}=\frac{2\times9}{3\times9}[/tex]Next, cancel the common number, 9:
[tex]\frac{18}{27}=\frac{2\times\cancel{9}}{3\times\cancel{9}}=\frac{2}{3}[/tex]The simplified form of the fraction is 2/3.
Option B is correct.
Which point does not lie on the circle with center (0,0) and radius 10? A. (6,8) B. (0, 10) C. (10,-8) O D. (-8, 6)
Answers
Answer:
C. (10, -8)
Explanation:
The equation of a circle is given as:
[tex]\begin{gathered} (x-h)^2+(y-k)^2=r^2 \\ \text{Centre}=(h,k) \\ \text{Radius}=r \end{gathered}[/tex]Given a circle with centre (0,0) and radius 10, its equation is:
[tex]\begin{gathered} x^2+y^2=10^2 \\ x^2+y^2=100\ldots(1) \end{gathered}[/tex]To determine the point that does not lie on the circle, we check the point that does not satisfy the equation (1) derived above:
• Point (6,8)
[tex]\begin{gathered} x=6,y=8 \\ 6^2+8^2=100 \\ 36+64=100 \\ 100=100 \end{gathered}[/tex]• Point (0,10)
,•
• Point (10, -8)
[tex]\begin{gathered} x=10,y=-8 \\ 10^2+(-8)^2=100 \\ 100+64=100 \\ 164\neq100 \end{gathered}[/tex]It is clear that point (10,-8) does not lie on the circle since the outcome is False.
The correct choice is C.
The profit when selling 'x pairs of shoes is defined by the function P(x) = 6x2 – 60x + 126. How many pairs of shoes should be sold to maximize the profit? 04 pairs of shoes O 7 pairs of shoes 6 pairs of shoes O 5 pairs of shoes
Answers
Hello!
We have a quadratic equation:
[tex]P(x)=6x^2-60x+126[/tex]As the coefficient a is positive, the concavity of the parabola will face upwards (so it will have a minimum point).
Look at the graph below:
Now, let's analyze the graph and the alternatives:
• when x = 5, profit is as small as possible
when x = 4 or x = 6, the profit
The function P(t) = 1001.05)×
Answers
we have the function
P(t)=100(1.05^t)
For the year 2010
t=2010-1992=18 years
substitute
P(t)=100(1.05^18)
P(t)=240.66 -------> is in thousands
so
P(t)=240,660
therefore
the answer is'option B (241,000)Evaluate each expression using the Order of Operations. 3. (122 – 48) x 2
Answers
The given expression is
[tex](122-48)\times2[/tex]First, we subtract.
[tex]74\times2[/tex]Then, we multiply.
[tex]148[/tex]Hence, the answer is 148.1 pointA construction company plans to invest in a housing project. There is a 30% chance that the company will lose $200,000 a 30% chance it will lose $100,000, and a 40% chance ofa $300,000 profitBased ONLY on this information, what should the company do?O The expected value is $60,000, so the company should proceed with the project.O The expected value is $30,000, so the company should proceed with the project.The expected value is $90,000, so the company should proceed with the projectThe expected value is -$90,000, so the company should not proceed with the project.1 point0
Answers
It is given that the company has 30% chance to lose $200,000 , 30% chance to lose $100,000 and 40% chance of
a $300,000 profit.
We have to determine the expected value which is
[tex]E(x)=xP(x)[/tex]Determine the expected value using the data given in the question.
[tex]E(x)=-200,000\times0.3-0.3\times100,000+0.4\times300,000[/tex][tex]E(x)=-60,000-30,000+120,000=30,000[/tex]The line through (2, -1) and parallel to x - 2y = 6 has the equation:
Answers
1) Parallel lines have the same slope. So our new slope has the slope m=1
Looking at the options, we have two options with the same slope. So let's test them x-2y=4 and x+2y=0 plugging in the point to verify
(2,-1)
x+2y=0
-1+2(2) = 0
3 = 0 False
x- 2y =4
2 -2(-1) =4
2+2=4
2) So x -2y =4 and x+2y=0 are parallel to x-2y=6
But only x -2y =4 passes through the point (2,-1)
x-2y=4 is the answer
if one face of a die has an area of 144 square centimeters. find the volume of the die
Answers
I need help with this problem. None of the above is a answer as well it just doesn’t pop up.
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Given the following radical:
[tex]\sqrt{64}[/tex]We will find the principal square root which will be the positive value of the root
As we know, 64 = 8 x 8
So, the principal square root = 8
So, the answer will be: 3 only
Your chores for the week are to cut the grass, wash the car, clean your room, clean the garage, and shine your shoes. You are to do 1 chore each day from Monday through Friday. You can do each chore on whatever day you want, except that you must wash the car either Thursday or Friday. In how many different orders can you perform your chores?
Answers
The chores
cut the grass
wash the car -- Thursday or Friday
clean your room
clean the garage
shine your shoes
We have 5 days in the week to do the chores.
[tex]\begin{gathered} 2*\mleft(4*3*2*1\mright)=48 \\ \end{gathered}[/tex]You can perform the chores in 48 different orders
What is 16/12 reduced?16/121 2/61 1/31 4/12
Answers
Explanation
Step 1
numerator and denominator divided by 2
[tex]\begin{gathered} \frac{16}{12} \\ \frac{16}{12}\rightarrow\frac{16\colon2}{12\colon2}\rightarrow\frac{8}{6} \\ \text{again, divide by 2} \\ \frac{8}{6}\Rightarrow\frac{8\colon2}{6\colon2}\Rightarrow\frac{4}{3} \\ \frac{16}{12}=\frac{4}{3} \end{gathered}[/tex]and
so,the answer is
[tex]\frac{4}{3}=1\frac{1}{3}[/tex]I hope this helps you.
if the radius (r) of the circle is 3 what in terms of pu
Answers
Answer
Option C is correct.
