To see the Polar form and Rectangular form of quotient as a complex number.
Polar numbers can be written in two different forms: rectangular and polar. Rectangular complex numbers are given by a + bi, where a represents the real part of the complex number and b represents the imaginary part of the complex number. Polar complex numbers are given by its absolute value,
r = [tex]\sqrt{x^{2} +y^2}[/tex] and the argument, θ, which can then be determined through the following relations:
x = rcosθy = rsinθ[tex]z=re^i^\theta[/tex]
If the complex numbers are in polar form, then the quotient is given by:
[tex]\frac{z_1}{z_2}=\frac{r_1e^i^\theta_1}{r_2e^i^\theta_2}[/tex] [tex]= \frac{r_1}{r_2}e^i^(^\theta_1^-^\theta_2^)[/tex]
If the complex numbers are in rectangular form, then you will have to use the complex conjugate to remove the complex number in the denominator:
[tex]\frac{a_1+b_1i}{a_2+b_2i}[/tex] [tex]=\frac{(a_1+b_1i)(a_2-b_2i)}{(a_2+b_2i)(a_2-b_2i)}[/tex].
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La figura muestra dos rectas cortadas por una transversal. (5b + 16)° (b + 32)°
The value of {b} is equivalent to 4.
What is geometry?Geometry is a branch of mathematics that deals with shapes, sizes, angles, and dimensions of objects.
Given are two intersecting lines.
The two intersect to create two pairs of vertically opposite angles. We can write -
(5b + 16)° = (b + 32)°
5b - b = 32 - 16
4b = 16
b = 4
Therefore, the value of {b} is equivalent to 4.
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{Question in english is given below -
The figure shows two lines cut by a transversal. (5b + 16)° (b + 32)°.}
dfm
KS3/4→Data Handling & Probability Probability
K259a: Draw a tree diagram to represent successive independent events.
b
Louis has 6 blue counters, 3 red counters and 1 black counter in a box.
Louis takes one counter at random from the box, puts it back, and takes another counter from the box.
Complete the tree diagram
Answer:
KS24 add K27ab4 6b3r1b olps
i have the awnser in the box
What is the domain and range of greatest integer function Class 11?.
For greatest integer function,
domain : set of all real numbers (ℝ)
range : set of all integers (ℤ)
We know that the greatest integer function of any real number n is the integer which is less than or equal to the given number n.
The mathematical definition of greatest integer function is :
f(x) = minimum { p ∈ Z ; p ≤ x }
where Z is the set of integers.
A greatest integer function is also known as the floor function.
The symbol to represent greatest integer function is ⌊ ⌋.
We can write greatest integer function for x as ⌊x⌋
The for x = 1.98,
⌊1.58⌋ = 1
From above definition of floor function, we can say that the domain of greatest integer function is the set of all real numbers (R) whereas the range of is the set of all integers (Z).
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If ∠1 and ∠2 form a linear pair and m∠1 is eighteen less than twice m∠2, find the difference in the measures of the two angles.
Difference in the measures of two angles is 48°.
What are Linear Pair of Angles?Linear pair of angles are the pair of angles formed when two lines are intersected at a point.
Sum of the measures of angles of linear pair is supplementary or 180°.
Given that ∠1 and ∠2 form a linear pair.
m∠1 + m∠2 = 180°
Also, m∠1 is eighteen less than twice m∠2.
m∠1 = (2 × m∠2) - 18
Substituting second equation in first equation,
(2 × m∠2) - 18 + m∠2 = 180°
3 × m∠2 - 18 = 180
3 × m∠2 = 198
m∠2 = 198 / 3
m∠2 = 66°
m∠1 = (2 × 66) - 18 = 114°
Difference of the angles = m∠1 - m∠2 = 114° - 66° = 48°
Hence there is 48° difference in the measures of angles.
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The box-and-whisker plot below represents some data set. What is the maximum value of the data?
What is the importance of knowing the concept of parallel and perpendicular lines?.
These concepts can be used in many buildings like skyscrapers because they may fall over if they are not parallel to the ground. In the case of a cabinet, you need the sides to be perpendicular to each other and to the top and bottom. Otherwise, the door may not open and close accurately.
