Classify the following polynomials by the number, №17886440, 04.06.2020 00:12

Mathematics

Classify the following polynomials by the number of terms and degrees -2x^3 +5x

Step-by-step answer

P Answered by PhD

A third degree binomial

Step-by-step explanation:

The highest power is 3 hence the 3rd degree

and there are 2 terms hence the bi-nomial

Mathematics

Explain what a polynomials is and identify the different parts of a polynomial. explain the different labels used to categorize polynomials explain how addition and subtraction of polynomials is accomplished when multiplying polynomials, we are taking every term of one polynomial and multiplying them by every term of the second polynomial, then collecting like terms explain how foil us to accomplish this and what category of polynomials foil applies to. there are tow special case products where the product takes on special patterns explain these

Step-by-step answer

P Answered by PhD

3

A polynomial is an expression of more than one term. An expression is considered a polynomial when is has more than one term, otherwise, it would be called a monomial. These can be combined together through multiplication, addition and subtraction only. (Meaning no division or fractions)

Ex.

$3 x^{2} + 3x - 3$

x is a variable (There can be more than 1 variable in a term. Ex. 3xy, 4xyz, 4ab)

*A variable may be represented by letters.

2 is an exponent

3 is a constant

Those are the parts of a polynomial.

Polynomials can be categorized depending on the number of terms and their degree.

A polynomial with two terms is called a binomial. If it has three terms it is called a trinomial. If the expression has more than 3 terms, they are generally called polynomials.

A polynomial can be categorized by degree as well. You can determine the degree of a polynomial by looking at the term that has the highest exponent.

Using the example above, you can categorize the polynomial as a 2nd degree trinomial because 2 is the highest exponent and it has three terms.

When you add and subtract polynomials you need to take note of the variables. You can only subtract and add like terms, which means that the variables and the exponents are the same.

Ex.

$( 2x^{3} + 2 y^{2} + x + 1) + (4 x^{2} - y^{2} + y + 2)$

When you add these two polynomials, you can disregard the parentheses because according to the associative property of addition, no matter how you group the terms, the answer will be the same.

Like mentioned before you can only add and subtract like terms. It would be easier if you just group like terms together by rearranging the expression. Do not forget that the sign or operation comes along with them.

$2 x^{3} + 2 y^{2} - y^{2} + 4 x^{2} + x + y + 2 + 1$

Now combine the like terms.

$2 x^{3} + y^{2} + 4 x^{2} + x + y + 3$

Notice that we retained the terms $2 x^{3}$ , $4 x^{2}$ , x and y, this is because they have no similar terms.

FOIL method is used when multiplying 2 BINOMIALS. Remember that a binomial is an expression with 2 terms.

FOIL means:

FIRST term: first terms of each binomial.

OUTSIDE term: The two outer terms when taking the equation as a whole.

INSIDE term: The two inner terms when taking the equation as a whole.

LAST term: Last term of each binomial (2nd term of each binomial)

To get the answer, you need to multiply them with their corresponding term.

Ex. (2x+3)(x-4)

F: 2x and x (2x)(x) = $2x^{2}$

O: 2x and -4 (2x)(-4) = -8x

I: 3 and x (3)(x) = 3x

L: 3 and -4 (3)(-4) = -12

Resulting expression:

$2 x^{2} - 8x + 3x -12$ -8x and 3x are similar or like terms, so you can combine them

$2 x^{2} - 5x -12$

When doing multiplication with binomials, there are two special cases you can consider doing, which follow a pattern. The first is multiplying sum and difference.

The condition where you can apply the first special case is the first term needs to be the same and the second term are additive inverses.

(a+b)(a-b)

The resulting expression follows this pattern $a^{2} - b^{2}$

Ex. (x+3)(x-3) = $x^{2} - 3^{2}$ or $x^{2} -9$

You can use FOIL to check your

F: (x)(x) = $x^{2}$

O: (x)(-3) = -3x

I: (3)(x) =

L: (3)(-3) =

Arrange the expression:

$x^{2} - 3x + 3x - 9$ Combining -3x+3x = 0

$x^{2} -9$

The next special case is squaring a binomial and there are two scenarios that you can consider.

