Answer:
This looks like an addition (summation is the technical term) series. Here we need to figure out the difference between each number in the series. The difference between 6.75 and 9.25 is 2.5 (You can subtract the larger number from the smaller number to find out). So, 6.75-2.5=4.25.
So the correct answer in the blank is 4.25, and the rule is +2.5.
Step-by-step explanation:
Given the expression: 6x^10 - 96x^2Part A: Rewrite the expression by factoring out the greatest common factor.Part B: Factor the entire expression completely. Show the steps of your work.
According to factorization method, we have find out that by factoring out the greatest common factor, we can rewrite the expression as [tex]6x^{2}(x^{8}-16)[/tex] and by factoring the entire expression by difference of squares concept, we can rewrite the expression as [tex]6x^{2}(x^{4}+4)(x^{2}+2)(x^{2} -2)[/tex].
The given expression is -
[tex]6x^{10}-96x^{2}[/tex] ---- (1)
We have to -
A: Rewrite the expression by factoring out the greatest common factor.
B: Rewrite the expression by factoring the entire expression completely.
Solving for Part A:
From equation (1), we have
[tex]6x^{10}-96x^{2}[/tex]
Applying the distributive property, we can rewrite this as -
[tex]6x^{10}-96x^{2}\\=6x^{2} x^{8}-6x^{2}*16\\=6x^{2}(x^{8}-16)[/tex]------- (2)
This is the expression obtained by factoring out the greatest common factor that is [tex]x^{2}[/tex].
Solving for Part B:
From equation (2), we have
[tex]6x^{2}(x^{8}-16)[/tex]
Applying the difference of squares concept, we can rewrite this as -
[tex]6x^{2}(x^{8}-16)\\=6x^{2}[(x^{4} )^{2} -4^{2} ]\\=6x^{2}(x^{4}+4)(x^{4}-4)\\=6x^{2}(x^{4}+4)[(x^{2} )^{2}-2^{2} ]\\=6x^{2}(x^{4}+4)(x^{2}+2)(x^{2} -2)[/tex]
This is the expression obtained by factoring the expression completely.
Therefore, according to factorization method, we have find out that by factoring out the greatest common factor, we can rewrite the expression as [tex]6x^{2}(x^{8}-16)[/tex] and by factoring the entire expression by difference of squares concept, we can rewrite the expression as [tex]6x^{2}(x^{4}+4)(x^{2}+2)(x^{2} -2)[/tex].
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two angles of a triangle measure 59° and 63°. if the longest side measures 28 cm find the length of the shortest side. round answers to the nearest tenth
The two angles of triangle are : 59 and 63 degrees.
Length of the longest side of traingle is 28cm.
In a triangle,
the smallest side is always opposite to the smallest angle of the triangle,
and the largest side is always opposite to the largest angle
The sum of all the angles in atriangle is 180 degree
let the other angle is x so,
x+63+59=180
x=180-59-63
x=58
So, the longest side of traingle is in the opposite of angle 63
the smallest side of triangle is in opposite of angle 58
Apply the Sine rule to find the side of the traingle which has an opposite angle of 58.
Sine formula is expressed as:
[tex]\frac{a}{\sin A}=\frac{b}{\sin B}^{}=\frac{c}{\sin C}[/tex]Let a be the smallest side and b be the longest side of triangle so,
A=58, B=63, b=28cm
Substitute the values, and solve for a,
[tex]\begin{gathered} \frac{a}{\sin58^{\circ}}=\frac{28}{\sin63^{\circ}} \\ a=\frac{28\times\sin58^{\circ}}{\sin63^{\circ}} \\ a=\frac{28\times(0.84804809)}{0.89100652} \\ a=\frac{23.74534652}{0.89100652} \\ a=26.6500 \\ a=26.7\operatorname{cm} \end{gathered}[/tex]The shortest length is 26.7cm
Which of the following describes the image of the transformation graphed to the right?A.) T(x,y)=(x+7,y-7)B.) T(x,y)=(x-7,y-7)C.) T(x,y)=(x+7,y+7)D.) T(x,y)=(x-7,y+7)
Answer:
D.) T(x,y)=(x-7,y+7)
Step-by-step explanation:
Function f(x,y).