The characteristics of parallel lines are;
1. The shortest distance between the lines that are parallel is the same at all points along the lines.
2. The lines are on the same plane.
3. As the plane is extended indefinitely, the lines do not intersect.
The characteristics of perpendicular lines are;
1. Perpendicular lines are lines that intersect at a point.
2. Where two lines are perpendicular, the slope of one line is the negative reciprocal of the other, such that the product of the slopes of two lines that are perpendicular is -1.
3. The angle formed at the intersection of perpendicular lines is 90°.
4. A horizontal and a vertical line on the same plane are perpendicular to each other.
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Can someone please explain to me how I’m suppose to solve it?
To solve the equation 6 - square root of x-2 = x + 4, you can follow these steps:
Start by isolating the square root term on one side of the equation. Subtract x + 4 from both sides to get:
6 - x - 4 - square root of x-2 = 0
Next, combine like terms:
2 - square root of x-2 = 0
Add square root of x-2 to both sides to get:
2 = square root of x-2
Now square both sides of the equation:
4 = x - 2
Finally, add 2 to both sides:
x = 6
So the solution is x = 6
Are parallel lines undefined or no solution?.
If the lines are parallel, then the pair of equations representing parallel lines have no solution.
Parallel lines are straight lines which, existing in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. Another term involved is the "plane". We keep the Plane as an undefined term. The only thing is that we can represent it intuitively or explain it with the help of a physical model.
As the parallel lines never intersect at any point, hence the pair of equations of parallel lines will not have any solution, i.e. they have no solution.
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If there are 50 dogs and 30 cats at a pet daycare, fill out all of the possible ratios of dogs to cats that could be made.
Answer:
50/30, 25/15, 10/6, 5/3
Step-by-step explanation:
To answer the question, you need to decompose the numerator and denominator into simple factors and see what combinations are possible.
50:2 = 25
25:5 = 5
5:5 = 1
Multipliers for 50: 2, 5, 5
30:2 = 15
15:3 = 5
5:5 = 1
Multipliers for 30: 2, 3, 5
Common multipliers for dogs and cats: 2 and 5
So we can divide the our ratio by 2, by 5 and by their product 2 x 5 = by 10.
We get the following series:
50/30, 25/15, 10/6, 5/3
A bakery ell pie and cake. In November, they old 400 pie and 250 cake totaling $8,550. In December, they old 300 pie and 500 cake totaling $11,100. What i the price of the pie?
The price of the Pie will be $12 and the price of the cake will be $15.
We will solve this problem with the help of equations.
Let the price of Pie be $X and the price of Cake be $Y.
According to a given question in November, they sold 400 Pie and 250 Cake and the total amount is $8550.
So our first equation will be,
400X + 250Y = 8550.
And in December they sold 300 pies and 500 cakes respectively totaling $11100.
So our second equation will be:
300X + 500Y = 11100.
Taking the value of X from our first equation we get:
X = (8550-250Y)/400
Putting it in our second equation we get:
300(8550-250Y)/400 + 500Y = 11100
On solving this:
3/4(8550-250Y)+500Y = 11100
25650/4 - 750/4 Y + 500Y = 11100
1250/4 Y = 18750/4
Y = 15
So the price of the Cake is $15.
Similarly, we can find the price of Pie as (8550-250*15)/400 = 12
So the price of the pie is $12.
Therefore the price of the pie is $12 and the cake price is $15.
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The amount of revenue for a business can be modeled by the function
L(t) = 7300(1.012) 4t. Write an equivalent function of the form L(t) = abt.
Round your final values to 4 decimal places.
Answer:
a=7300, b=1.0108.
L(t) = 7300(1.0108)^t
Step-by-step explanation:
First, we can see that L(t) = 7300(1.012)^(4t) can be rewritten as L(t) = 7300e^(4tln(1.012))
Using the properties of logarithms, we can simplify this to L(t) = 7300e^(4t(ln(1.012))
Now, we can see that the function L(t) is in the form L(t) = ab^t, where a = 7300 and b = e^(ln(1.012))
Therefore, we can write the equivalent function as L(t) = 7300e^(0.048t)
The final values rounded to 4 decimal places are a=7300, b=1.0108.