$(a+b)^{2}$ and $(a-b)^{2}$

The resulting expression follows a certain pattern for each:

$(a+b)^{2}$ = $a^{2} + 2ab + b^{2}$

$(a-b)^{2}$ = $a^{2} - 2ab + b^{2}$

Let's use an example of each to demonstrate this and check with FOIL:

$(a+b)^{2}$

$(2x+4)^{2}$

a = 2x b = +4

Let's insert that into our pattern:

$a^{2} + 2ab + b^{2}$

$2x^{2} + 2(2x)(4) + 4^{2}$

Simplify the expression:

$2x^{2} + 16x + 4^{2}$

$4x^{2} + 16x + 16$

Let's check with FOIL

$(2x+4)^{2}$ = (2x+4)(2x+4)

F: (2x)(2x) = $4x^{2}$

O: (2x)(4) =

I: (4)(x) =

L: (4)(4) =

Let's arrange the terms:

$4x^{2} + 8x + 8x + 16$ Combine the like terms
$4x^{2} + 16x + 16$ It's the same.

Now let's use the second scenario:

$(a-b)^{2}$
$(2x-4)^{2}$

a = 2x b = -4

Let's insert that into our pattern:

$a^{2} - 2ab + b^{2}$

$2x^{2} - 2(2x)(-4) + (-4)^{2}$

Simplify the expression:

$2x^{2} - 16x + (-4)^{2}$

$4x^{2} - 16x + 16$

Let's check with FOIL

$(2x+4)^{2}$ = (2x-4)(2x-4)

F: (2x)(2x) = $4x^{2}$

O: (2x)(-4) = -8x

I: (-4)(x) = -8x

L: (-4)(-4) =

Let's arrange the terms:

$4x^{2} - 8x - 8x + 16$ Combine the like terms
$4x^{2} - 16x + 16$ It's the same.

Mathematics

Explain what a polynomials is and identify the different parts of a polynomial. explain the different labels used to categorize polynomials explain how addition and subtraction of polynomials is accomplished when multiplying polynomials, we are taking every term of one polynomial and multiplying them by every term of the second polynomial, then collecting like terms explain how foil us to accomplish this and what category of polynomials foil applies to. there are tow special case products where the product takes on special patterns explain these

Step-by-step answer

P Answered by Master

1

Question 1
Polynomials are expressions that include constants and variables. We can use addition, subtraction, multiplication and we can use exponents but only with positive integers. In polynomials, you cannot use division by a variable. This means that you can't have terms like:
$\frac{a}{x^n}$
Where a is constant, x is a variable and n is a positive integer.
Here is an example of a polynomial:
$5x^3+3x^2+\frac{x}{2}+9$
x would be is a variable, please keep in mind that you can have polynomials with more than one variable. Numbers before variables are called constants.
Here are some example of constants:
$2;2/3;\pi;\sqrt(2)$
And we also have exponents. Exponents are numbers written in the upper-right corner of a variable. These must be positive integers.
Question 2
We can name polynomials based on their degree and number of terms.
A degree is largest exponent. Examples:
$x^3+2x+3; degree=3\\ x^2+2; degree=2\\ 21;degree=0$
You can have polynomial without variable.
Here is a chart showing you special names for polynomials based on their degree:
$0\ constant\\ 1 \ linear\\ 2 \ quadratic\\ 3 \ cubic\\ 4 \ quartic\\ 5 \ quintic\\$
There are special names for polynomials with 1, 2, 3 terms. These are monomial, binomial, trinomial. For polynomials that have more than 3 terms, we simply say polynomial of n terms.
Question 3
When adding or subtracting polynomials we simply subtract/add like terms.
Like terms are those that have the same exponent. Here is an example:
$4x^3+2x^2+x+5\\ x^4 -x^3+3x^2\\$
Let us add these two polynomials:
$4x^3+2x^2+x+5\\ x^4 -x^3+3x^2\\ x^4+(4x^3-x^3)+(2x^2+3x^2)+x+5=x^4+3x^3+5x^2+x+5$
Question 4
Foil stands for First, Outer, Inner, Last.
Foil can be used only to multiply two binomials( polynomials that have 2 terms).
Here is an example:
$(2x+3)(x-2)= (2x\cdot x) (First)+(2x\cdot (-2)) (Outer)+(3\cdot x)(Inner)\\ + (3\cdot (-2)) (Last);\\ (2x+3)(x-2)=2x^2-2x+3x-6=2x^2+x-6$
Question 5
First special case is square of a sum/difference. This happens when we want to multiply two identical polynomials. Example:
$(x+2)(x+2)\\ (x^2-x)(x^2-x)$
Second special case is a product of a sum and a difference.
Here is an example:
$(x-2)(x+2)\\
(x^2-x)(x^2+x)$
Question 6
For these two special cases we can use following formulas:
$(a\pm b)^2=a^2 \pm2ab+b^2\\
(a+b)(a-b)=a^2-b^2$
Let me explain first formula:
$(x+3)^2=x^2+6x+9\ Using the formula\\
(x+3)(x+3)=x^2+3x+3x+9=x^2+6x+9;Using FOIL
$
$(x-3)^2=x^2-6x+9\ Using the formula\\ (x-3)(x-3)=x^2-3x-3x+9=x^2-6x+9;Using FOIL$
Here is an example for second formula:
$(x-3)(x+3)=x^2-9=x^2-(3)^2; Using formula\\
(x-3)(x+3)=x^2+3x-3x-9=x^2-9=x^2-(3)^2$
You can see that we get the same result, so when you have these special cases you can use formulas as a schortcut.