Transformation along the x axis:
Moving a units to the left: We have f(x+a,y).
Moving a units to the right: We have f(x - a,y).
Transformation along the y axis:
Moving a units up: We have f(x, y-a).
Moving a units down: We have f(x, y+a).
In this question:
The function is moved 7 units to the right among the x-axis(x - 7) and 7 units down along the y axis(y + 7).
So the answer to this question is:
D.) T(x,y)=(x-7,y+7)
need this asap What is the x-coordinate of the solution to this system of equations: 4x + y = 2 x - y = 3
Given the following system of equations:
[tex]\begin{cases}4x+y=2 \\ x-y=3\end{cases}[/tex]to find the x-cooridnate of the solution, we can solve for 'y' the first equation to get the following:
[tex]\begin{gathered} 4x+y=2 \\ \Rightarrow y=2-4x \end{gathered}[/tex]next, we use this value on the second equation to get the following expression:
[tex]\begin{gathered} x-y=3 \\ \Rightarrow x-(2-4x)=3 \end{gathered}[/tex]simplifying we get the following:
[tex]\begin{gathered} x-(2-4x)=3 \\ \Rightarrow x-2+4x=3 \\ \Rightarrow x+4x=2+3 \\ \Rightarrow5x=5 \\ \Rightarrow x=\frac{5}{5}=1 \\ x=1 \end{gathered}[/tex]therefore, the x-coordinate of the solution to the system of equations is x = 1
17. Felicia bought 5 pieces of candy for 75 cents. Write a proportion youcould solve to find out how much would 8 pieces cost? *
Given the information on the problem, we can write the 5 pieces of candy for 75 cents like this:
[tex]\frac{75\text{ cents}}{5\text{ pieces}}[/tex]then, for 8 pieces, we would have the following proportion:
[tex]\frac{x\text{ cents}}{8\text{ pieces}}[/tex]then, we can equate both proportions to get the following:
[tex]\begin{gathered} \frac{75\text{ cents}}{5\text{ pieces}}=\frac{x\text{ cents}}{8\text{ pieces}} \\ or \\ \frac{75}{5}=\frac{x}{8} \end{gathered}[/tex]thus, solving for x, we get:
[tex]\begin{gathered} \frac{x}{8}=\frac{75}{5} \\ \Rightarrow x=\frac{75}{5}\cdot8=120 \\ x=120\text{cents} \end{gathered}[/tex]therefore, 8 pieces cost $1.20
f (t) = 44 – 5 g(t) = 2t - 1 Find (f - g)(t)
To solve the given problems, we have to subtract the functions.
[tex](f-g)(t)=4t-5-(2t-1)[/tex]Then, we simplify
[tex](f-g)(t)=4t-5-2t+1=2t-4[/tex]Hence, the resulting function is 2t-4.: + = 27 = + 3. (15,12). (12,15). (6,21).
(12, 15) (option B)
Explanation:x + y = 27 ..equation 1
y = x + 3 ...equation 2
Using substitution method:
We would substitute for y = x + 3 in equation 1
x + (x + 3) = 27
x + x + 3 = 27
2x + 3 = 27
2x = 27 - 3
2x = 24
x = 24/2
x = 12
Substitute 12 for x in equation 2:
y = 12 + 3
y = 15
The solution to both equations (x, y):
(12, 15) (option B)
Draw the circle ( x − 3 ) 2 + y 2 = 1 .
A drawing of this equation of a circle (x − 3)² + y² = 1 is shown in the image attached below.
What is the equation of a circle?Mathematically, the standard form of the equation of a circle is represented by this mathematical expression;
(x - h)² + (y - k)² = r² ....equation 1.
Where:
h and k represents the coordinates at the center of a circle.r represents the radius of a circle.From the information provided, we have the following equation of a circle:
(x − 3)² + y² = 1 .........equation 2.
By comparing equation 1 and equation 2, we can logically deduce the following parameters:
Coordinate at the center, h = 3.Coordinate at the center, k = 0.Radius of circle, r = 1.In conclusion, the given equation of a circle (x − 3)² + y² = 1 was plotted by using an online graphing calculator.