L(t) = 7300(1.0108)^t
Answer
L(t)=
7300
1.0489
7300(1.0489)
Step-by-step explanation:
Is every polynomial is binomial?.
No, a polynomial is not always a binomial. A binomial is a polynomial of two terms, while a polynomial can have any number of terms.
Since polynomials can have more than two terms, they are not all binomials.
For instance
A polynomial but not a binomial is 2x2 + 3x + 4.
A polynomial and a binomial, 2x2 + 6
The assertion thus is false.
For some whole integer n, a polynomial is the sum of n monomials. As a result, polynomials are a general term for monomials, binomials, and trinomials.
You can have as many terms as you like in a polynomial. The exponents of all a monomial's variables are added to determine its degree.
The binomial theorem or, alternatively, Pascal's triangle can be used to extend a binomial raised to the nth power, denoted as (x + y)n.
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What is the value of k in the quadratic polynomial 3x²?.
Since 3* + 4x + 2k equals the value of k= -2, the polynomial's zero is 2. A polynomial is an equation with variables, constants, and other terms.
A polynomial is a mathematical expression made up of variables (also known as indeterminates) and coefficients. It can be expressed using only the operations addition, subtraction, multiplication, and non-negative integer exponentiation of variables. A polynomial is an equation that contains variables, constants, and exponents and is based on the operations of addition, subtraction, multiplication, and division of numbers (No division operation by a variable).
Given that -2 is a zero in the equation
f(x)=3a*+4x+2
k, f(-2)=0 :3(-2) +4 (-2) +2k=0\s
=>12-8+2k=0\s
=>2k=-4\s
=>k=-2
Since 3z + 4x + 2k is the value of k = -2, the polynomial's zero is 2.
Complete question:
What is the value of k in the quadratic polynomial 3x². The polynomial 3x2 4x 2k has a zero for what value of k(-2).
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The quadratic polynomial 3x² does not include a term with the variable k.
The expression 3x² is a second degree polynomial, also known as a quadratic, but it does not include a term with the variable k. The general form of a quadratic polynomial is ax² + bx + c, where a, b, and c are coefficients and x is the variable.
In this case, the coefficient of x² is 3, the coefficient of x is 0, and the constant term is 0. The variable k is not present in the expression 3x², so it is not possible to determine its value.
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Complete Question:
What is the value of k in the quadratic polynomial 3x²?.
write 2 equations that relate distance and time
I'm been doing this question about an hour still can't solve so I really need your help! Really appreciate if u do :D
Explanation:
1 hat = 45 dollars
6 hats = 6*45 = 270 dollars
1 phone = 40 dollars
4 phones = 4*40 = 160 dollars
total = $270 + $160 = $430
Felicia spent a total of $430. Since this amount of money leaves her account, this means we write a negative sign out front to end up with the final answer of -430
In other words, her account balance went down by $430 which is why we use a negative. If someone gave her $430, then the answer would be positive.
a+b-2c=5 2a+2b-4c=1 2a-b+2c=3
The system of equations does not have a solution.
What is the simultaneous equation?Generally, These are three equations with three variables (a, b, and c). To find the solution, you can use a method such as substitution, elimination, or graphing.
This is a system of linear equations. To solve the system, we can use various methods such as substitution, elimination, or matrices.
One way to solve this system is by using the elimination method. We can start by adding the first and second equations, which will cancel out the b and -2c terms, leaving us with 3a = 6. Therefore, a = 2.
Next, we can substitute this value of a into the other equations and solve for b and c.
For example, using the first equation we can get:
a + b - 2c = 5
2 + b - 2c = 5
b - 2c = 3
Thus, b = 3 + 2c
Using the last equation we can get:
2a - b + 2c = 3
2(2) - (3 + 2c) + 2c = 3
4 - 3 - 2c + 2c = 3
1 = 3
This is a contradiction, so the system of equations does not have a solution.
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How do you find the length of an acute triangle?.
The longest side c of an acute triangle is the one opposite the largest angle γ. To determine its length, use the law of cosines: c = √(a²+ b² - 2ab cos(γ)), where a and b are the two shorter sides of the triangle.