Area of the circle = 9π
Explanation
Question 4
Area of a circle = πr²
where
π = pi
r = radius of the circle = 3 feet
Area of this circle = πr²
Area of this circle = π (3²) = 9π
Hope this Helps!!!
Pure sounds produce single sine waves on an oscilloscope. Find the amplitude and period of the sine wave graph. On the vertical scale, each square represents 0.5; on the horizontal scale, each square represents 30° or pi/6.
Answers
To solve the question, we will need to understand some definition
PERIOD: The period is the time it takes for one complete cycle of a harmonic oscillation
AMPLITUDE: The amplitude is the maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position.
To find the amplitude, we will have to count how many boxes there are from the equilibrium or mean position on the vertical axis and then multiply by 0.5.
The number of boxes is 8
Therefore, the amplitude will be
[tex]\begin{gathered} 0.5\times8=4 \\ =4 \end{gathered}[/tex]Amplitude = 4 units
For the period, we will have to count the number of boxes within one period
There are four boxes within one period. Since we know that each box on the horizontal axis is 30 degrees
or pi/6
The period will be
[tex]\begin{gathered} 30^0\times4=120^0 \\ or \\ \frac{\pi}{6}\times4=\frac{2}{3}\pi \end{gathered}[/tex]Fill in the blanks below with the correct units. (a) Lamar went skiing on a mountain that is about 3 ? (b) To wash his car, Omar used 20 ? of water. (c) Melissa ate an apple that had a mass of about 150 ?
Answers
Solution:
a) Lamar went skiing on a mountain that is about 3 kilometers high. This is because the other units are too small to measure the height of a mountain.
b) To wash his car, Omar used 20 liters of water. This is because 20 milliliters of water is too small to wash a car.
c) Melissa ate an apple that had a mass of about 150 grams. An apple can not be as heavy as 150kg.
Priya and Mai used the diagram below to find the value of 1/3 ÷ 4
Answers
1) Examining both diagrams, we can state that both diagrams represent
The value of
Prya's:
1/3 : 4
Mai's
1/4 x 3
The formula v= 42.5r can be used to estimate the maximum safe velocity, v, in miles per hour, at which a car can travel if it is driven along a curved roadius of curvature r in feet. Find the maximum safe speed to the nearest whole number if a cloverleaf exit on an expressway has a radius of curvature of 290
Answers
Given:
[tex]\begin{gathered} V\text{ = 42.5r} \\ where\text{ V = maximum safe velocity} \\ r\text{ = radius of curvature} \end{gathered}[/tex]To find:
the maximum safe speed when the radius of curvature is 290
To determine the velocity, we will substitute the values into the formula:
[tex]\begin{gathered} V\text{ = 42.5\lparen290\rparen} \\ \\ V=\text{ 12325 miles per hour} \end{gathered}[/tex]What expression can be used to subtract 2/3 - 4/9A. 8/12 - 4/12B. 2/12 - 4/12C. 2/9 - 4/9D. 6/9 - 4/9
Answers
What expression can be used to subtract 2/3 - 4/9?
Lowest Common Mutiple of 3 and 9 = 9
6/9 - 4/9 (D)
translate the sentence into an equation four more than the quotient of a number and 6 is 2 use the variable b for the unknown number
Answers
Consider each part of the sentence:
four (4) more (+) than the quotient of a number and 6 (b/6) is 2 (= 2):
Hence, you have:
4 + b/6 = 2
Graph the line: 3/4y -2x=6
Answers
Given the equation of the line:
[tex]\frac{3}{4}y-2x=6[/tex]To graph the line, we need two points lying on the line
We will graph the line with the help of the intercepts
when x = 0
[tex]\begin{gathered} \frac{3}{4}y=6 \\ y=\frac{4\cdot6}{3}=\frac{24}{3}=8 \end{gathered}[/tex]so, the y-intercept = (0, 8)
When y = 0
[tex]\begin{gathered} -2x=6 \\ x=\frac{6}{-2}=-3 \end{gathered}[/tex]So, the x-intercept = (-3, 0)
The line passes through the points (-3, 0) and (0, 8)
The graph of the line will be as shown in the following picture:
Find the derivative of the function at the given numberf(x)=x^2 - 5 when x=3 using the difference quotient f(x+h) - f(x) over h
Answers
Derivative
Given a function f(x), the derivative of f(x) in a point x = a can be found by using the definition:
[tex]f^{\prime}(a)=\lim _{h\to0}\frac{f(a+h)-f(a)}{h}[/tex]Given:
[tex]\begin{gathered} f(x)=x^2-5 \\ a=3 \end{gathered}[/tex][tex]\begin{gathered} f^{\prime}(3)=\lim _{h\to0}\frac{f(3+h)-f(3)}{h} \\ f^{\prime}(3)=\lim _{h\to0}\frac{(3+h)^2-5-(3^2-5)}{h} \end{gathered}[/tex]Operating:
[tex]\begin{gathered} f^{\prime}(3)=\lim _{h\to0}\frac{9+6h+h^2-5-4}{h} \\ f^{\prime}(3)=\lim _{h\to0}\frac{6h+h^2}{h} \\ \text{Factoring:} \\ f^{\prime}(3)=\lim _{h\to0}\frac{h(6+h)}{h} \\ \text{Simplifying:} \\ f^{\prime}(3)=6 \end{gathered}[/tex]