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Complete Question:
Draw the circle (x − 3)² + y² = 1.
Quicy has 1/2 box of cereal to eat over 6 days if he splits the 1/2 box into equal portions how much will he eat each day
ANSWER:
1/12 of the cereal box
STEP-BY-STEP EXPLANATION:
In this case we must divide the amount that we have, that is, half a box of cereal by the number of portions that are desired equal, that is, 6
Therefore, we are left with:
[tex]\frac{\frac{1}{2}}{6}=\frac{1}{2\cdot6}=\frac{1}{12}[/tex]That is, for each day, you will have to eat 1/12 of the cereal box
The area of the trapezoid is 14 square feet. Write an equation that you can use to find the value of x
Explanation
the area of a trapezoid is equal to half the product of the height and the sum of the two bases.
[tex]A=(\frac{base1+base2}{2})\cdot\text{heigth}[/tex]then
Step 1
Let
base1=2x
base2=x
height= 2 ft
area= 14 square feet
replace,
[tex]\begin{gathered} A=(\frac{base1+base2}{2})\cdot\text{heigth} \\ 14=(\frac{2x+x}{2})\cdot\text{2 } \\ 14=(\frac{3x}{2})\cdot2 \\ 14=3x\rightarrow equation \\ divide\text{ both sides by 3} \\ \frac{14}{3}=\frac{3x}{3} \\ \frac{14}{3}=x \\ x=\frac{14}{3} \end{gathered}[/tex]then put 3x in the box.
I hope this helps you
Determine whether each data set has a positive relationship, negative relationship or no relationship.
Two groups of numbers have a positive relationship, when we can identify a pattern where the two variables increase together, a negative relationship when the two variables decrease together and no relationship when we can't identify any pattern.
For the first graph we can identify that as the temperature grows, the number of chirps also grows, therefore they have a positive relationship.
For the second graph we can't clearly identify any relationship between the two variables, therefore it has no relationship.
h e l p please. can't solve. picture explains, and i know the answer, but i cannot figure out how to make the equation to solve
We want to find how many miles Jodi traveled if her ride costed $12.50. We also know that a taxi ride has an initial fee of $3.75 plus $1.25 for every 1/2 mile traveled.
This means that for every mile, the fee will be:
[tex]2(1.25)=2.50[/tex]And the expression for the cost on terms of the miles traveled will be:
[tex]\begin{gathered} 3.75+2.50x \\ \text{where }x\text{ represents the number of miles traveled } \end{gathered}[/tex]As we know that Jodi spent $12.50 on a ride, we equal the value to the above expression:
[tex]3.75+2.50x=12.50[/tex]And we solve for x, as we want to know the number of miles:
[tex]\begin{gathered} 2.50x=12.50-3.75 \\ 2.50x=8.75 \\ x=\frac{8.75}{2.50}=3.5 \end{gathered}[/tex]This means that the number of miles that Jodi traveled was 3 and a half.
Find the prime factorization and match your result to the correct answer below 231.Select one:a. 21•11b. 2•3•7•11c. 3•7•11d. 3•77
Solution
Hence, the prime factorization of 231 is 3•7•11
Hence, the correct option is C.
An equilateral triangle and a square have equal perimeters. The side of the triangle measures 8cm. What is the area of the square, in square centimeters?
Answer
Area of the square = 36 square cm
Step-by-step explanation:
Given:
The perimeter of an equilateral triangle = the perimeter of a square
In an equilateral triangle, all sides are equal
Each side of the equilateral triangle = 8cm
Perimeter of the equilateral triangle = 3 * 8cm
Perimeter of the equilateral triangle = 24cm
Since, perimeter of square = perimeter of an equilateral triangle
Perimeter of a square = 24cm
We need to find each side of the square
Perimeter of a square = 4s
24 = 4s
Divide both side by 4
24/4 = 4s/4
s = 6cm
Since the length of each side of the square is 6cm
Therefore, area = l^2
Area = 6 * 6
Area of a square = 36 square cm
Solve the right triangle. Round decimal answers to the nearest tenth. find RSfind RTfind angle T
Δ RST shown in the picture is a right triangle, we know the measure of ∠R and ∠S, and the length of side ST.