A triangle is a polygon with three edges and 3 vertices. The terminology for categorizing triangles is more than two thousand years vintage, having been defined on the first actual page of Euclid's elements.
In Euclidean geometry, any three factors, while non-collinear, decide a completely unique triangle and simultaneously, a unique plane (i.e. a -dimensional Euclidean space). In different words, there's best one plane that consists of that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is the best one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is not real. this text is ready triangles in Euclidean geometry, and in particular, the Euclidean plane, except wherein otherwise stated.
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solve the system if equations using the substitution method.
2x+y=5
6x+2y=17
A larger number is double the sum of 3 and a smaller number. The larger number is 2 less than 3 times the smaller number. If y represents the larger number and x represents the smaller number, which equations model the situation? check all that apply. Y = 3 x minus 23 x minus y = 23 x minus y = negative 2y = 2 minus 3 xy = 2 (x + 3).
The Required Equations are y=2(x+3) and y=3x-2
Given:
'y' represents the larger number and
'x' represents the smaller number
Consequently, a greater number is double the sum of three, and a smaller number is larger no = double of ( 3 and smaller number)
The necessary equation for the first condition is y=2(x+3).
The bigger figure is now two less than three times the smaller one.
For the first condition, the equation is larger number Equals three times smaller number and two less, or y=3x-2.
Therefore, y=2(x+3) and y=3x-2 are the Required Equations.
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What are the formulas in algebraic expressions Class 8?.
In class 8, students typically learn the following formulas in algebraic expressions:
The distributive property: This formula states that a(b + c) = ab + ac. This formula is used to simplify expressions by breaking down a product of a sum into separate products.
The commutative property: This formula states that a + b = b + a and a x b = b x a. This formula is used to change the order of the terms in an expression without changing its value.
The associative property: This formula states that (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c). This formula is used to group the terms in an expression in a different way without changing its value.
The identity property: This formula states that a + 0 = a and a x 1 = a. This formula is used to simplify expressions by canceling out certain terms.
The inverse property: This formula states that a + (-a) = 0 and a x (1/a) = 1. This formula is used to simplify expressions by canceling out certain terms.
The order of operations: This formula, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is used to determine the order in which mathematical operations should be performed.
Therefore, These are some of the basic formulas in algebraic expressions that students learn in class 8, they are important to understand and apply in solving algebraic equations and problems.
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A man got a 10% increase in his salary. If his new salary is rupee 1,54,000, find his original salary
1,40,000 Rs. is his original Salary.
Let the original Salary of the man be = X
His New Salary = 1,54,000 Rs. (given)
X + (10/100 × X ) = 154000
X + (X/10) = 154000
11X / 10 =154000
X = 154000 × (10/11)
x = 140000
Hence we can say that his original Salary was 1,40,000 Rs.
What are the rules of multiplication and the rules of addition?.
The rules of multiplication are Associative Property, Commutative Property, Distributive Property, and Identity Property, and the rules of addition are Addition of two positive numbers is always positive, and the addition of two negative numbers is always negative.
The properties of multiplication are particular rules that are used while multiplying numbers. These properties help simplify expressions easily and, hence, have a significant role in solving all kinds of mathematical expressions, whether algebraic expressions, fractions, or integers.
Associative Property: (P × Q) × R = P × (Q × R)
For example, (4 × 5) × 3 = 4 × (5 × 3) = 60.
Commutative Property: P × Q = Q × P
For example, 3 × 4 × 2 = 2 × 3 × 4 = 24.
Distributive Property: P(Q + R) = PQ + PR; P(Q - R) = PQ - PR
For example, 3(2 + 4) = (3 × 2) + (3 × 4) = 6 + 12 = 18.
Identity Property: P × 1 = P
For example, 4 × 1 = 4, or 1 × 27 = 27.
The addition means summing up two or more numbers or values to get another number.
