how does (4x + 5)(3x + 10) demonstrate the closure property for polynomials?

Answers

Answer 1

Answer:

Step-by-step explanation:

A polynomial is a finite sum of terms in which all variables have whole number exponents and no variable appears in a denominator.

box4aa

box1

rembox22

The leading term in a polynomial is the term of highest degree.

The constant term in a polynomial is the term without a variable.

Polynomials: YES or NO

3x½

NO - exponent is not a whole number

polyd

NO - the variable is in the denominator

-3x2

YES - satisfies the polynomial definition

polyd1

NO - the square root of x can be written with a fractional exponent of x.

2x-3

NO - exponent is not a whole number.

Negative exponent places x3 in denominator.

polyrad3

YES - the exponent of x is 1 which is OK.

The coefficient being radical 3 is not a problem.

 

Classification by Terms

monomial

one term: 12, 4x, x2, -5xy

binomial

two terms: 2x - 1, x2 - 4

trinomial

three terms: x2 + 2x + 1

polynomial - one or more terms: polynomial means "many", but it can also be one term.

The ending of these words "nomial"

is Greek for "part".

Classification by Degree

Linear - degree of 1 or 0:    3x + 1  or  12

Quadratic - degree of 2:   2x2 - x + 7

Cubic - degree of 3:   3x3 + 4x2 + 3x + 5

definition The degree of a term with whole number exponents is the sum of the exponents of the variables, if there are variables. Non-zero constants have degree 0, and the term zero has no degree.  Example: 6x2 has a degree of 2;   4x2y3 has a degree of 5 (the sum of 2 and 3).

definition

The degree of a polynomial is the highest degree of its terms.

Example: 3x2 + 4x + 1 has a degree of 2;  x3 - x2 + 5x - 2 has a degree of 3

The standard form of a polynomial is when all like terms are combined and the degrees are arranged in descending order.

Polynomial: 2x + 3x5 + 4x3 - 8

Standard form: 3x5 + 4x3 + 2x - 8

3x5 + 4x3 + 2x1 - 8x0

boxdegree1

Will we ever really use the degree?

Knowing the "degree" of a polynomial function will let you determine the most number of solutions the function may have, and the most number of times the function will cross the x-axis.

FYI: The standard form of a polynomial is formally written as anxn + an - 1xn - 1 + ... + a2x2 + a1x + a0, where n is a non-negative integer, and a0, a1, a2, ..., an are real number constant coefficients with an ≠ 0.

bullet Polynomials and Closure:

Polynomials form a system similar to the system of integers, in that polynomials are closed under the operations of addition, subtraction, and multiplication.

CLOSURE: Polynomials will be closed under an operation if the operation produces another polynomial.

• Closure under Addition: (2x2 + 3x + 4) + (x2 - 5x - 3) = 3x2 - 2x + 1

When adding polynomials, the variables and their exponents do not change. Only their coefficients will possibly change. This guarantees that the sum has variables and exponents which are already classified as belonging to polynomials. Polynomials are closed under addition.

• Closure under Subtraction: (2x2 + 3x + 4) - (x2 - 5x - 3) = x2 + 8x + 7

When subtracting polynomials, the variables and their exponents do not change. Only their coefficients will possibly change. This guarantees that the difference has variables and exponents which are already classified as belonging to polynomials. Polynomials are closed under subtraction.

• Closure under Multiplication: (x + 1)(x2 + 4x + 3) = x3 + 5x2 + 7x + 3

When multiplying polynomials, the variables' exponents are added, according to the rules of exponents. Remember that the exponents in polynomials are whole numbers. The whole numbers are closed under addition, which guarantees that the new exponents will be whole numbers. Consequently, polynomials are closed under multiplication.

What about division?

Polynomials are NOT closed under division. Look at this simple counterexample:

closure1

When dividing variables with exponents, there is the possibility of creating a negative exponent. Negative exponents are not allowed in polynomials.


Related Questions

Debbie has at most $60 to spend on clothes. She wants to buy a pair of jeans that cost $22 and spend the rest on t-shirts. Each t-shirt costs $8. please show the process

Answers

That’s the answer
Step:

Answer:

Jeans + Shirts can’t be more than $60.