To determine the length of the missing angle, you have to remember that the sum of the inner angles of a triangle add up to 180º
Then we can calculate the measure of ∠T as follows:
[tex]\begin{gathered} \angle R+\angle S+\angle T=180 \\ 57+90+\angle T=180 \\ 174+\angle T=180 \\ \angle T=180-174 \\ \angle T=33º \end{gathered}[/tex]The measure of the missing angle is ∠T=33º
To determine the lengths of the missing sides RS and RT, you have to use the trigonometric ratios of sine, cosine, and tangent. These ratios are defined as follows:
[tex]\begin{gathered} \sin \theta=\frac{opposite}{hypothenuse} \\ \cos \theta=\frac{adjacent}{hypothenuse} \\ \tan \theta=\frac{opposite}{adjacent} \end{gathered}[/tex]"θ" represents the angle of interest.
Using ∠R as a point of reference, side ST is across this angle so it is the side of the triangle "opposite" to it.
Side RS is next to ∠R, so it is the side "adjacent" to it.
The trigonometric ratio that shows the relationship between the opposite and adjacent sides to an angle is the tangent. Using the definition of the tangent you can determine the length of RS as follows:
[tex]\tan R=\frac{RS}{ST}[/tex]-First, write the expression for RS, which means that you have to pass "ST" to the left side of the equal sign by applying the opposite operation to both sides of it.
[tex]\begin{gathered} ST\tan R=ST\cdot\frac{RS}{ST} \\ ST\tan R=RS \end{gathered}[/tex]-Replace the expression with the values of ∠R and ST:
[tex]\begin{gathered} 15\cdot tan57=RS \\ 23.09=RS \\ RS\approx23.1 \end{gathered}[/tex]Side RS measures 23.1
Side RT is the longest side of the triangle, and thus, its hypothenuse. To determine its length using ∠R and side ST, you have to apply the definition of the sine:
[tex]\sin R=\frac{ST}{RT}[/tex]-Pass the term RT to the other side of the equation to take the term out of the denominators place:
[tex]\begin{gathered} RT\cdot\sin R=RT\frac{ST}{RT} \\ RT\cdot\sin R=ST \end{gathered}[/tex]-Next, divide both sides by the sine of R to write the expression for RT
[tex]\begin{gathered} RT\cdot\frac{\sin R}{\sin R}=\frac{ST}{\sin R} \\ RT=\frac{ST}{\sin R} \end{gathered}[/tex]-Now replace the expression with the values of ∠R and ST to determine the length of RT
[tex]\begin{gathered} RT=\frac{15}{\sin 57} \\ RT=12.58 \\ RT\approx12.6 \end{gathered}[/tex]Side RT measures 12.6
round to the nearest ten-thousandth -7(10^x)=-54
The value x in the given expression is 0.8873.
Define logarithm.The power to which a number must be raised in order to get additional values is known as the logarithm. This is the simplest way to express really large numbers. The fact that addition and subtraction logarithms can also be expressed as multiplication and division of logarithms is shown by a number of significant properties of a logarithm.
The other technique to write exponents in mathematics is using logarithms. The base-based logarithm of a number equals another number. The exact opposite function of exponentiation is carried out by a logarithm.
Given expression -
-7(10^x)=-54
We can also write it as
7(10^x) = 54
Using the concept of logarithm, apply the log on both sides
log(7*[tex]10^{x}[/tex]) = log 54
Applying the identity of logarithm,
log 7 + log [tex]10^{x}[/tex] = log 54
x log 10 = log 54 - log 7
x log 10 = 0.8873
As value of log 10 = 1,
x = 0.8873
Hence the value x in the given expression is 0.8873.