Positive + Positive Addition (Sign will be Positive) 3 + 4 = 7
Negative + Negative Addition (Sign will be negative) – 3 + (-4) = -7
Positive + Negative Subtraction (Sign of greater number) 3 + (-4) = -1
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NEED TO FINISH BY 5!!! WILL MARK BRAINLIEST
Answer:
10 1/8
Step-by-step explanation:
good luck to you :)
Answer:
10 1/8
Step-by-step explanation:
hope this helps good luck!
Which relationships can be represented by the equation y = 1/5x
The relationship is proportional because the equation graphs to a straight line that goes through the origin.
The y-value at x = 1 is 1/5, and this value serves as the proportionality constant.
The price of the carpet is proportional to the square footage, for instance, if each square foot of carpet costs $1.50. In this circumstance, the proportionality constant is 1.5.Two numerical sequences, frequently experimental data, are said to be proportional or directly proportional in mathematics if their corresponding elements have a fixed ratio, known as the coefficient of proportionality or proportionality constant.An equation of the form y = kx, where k is the proportionality constant, or a ratio table, which plots a straight line through the origin, can be used to represent a proportional relationshipTo know more about equation here
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What is the graph of the solution set?.
The set of all possible solutions to a linear equation with two variables is called the solution set. Specifically, the collection of all ordered pairs that are equivalent to the equation.
What is the graph of the inequality's solution set?A region always appears on the graph of the solution to a linear inequality. However, that set might not always include the boundary. The "or equal to" part of the inclusive inequality made the line a part of the solution set in the previous example.
What is the solution set?Any value of a variable that makes the equation true is a solution. The set of all variables that makes the equation true is called a solution set. Because 2(4) + 6 = 14, the solution set for 2y + 6 = 14 is 4.
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4. Given f(x) = ¹ + 10x +32
x+5
a. Find algebraically the values of x for which
f(x) = 8.
b. Show algebraically that f(x) never equals 5.
c. Does f(x) ever equal -5? Justify your answer.
d. Confirm the results of parts a, b, and c by
plotting the graph of function fon your
grapher and sketching the result.
(a). The values of x for which f(x) = 8, are x = (-1 + √17) / 2 and x = (-1 - √17) / 2.
(b). The equation has no real roots, and f(x) never equals 5.
(c). The equation has no real roots, and f(x) never equals -5.
(d). The graph will confirm that there are two x-intercepts corresponding to the two solutions found.
What is algebraic value?
A general rule is that the algebraic expression should take any of the following forms: addition, subtraction, multiplication, and division. Bring the variable to the left side and the other values to the right side in order to find the value of x.
a.
To find the values of x for which f(x) = 8,
we need to solve the equation x² +10x +32 / x+5 = 8.
We can start by multiplying both sides of the equation by (x+5) to get rid of the fraction:
x² + 10x + 32 = 8(x + 5)
Expanding the right side:
x² + 10x + 32 = 8x + 40
Subtracting 8x from both sides:
x² + 2x + 32 = 40
Subtracting 32 from both sides:
x² + 2x = 8
Dividing both sides by 2:
x² + x = 4
Subtracting 4 from both sides:
x² + x - 4 = 0
We can use the quadratic formula to solve this equation:
x = (-b ± √(b² - 4ac)) / 2a
where a = 1, b = 1, and c = -4.
So,
x = (-1 ± √(1² - 4 * 1 * -4)) / 2 * 1
x = (-1 ± √(1 + 16)) / 2
x = (-1 ± √17) / 2
Thus, the two solutions are x = (-1 + √17) / 2 and x = (-1 - √17) / 2. These are the values of x for which f(x) = 8.
b. To show that f(x) never equals 5, we need to show that there is no solution to the equation (x² + 10x + 32)/(x + 5) = 5.
Suppose there is such a solution, say x = a.
Then,
5(x + 5) = x² + 10x + 32
5x + 25 = x² + 10x + 32
-x² + 5x - 7 = 0
This is a quadratic equation and can be solved using the quadratic formula. However, we can see that the equation has no real solutions because the discriminant, b² - 4ac, is negative. Therefore, the equation has no real roots, and f(x) never equals 5.
c. To determine whether f(x) ever equals -5, we can follow a similar approach as in part b.