Jeans + Shirts could equal $60.

$22 pair of jeans + $8 per t-shirt ≤ $60

3,735 ÷ 45 = ________.

Answers

Answer:

83

Step-by-step explanation:

3735/45=83, 83x45=3735

Suppose 30=a×60.
What is 30a?

Answers

Answer:

30 x2 a=2

Step-by-step explanation:30 / 60 = 2 30 x 2 = 60

Answer:

a = 0.5

Step-by-step explanation:

a=0.5 because 0.5×60=30

Julie has 1.54 L of water in a large bottle for a science experiment. She pours the water into beakers. Each beaker holds 0.22 L.

If each beacker gets 1.54 L of water, how many beakers will she fill?
How many beakers does it take to hold the 1.54 L of water?

0.3388

14.28

0.7

7

Answers

Answer: the answer is 7 hope that helped and have a good and always good luck

Step-by-step explanation:

7. 0.22 times 7 equals 1.54. Good luck

a and b are natural numbers. The diameter of cell a is 2.3 , and the diameter of cell b is 2.3 .
Which is a true statement?
A. If a = 4 and b = 6, then cell a is 100 times greater than cell b.
B. If a = 4 and b = 6, then cell b is 100 times greater than cell a.
C. If a = 4 and b = 6, then cell a is 2 times greater than cell b.
D. If a = 4 and b = 6, then cell b is 2 times greater than cell a.

Answers

Answer:

I think its A

can someone help with this pls? i don’t understand math at all

Answers

it’s 15t + 19r=2
because these both aren’t related at all as 15 as ‘t’ and 9 has ‘r’ whereas 2 is a coefficient.. only likes terms will be simplified
First one is the one

The image shows the linear graph for a family that hired a babysitter for a day. The family gave the babysitter some money up front to buy herself some lunch, and then paid her a constant hourly rate for the hours she worked. Click the thumbnail image to see the full size image.


Which 2 of the following statements are TRUE about the EQUATION of this line? Select the TWO statements that are accurate.

A. The equation of this line is y = 10x + 10.

B. The equation of this line is y = 20x.

C. The equation of this line shows a proportional relationship between the cost and the time.

D. The equation of this line shows a non-proportional relationship between the cost and the time.

Answers

Answer:

A & C

Step-by-step explanation:

If you look at the graph you will see what I am talking about.

Answer:

A and C

Step-by-step explanation:

y=10x(ten dollars per hour-x)=10(the lunch money)

the equation is a proportional relationship because for each hour there is a raise of ten dollars. proportional relationships are when one variable is a constant value times another(10 times the hour)

What is the GCF using distributive property of 40 and 25?

Answers

Answer:

Great Common Factor ♡•♡

Answer:

5 :)

Step-by-step explanation:

7th GRADE MATH RATIOS HELP TIME ALMOST OUT

Answers

Answer:

8, 40, 28.6, 40, 80, 36.6

Step-by-step explanation:

First find out what the ratio of tulips and roses is for each garden.

Ben: 40÷(4+1) = 8 so 8×4=32 and 8×1= 8

Ina: 40÷(2+5) =5.7 so 2×5.7=11.4 and 5×5.7= 28.6

Use these numbers accordingly.

Seems odd that there would be decimals in a problem like this, consult your teacher.

Answer:

8, 40, 28.6, 40, 80, 36.6

Step-by-step explanation:

i hope you got it in time!

plz help me with this math

Answers

Answer: -6 4/9 -3 2/9 - 8 2/9 =-17 8/9

-12.48-(-2.99)-5.62= 15.11

-19 2/9 - 4 1/9 - (-3 4/9)= -19 8/9

-353.92- (-283.56) - 131.29 = 201.65

83 1/5- 108 2/5 -(-99 1/5) = 74

Step-by-step explanation:

PPPPPPPPlease help me (20 points!!!!)

