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

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.

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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

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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.

1. For each event, write the sample space and tell how many outcomes there are.
Spin these 3 spinners.
Answers: A. 14/ B. 16/ C. 12
2. A simulation is done to represent kicking 5 field goals in a single game with a 72% probability of making each one. A 1 represents making the kick and a 0 represents missing the kick. Based on these results, estimate the probability that 3 or more kicks are made.
Answers: A. 2/5 B. 3/5 C. 4/5
I will type the trial in the comment section.

PLZ HELP!!! ALL ANSWERS THAT ARENT DETAILED AT ALL WILL BE DELETED JUST MAKE IT SORT OF DETAILED LOL PLS- Solve the equation 7x + 3x − 18 = 6(x + 3), and show your work. (5 points)

Part B: Use this list of properties to justify each step of your math work. Justifications may be used more than once if needed. (5 points)


Distributive Property
Subtraction Property of Equality
Combine Like Terms
Multiplication Property of Equality
Division Property of Equality
Given
Addition Property of Equality

Please answer!!! Will award brainliest!!!! TY

What are the dimensions of this matrix?


2
3
6
9

i need answers quick 20 points

PLEASE SOLVE RXU FOR BRAINLIEST :)

Need help. thanks. please answer within 45 minutes. I think i have selected the right ones but please confirm

help plzzzzzzzzzzzzz

pls help due in 10 mins

Find the 13th term of the following geometric sequence.
3,6,12,24, ...

What is the mode interval for the set of data?

1. Consider the surface area of the following pyramid.

Calculate the total surface area of the pyramid. PLZZZZ SHOW YOURE WORK!

20 points! need these answers asap!
1.) The ratio of the
Choose...
(diameter, circumference, or radius)
of a circle to its
Choose... (diameter, circumference, or radius)
is approximately 3.14.
2.) Fill in the blank with the correct response.
If the diameter of a circle is 12 feet, then the radius of the circle is ___ feet.
3.)What is the perimeter of the following figure in centimeters?
A. 7.5 cm
B. 57 cm
C. 28.2 cm
D. 5.7 cm
the picture goes for the 3rd question btw

PLEASE HELP I WILL REPORT FOR WRONG ANSWER SORRY

Can someone help me with this?

A fisherman is sitting 2 feet above the surface of a lake on a boat. The hook on his fishing pole is floating 6 feet below the lake's surface.



Part A: Which location has the greater absolute value, the fisherman's or his hook's? Explain your reasoning.

Part B: Are the locations of the fisherman and his hook opposites? Explain your reasoning.

10. A quilt is made with 10 square pieces of fabric. If the area of each square piece is 169 square inches, what is the length of each square piece?
f. 12 inches
g. 13 inches
h. 14 inches
j. 15 inches

A manager samples the receipts of every fifth person who goes through the line. Out of 50 people, 5 had a mispriced item. If 1,450 people go to this store each day, how many people would you expect to have a mispriced item?

Which situation can be represented by the equation y = 5x?
A. A restaurant sells x number of sandwiches for $5.00 each. The cost is y, for x number of sandwiches.
B. Manuel has driven a car for x years. This is 5 times y, the number of years his mother has driven a car.
C. Meg is x, years old. Her age is 5 years greater than y, her brother’s age in years.
D. Wally spent x dollars to buy a pizza for his sister. He gave the cashier y dollars and received $5 back in change.

Just fill in the boxes please

Ahmad's mom prepares a plate with 5 peaches and 4 apples for the 6 students in his study group. What is the ratio of fruit to students?​

No absurd answers or links. Please. It's standard to slope intercept form. I will give brainliest.

Which quadratic function is represented by the graph?

Plz plz ans this with explanation

Two word problems :)
BTW.. if you cant read it, the numbers are 56 out of 400, and 7 out of 50, i wrote over them because my teacher said to so yea- good luck
oh and i need help with the second one
If you have any questions, on what something says, just ask :-)

please help 14 math problem

The scatter plot shows the temperature and the number of frozen yogurt bars sold during 10 baseball games. Using the trend in the scatter plot, approximately how many yogurt bars can the home team plan to sell when the temperature reaches 80°F?

Mr. Diego deposits $2000 into an account with 4 1/2% interest compounded twice per year. How much money will be in the account in two years?

Watch help video
Hannah is deciding between two truck rental companies. Company A charges an
initial fee of $30 for the rental plus $2 per mile driven. Company B charges an initial
fee of $100 for the rental plus $1 per mile driven. Let A represent the amount
Company A would charge if Hannah drives miles, and let B represent the amount
Company B would charge if Hannah drives s miles. Write an equation for each
situation, in terms of , and determine the interval of miles driven, I, for which
Company A is cheaper than Company B.

would anyone help really quick. ty.