Suppose there is a solution to the equation (x² + 10x + 32)/(x + 5) = -5. Then,
-5(x + 5) = x² + 10x + 32
-5x - 25 = x² + 10x + 32
x² - 15x + -57 = 0
This is a quadratic equation and can be solved using the quadratic formula. However, we can see that the equation has no real solutions because the discriminant, b² - 4ac, is negative. Therefore, the equation has no real roots, and f(x) never equals -5.
d. To confirm the results of parts a, b, and c, we can plot the graph of the function f(x) = (x² + 10x + 32)/(x + 5) and sketch the result.
The graph of the function will show the x-intercepts and the y-intercepts and will also show any asymptotes.
The graph will confirm that there are two x-intercepts corresponding to the two solutions found.
Hence, (a). The values of x for which f(x) = 8, are x = (-1 + √17) / 2 and x = (-1 - √17) / 2.
(b). The equation has no real roots, and f(x) never equals 5.
(c). The equation has no real roots, and f(x) never equals -5.
(d). The graph will confirm that there are two x-intercepts corresponding to the two solutions found.
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If cosine of x degrees equals three-fifths, what is the value of b?
triangle LMN in which angle M measures 90 degrees, angle L measures x degrees, LN measures 20 units, and LM measures 3b units
b = 4
b = 5
b = 6
b = 7
If cosine of x degrees equals three-fifths and angle L measures x degrees, LN measures 20 units, and LM measures 3b units. The value of b is 11.
What is the value of angle b?Value of angle b can be determine by dividing the LN measures by the LM measures.
Given data:
LN measures 20 units
LM measures 3b units
Using this formula to find the value of b
Cosine(x°) = LN/NM
3/5 = 20/(3b)
b = 20×5/ (3×3)
b = 100 /9
b = 11
Therefore we can conclude that the value of b is 11.
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Answer:
b = 4
Step-by-step explanation:
cos x = 3/5 = 3b/20
20/5 = 4
b/4 = 3
4 x 3 = 12
12/20 = 3/5
So, b = 4
The output of a function i 6 more than 2 time the input find the input when the output i -10
The value of input is -8 when the output is -10.
What is function?A function is a mathematical relation between a set of inputs (also called independent variables) and a set of outputs (also called dependent variables). It assigns a unique output to each input in its domain. Functions are often represented by mathematical expressions, graphs, or tables. In computer science, functions are used to organize and structure code by breaking it down into smaller, more manageable tasks.
The function describes that the value of function is 6 more than 2 times tha input
so f(x) = 2x + 6
so this is required function according to the description.
given that output of the function is -10
so f(x) = -10
=> 2x+6 = -10
=> 2x = -16
=> x = -16/2
=> x = -8
so the value of input is -8 for which the output is 10.
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Find each product mentally show the steps you used 11x 27
Work Shown:
11 = 10+1
27*11 = 27*(10+1) = 27*10 + 27*1 = 270+27 = 297
I used the distribution rule a(b+c) = ab+ac
The 11 was broken up into 10+1 since 10 is easy to work with.
I need help, it’s the Remainder Theorem
A nameless polynomial p(x), or simply "some polynomial p whose variable is x," serves as the foundation of the Remainder Theorem. The Theorem then talks about dividing that polynomial by some linear factor, x a, where an is just a number.
What is The Remainder Theorem?Although it might not appear so, at least at first glance, the Remainder Theorem is useful for evaluating polynomials at a given value of x.The good news is that all you really need to know about the theorem is how to utilize it; you don't "have" to comprehend how it was proved.Once the lengthy polynomial division is complete, you are left with a polynomial answer q(x), where "q" stands for "the quotient polynomial," and a remainder r(x), where "r" stands for "the residual, after division." This balance could either be a simple integer or a polynomial with a correct variable.Examples :
Factor 4x² - x - 3 : (4x + 3 ) (x - 1)
4x² - x - 3
=(4x² + 3x) + (-4x -3)
Factor out x from 4x² + 3x : x(4x + 3 )
Factor Out -1 from -4x -3: -(4x + 3)
=x(4x + 3) - (4x +3)
Factor out common team (4x + 3): (4x + 3) (x - 1)
=(4x + 3) (x - 1).
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