Answers

[tex]\longrightarrow{\mathfrak{\frac{-9}{6}÷\frac{3}{-2}}}[/tex]

[tex]\longrightarrow{\sf{\frac{-\cancel{9}}{\cancel{6}}\times\frac{-\cancel{2}}{\cancel{3}}}}[/tex]

[tex]\longrightarrow{\sf{\frac{3}{3}}}[/tex]

[tex]\longrightarrow{\boxed{\bf{1}}}[/tex]

Identity applied -

[tex]\star{\:\:\:\:\:\:\boxed{\bf{\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}}}}[/tex]

The answer is B
When dividing fractions you flip the numerator and the denominator of the first fraction making it go from -9/6 to 6/-9 then you multiply the fractions across, in the end the answer is 1 whole

Again, I need help with another problem...(I'm going to post 2 more DIFFERENT problem after this one is answered)

Answers

Answer:

y = 4x + (-9)

Step-by-step explanation:

You can observe that the line crosses the y-axis at -9, so that is our y-intercept.

For the slope, just take the y-intercept for example, and count how many squares the line goes up before reaching 1. In this problem you can see that the line goes 4 squares up in the time it takes to go 1 square to the right, so the slope is y over x, meaning 4 over 1, or just 4.

Answer:

The answer should be y = 4x + (-9).

Compare the values of the two expressions using -3(-2) + 3 __3(-4)+9​

Answer:

>

Step-by-step explanation:

expression 1: -3(-2) + 3 = 9

expression 2: 3(-4) + 9 = 3

Answers

Answer:

I think its number 1

Step-by-step explanation:

3 is a factor of 9. 3 x 3 = 9

May someone help me with this. The question is in the picture.

Answers

6,000 students will enroll

Answer:

first we have to find oput how many they accepted, 60% of 10,000 is 6000. Next we have to find out how many of those 6000 will actually enroll.

to find 40% of 6000 we can first find 10% which is 600, so 600x4=2400 or 40% of 6000.

2400 will actually enroll

Mike jogged a total distance of 5 1/3 miles during the months of October and November. If Mike only jogger 1/6 mile every day, Which expression shows the number of days in which he went jogging?

Answers

Answer:

D

Step-by-step explanation:

If he jogged [tex]x[/tex] miles every day, and he jogged a total distance of [tex]y[/tex] miles, then he runs for [tex]y/x[/tex] days. This is an extension of the [tex]d=rt[/tex] or distance equals rate times time.

We plug in for y and x to get that he runs for [tex]5~1/3~~/~~1/6[/tex].

So, the answer is [tex]\boxed{D}[/tex] and we're done!

Answer:

d

im always right

Graph the ordered pair (-4,-1)

Answers

in the 3rd quadrant, so go to the left for spaces and go down one space
you start from negative 4 and go down one point to negative 1

Answer.
What are the four key points of Dalton’s Atomic Theory?
a.
b.
c.
d.
2. In 1913, Neils Bohr proposed a new atomic model known as the _________ model, which is patterned after the solar system.

Answers

Answer:

a. all matter is made of atoms

b. atoms are indivisible and indestructible

c. atoms of given elements are identical in mass and properties

d. compounds are combination of 2> different types of atoms

2. it's called Planetary model

Answer:

1. All matter are made up of atoms, which are indivisible

2. All atoms of the same element are identical; different elements have different types of atom.

3. Atoms can neither be destroyed nor created

4. Compounds are combinations of two or more different types of atoms.

Rutherford-Bohr Model

Step-by-step explanation:

Hope this helps

PPPPPPPPLEASE HELP ME!!

Answers

Answer:

B.) -9.75 x 2

Step-by-step explanation:

Fee would be negative because it’s being taken away from the customer’s account. The 2 is just how many times it was taken away.

The answer is B I hoped it helps :3

Walmart marked up a bottle of nail polish 150% in order to make profit. if Walmart sells the nail polish for $8.75, what was the original cost of the polish?

Answers

The original price of the nail polish with a 150% markup sold at $8.75 is $3.50.

Let x be the original price of the polish.

Walmart marked up 150%.