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A rocket is fired from the ground. Its height, in feet, is represented by the function h(t)=-16^2 +48t, where t(in seconds) represents the amount of time in the air since takeoff. When does the rocket land on the ground?A. 2 secondsB. 3.5 secondsC. 3 secondsd. 4.5 seconds
Given the height to be represented by the function
[tex]h(t)=16^{(2+48t)}[/tex]5x+10=-25solving for x
start by substracting 10 on both sides
[tex]\begin{gathered} 5x+10-10=-25-10 \\ 5x=-35 \end{gathered}[/tex]divide by 5 on both sides
[tex]\frac{5x}{5}=-\frac{35}{5}[/tex][tex]x=-7[/tex]Assume that a varies directly as b.When the value of a is 5,the value of b is 18.When the value of a is 22,what is the value of b?
In this case we have the proportion 18:5, then we have the equation
[tex]\begin{gathered} \frac{18}{5}=\frac{b}{22} \\ b=\frac{18}{5}\times22 \\ b=\frac{396}{5} \end{gathered}[/tex]Then, when a is 22, the value of b is 396/5 that is also equal to 79.2 or
[tex]79\frac{1}{5}[/tex]zoo nutritionist orders 5 1/4 tons of apples and 7 2/4 tonsof bananas each year to feed theanimals. She orders 6 times as manytons of herbivore pellets than tons offruit. How many tons of herbivorepellets does the nutritionist order?
From the information provided, the zoo nutritionist orders the following quantity of fruits;
[tex]\begin{gathered} 5\frac{1}{4}\text{ tons of apples} \\ 7\frac{2}{4}\text{ tons of bananas} \\ \text{Total}=5\frac{1}{4}+7\frac{2}{4} \\ \text{Total}=\frac{21}{4}+\frac{30}{4} \\ \text{Total}=\frac{51}{4}\text{ tons} \\ \end{gathered}[/tex]Next, we are told that the nutritionist orders 6 times as many tons of pellets than tons of fruits.
This means for every ton of fruit ordered, there was 6 tons of pellets ordered.
Therefore;
[tex]\begin{gathered} \text{Fruits:Pellets}=1\colon6 \\ \text{When fruits are }\frac{51}{4}tons \\ \text{Pellets}=\frac{51}{4}\times6 \\ \text{Pellets}=\frac{306}{4} \\ \text{Pellets}=76\frac{1}{2}tons \end{gathered}[/tex]ANSWER:
[tex]\text{Pellets ordered}=76\frac{1}{2}tons[/tex]I need help on this!Greater than, less than or equal to?number 28
Explanation:
We can draw a number line to see the numbers and check which one is greater:
2.5 is on the left of 3, so it is less than 3.
Answer:
2.5 < 3
Karen owns a seafood restaurant. She orders trout from an online retailer.Each pound of trout costs $30, and the company charges a $2 fee forshipping the order. However, if Karen orders 10 or more pounds, the troutcosts only $24 per pound, but the shipping fee is $6.Which piecewise function models the cost of x pounds of trout?○ A. f(x) = {={O B. f(x) ={O c. f(x) = -O D. f(x) =30x + 2, 0< x≤ 1024x+6, x 1024x+6, 0
The piecewise function that models the cost of x pounds of trout is:
[tex]C =\left \{ {{30x + 2}\\ x < 10\atop {24x + 6 x \geq 10}} \right.[/tex]
Let the number of pounds of trout is x and the price is C.
For, x < 10 :
From the question, we have
Each pound of trout costs $30
C = 30x + 2 ....1 )
For, x ≥ 10 :
Each pound of trout costs $24.
C = 24x + 6 ....2 )
Therefore, the piecewise function that models the cost of x pounds of trout is:
[tex]C =\left \{ {{30x + 2}\\ x < 10\atop {24x + 6 x \geq 10}} \right.[/tex]
Piecewise Function :
A function with many parts of curves in its graph is said to be piecewise. It implies that based on the value of the input, it has many definitions. In other words, a piecewise function responds differently to various inputs. A piecewise function, or f(x), is a function with various definitions at various intervals of x. A piecewise function's graph is divided into sections that each correspond to one of its definitions. A very good illustration of a piecewise function is the absolute value function.
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For a field trip 22 students rode in cars and the rest filled 5 buses how many students were in each bus if 317 students were on the trip.
We have to find how many students were in each bus.
We can call this quantity "x".