Selling Price = Original Cost + Mark-up

therefore the selling price = x + 150% of x

we know the selling price is $8.75

Therefore, substituting the values:

8.75 = x + 150% of x

convert percentages to decimals:

8.75 = x + 1.5x

combine like terms:

8.75 = 2.5x

divide by 2.5 into both sides:

3.5 = x

use reflexive property:

x = $3.50

Thus, the original price of the nail polish is $3.50.

Learn more about markups:

brainly.com/question/28945726

Which best describes the error in solving the inequality or graphing the solution of the inequality?

use picture to answer question

Identify and graph the correct solution.

Answers

Answer:

111111111111ʕ•ﻌ•ʔʕ•ﻌ•ʔʕ•ﻌ•ʔʕ•ﻌ•ʔʕ•ﻌ•ʔ

Answer:

i do not know

Step-by-step explanation:

Drag the tiles with equivalent ratios into each box, two tiles have no match

Answers

1:8, 15:120 is the same have a good day
2:3 and 4:60 bot have the same and equivalent ratio

Line segment EF is shown on a coordinate grid:

A coordinate plane is shown. Line segment EF has endpoints negative 3 comma 2 and negative 1 comma 2.

The line segment is rotated 180 degrees counterclockwise about the origin to form E′F′. Which statement describes E′F?

Group of answer choices

E′F′ is perpendicular to EF.

E′F′ and EF are equal in length.

E′F′ is half the length of EF.

E′F′ is greater than twice the length of EF.

Answers

Answer:

E'F' and EF are equal in length.

Step-by-step explanation:

This is because it is a mirror translation over the y-axis, and the length is not changed.

If you rotate a line 180° it’ll be the same line but the endpoints will be swapped because you’re just flipping it over. This means they are the same in length, while they’re in the same position essentially, this is the only answer that fits because it’s not perpendicular, E‘ F‘ is not half the length of EF,And it’s not greater than twice the length either. This means your answer is answered number two, E’F’ and EF are the same length.

To be consistent with the mystery theme, a local movie theater reported the attendance at their mystery movie festival in the form of a problem to be solved. The report stated the following: “If there were 2 times more adults plus an additional 86 adults, a total of 638 adults would have been in attendance.”

Refer to the information above. If twice as many children as adults attended the festival, then how many children attended the mystery movie festival?
Group of answer choices

A. 276

B. 552

C. 724

D. 466

Answers

I think the answer is 724 if it’s right than make me brainliest

Brett is building a fence around his land. The perimeter of his land is 101,376 inches. Each picket of the fence is 5 inches across. How many pickets are needed for the fence?

Answers

Answer:

20,275.2

Step-by-step explanation:

PLEASE GIVE ME BRAINLIEST.          

20, 275 or 20, 276. I don't think you can buy a decimal of something. It's not like you can buy 0.7 apples. Or 0.3 televisions.

Marlye posted a story on her blog. On the first day there were 80 responses and on the second day there were 180 responses. What percent is the increase, 100, of the original number of responses, 80?
a. 55%
b. 80%
c. 100%
d. 125%

Answers

100/80 = 1.25 or 125%

the answer is D-125%

You can spend up to $35 on a shopping trip.

a. You want to buy a shirt that costs $14. Identify an inequality that represents the amount of money, x-
(in dollars), you will have left if you buy the shirt.

b. You notice that the shirt is on sale for 30% off. Identify the new inequality.


c. Do you have enough money to buy the shirt that is on sale and a pair of pants that costs $23? Which best explains your answer?

Answers

The answer for A is A. Answer for B is B. Answer for C is D (last circle).

Task 1:Different ways to represent Relations. Complete the table with the different ways.

Answers

Answer:

I dont know

Step-by-step explanation:

This i dont know

domain- -4, -3, -2, -1, 0

HELP I AM GOING TO BE ASKING QUESTIONS ALL DAY!!!!

Answers

Answer: yes, no. yes, yes

Step-by-step explanation:

-7.5238 X 378=?????????

Answers

Answer:

−2,843.9964

Step-by-step explanation:

Multiply

-7.5238 × 378

Hope it helps

Do any of u guys understand this?

Answers

Answer:

i dont

Step-by-step explanation:

a. 8 pounds to 12
b. 1.5 for 1 pound
c. 31.5
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