The total number of students is 317.
This quantity can be divided in the number of students that rode in cars, 22, and the rest went in 5 buses.
The quantity that rode in bus can be expressed as 5x.
Then we can write:
[tex]22+5x=317[/tex]We can solve this as:
[tex]\begin{gathered} 22+5x=317 \\ 5x=317-22 \\ 5x=295 \\ x=\frac{295}{5} \\ x=59 \end{gathered}[/tex]Answer: there were 59 students in each bus.
On average, there are six pages in every chapter of a Rodriguez Hernandez novel. Each book has Approximately 73 chapters. Rodriguez Hernandez has published 54 books. Approximately how many pages has Rodriguez Hernandez written?
Given:
There are given that 6 pages in every chapter and each book have 73 chapters.
Explanation:
According to the question:
We need to find the pages for 54 books.
So,
First, we need to find that 73 chapters mean one book.
Then,
[tex]\begin{gathered} 1ch\rightarrow6pages \\ 76ch\rightarrow6\times76 \\ =456pages \end{gathered}[/tex]Now,
We need to find the pages for 54 books:
So,
[tex][/tex]35) Use a formula to find the area of the figure.a15 in.7 in.20 in.
Given
height = 7 inches
base = 20 inches
Recall the formula for finding the area of the triangle
[tex]\begin{gathered} A_{\text{triangle}}=\frac{1}{2}bh \\ \text{where} \\ b\text{ is the base} \\ h\text{ is the height} \end{gathered}[/tex]Substitute the given dimensions of the triangle and we have
[tex]\begin{gathered} A=\frac{1}{2}bh \\ A=\frac{1}{2}(20\text{ in})(7\text{ in}) \\ A=\frac{1}{2}\cdot140\text{ in}^2 \\ A=70\text{ in}^2 \\ \\ \text{Therefore, the area of the triangle is }70\text{ in}^2. \end{gathered}[/tex]The function f(x)= 4x is one to one Find A and B
Given:
[tex]f(x)=4x[/tex]a)
[tex]\begin{gathered} x=4f^{-1}(x) \\ f^{-1}(x)=\frac{x}{4}\text{ , for all x} \end{gathered}[/tex]Option C is the final answer.
b)
[tex]\begin{gathered} f(f^{-1}(x))=f(\frac{x}{4}) \\ =4(\frac{x}{4}) \\ =x \end{gathered}[/tex][tex]\begin{gathered} f^{-1}(f(x))=f^{-1}(4x) \\ =\frac{4x}{4} \\ =x \end{gathered}[/tex][tex]f(f^{-1}(x))=f^{-1}(f(x))=x[/tex]I’m not sure how to solve this I just know I have to get 5 with an exponent of 7 I think
this is an addition of powers .
on sunday , Margos dad gives 5c, we will represent it as 5^1
therefore: sunday 5^1
Monday 5^1 x ( 5^1)
because bases are the same , we add exponents
this means that 5^1(5^1) = 5^2
Tuesday 5^2(5^1) = 5^3
wednesday 5^3(5^1) = 5^4
thursday 5^4( 5^1)= 5^5
Friday 5^5(5^1)= 5^6
saturday 5^6 (5^1)= 5^7
N. Students wanted to determine theheight of the flagpole. They measuredits shadow as 27 ft and the shadow of a 6ft tall student as 2 ft. How tall is theflagpole? Use the digit in the ones placeof your answer for N.
Given:
Shadow of flagpole = 27 feet
Height of the tall student = 6 feet
Shadow of the tall student = 2 feet
Required:
the height of the pole
Explanation:
Both the triangles are similar so the ratio of their corresponding sides are equal
[tex]\frac{height\text{ }of\text{ }flagpole}{height\text{ }of\text{ }student}=\frac{shadow\text{ }of\text{ }flagpole}{shadow\text{ }of\text{ }student}[/tex]Substituting the given values we get
[tex]\begin{gathered} \frac{height\text{ }of\text{ }flagpole}{6}=\frac{27}{2} \\ \\ height\text{ }of\text{ }flagpole=\frac{27\times6}{2}=27\times3 \\ \\ height\text{ }of\text{ }flagpole=81\text{ }feet \end{gathered}[/tex]Final answer:
The height of the flagpole is 81 feet.
Use the factor theorem to find all real zeros for the given polynomial and one of it's factors.Polynomial: f(x)=3x^3+x^2-20x+12 Factor: x+3List the zero's from smallest to largest. If a zero is not an integer write it as a fraction.The zeros are Answer , Answer and Answer
the zeros are -3, 2/3, and 2
Explanation:[tex]\begin{gathered} f(x)=3x^3+x^2\text{ - 20x + 12} \\ We\text{ n}eed\text{ to test if x + 3 is a factor} \end{gathered}[/tex]x + 3 = 0
x = -3
We will susbtitute -3 for x in the polynomial:
[tex]\begin{gathered} f(-3)=3(-3)^3+(-3)^2\text{ - 20(-3) + 12} \\ f(-3)=\text{ 3(-27) + 9 + 60 + 12 } \\ f(-3)\text{ = 0} \end{gathered}[/tex]Since the remainder is zero, this means x + 3 is a factor of the polynomial
Using synthetic division to get the remaining factor after factoring (x + 3):
[tex]3x^3+x^2-20x+12=(3x^2\text{ - 8x + 4)(x + 3)}[/tex]Using the factor theorem to find other factors:
[tex]\begin{gathered} f(x)=3x^2\text{- 8x + 4} \\ \text{factors of 4 = }\pm1,\text{ }\pm2,\text{ }\pm4 \\ \text{Let's try x = }1 \\ f(1)\text{ = }3(1)^2\text{- 8(1) + 4 = 3(1) - 8 + 4 = -1} \\ f(2)\text{ = }3(2)^2\text{- 8(2) + 4 = 3(4) - 16 + 4 = 0} \\ \text{Since f(2) = 0} \\ x\text{ = 2} \\ x\text{ - 2 = 0 . As a result, (x - 2) is a factor of the polynomial} \end{gathered}[/tex]Using synthetic division:
[tex]3x^2\text{- 8x + 4 = (x - 2)(3x -2)}[/tex][tex]\begin{gathered} 3x^3+x^2-20x+12=(3x^2\text{ - 8x + 4)(x + 3)} \\ 3x^3+x^2-20x+12=(x-2)(3x-2\text{)(x + 3)} \end{gathered}[/tex][tex]\begin{gathered} \text{x - 2 = 0; x = 2} \\ 3x\text{ - 2; x = 2/3} \\ x\text{ + 3; x = -3} \\ \text{The zeros are 2, 2/3 and -3} \\ \end{gathered}[/tex]From the smallest to the largest, the zeros are -3, 2/3, and 2
Find MK. ML = 8, LK = x + 2, MK = 4x - 2
From the given description, it appears that M, L, and K are collinear and the length of MK is equal to the sum of ML and LK.
To be able to find the length of MK let's first find the value of x using the equation of the sum of the lines.
[tex]\text{ }\bar{\text{ML}}\text{ + }\bar{\text{LK}}\text{ = }\bar{\text{MK}}[/tex]Let's plug in the values given in the description.
[tex]\text{ (8) + (x + 2) = 4x - 2}[/tex][tex]\text{ 8 + x + 2 = 4x - 2}[/tex][tex]\text{ x + 10 = 4x - 2}[/tex][tex]\text{ x - 4x = -2 - 10}[/tex][tex]\text{ -3x = -12}[/tex][tex]\text{ }\frac{\text{-3x}}{-3}\text{ = }\frac{\text{-12}}{-3}[/tex][tex]\text{ x = 4}[/tex]Let's plug in x = 4 in the equation for the length of MK = 4x - 2.
[tex]\text{ }\bar{\text{MK}}\text{ = 4x - 2}[/tex][tex]\text{ }\bar{\text{MK}}\text{ = 4(4) - 2}[/tex][tex]\text{ }\bar{\text{MK}}\text{ = 16 - 2}[/tex][tex]\text{ }\bar{\text{MK}}\text{ = 1}4[/tex]Therefore, the length of MK is 14.