Mathematics

Explain what a polynomials is and identify the different parts of a polynomial. explain the different labels used to categorize polynomials explain how addition and subtraction of polynomials is accomplished when multiplying polynomials, we are taking every term of one polynomial and multiplying them by every term of the second polynomial, then collecting like terms explain how foil us to accomplish this and what category of polynomials foil applies to. there are tow special case products where the product takes on special patterns explain these

Step-by-step answer

P Answered by PhD

3

A polynomial is an expression of more than one term. An expression is considered a polynomial when is has more than one term, otherwise, it would be called a monomial. These can be combined together through multiplication, addition and subtraction only. (Meaning no division or fractions)

Ex.

$3 x^{2} + 3x - 3$

x is a variable (There can be more than 1 variable in a term. Ex. 3xy, 4xyz, 4ab)

*A variable may be represented by letters.

2 is an exponent

3 is a constant

Those are the parts of a polynomial.

Polynomials can be categorized depending on the number of terms and their degree.

A polynomial with two terms is called a binomial. If it has three terms it is called a trinomial. If the expression has more than 3 terms, they are generally called polynomials.

A polynomial can be categorized by degree as well. You can determine the degree of a polynomial by looking at the term that has the highest exponent.

Using the example above, you can categorize the polynomial as a 2nd degree trinomial because 2 is the highest exponent and it has three terms.

When you add and subtract polynomials you need to take note of the variables. You can only subtract and add like terms, which means that the variables and the exponents are the same.

Ex.

$( 2x^{3} + 2 y^{2} + x + 1) + (4 x^{2} - y^{2} + y + 2)$

When you add these two polynomials, you can disregard the parentheses because according to the associative property of addition, no matter how you group the terms, the answer will be the same.

Like mentioned before you can only add and subtract like terms. It would be easier if you just group like terms together by rearranging the expression. Do not forget that the sign or operation comes along with them.

$2 x^{3} + 2 y^{2} - y^{2} + 4 x^{2} + x + y + 2 + 1$

Now combine the like terms.

$2 x^{3} + y^{2} + 4 x^{2} + x + y + 3$

Notice that we retained the terms $2 x^{3}$ , $4 x^{2}$ , x and y, this is because they have no similar terms.

FOIL method is used when multiplying 2 BINOMIALS. Remember that a binomial is an expression with 2 terms.

FOIL means:

FIRST term: first terms of each binomial.

OUTSIDE term: The two outer terms when taking the equation as a whole.

INSIDE term: The two inner terms when taking the equation as a whole.

LAST term: Last term of each binomial (2nd term of each binomial)

To get the answer, you need to multiply them with their corresponding term.

Ex. (2x+3)(x-4)

F: 2x and x (2x)(x) = $2x^{2}$

O: 2x and -4 (2x)(-4) = -8x

I: 3 and x (3)(x) = 3x

L: 3 and -4 (3)(-4) = -12

Resulting expression:

$2 x^{2} - 8x + 3x -12$ -8x and 3x are similar or like terms, so you can combine them

$2 x^{2} - 5x -12$

When doing multiplication with binomials, there are two special cases you can consider doing, which follow a pattern. The first is multiplying sum and difference.

The condition where you can apply the first special case is the first term needs to be the same and the second term are additive inverses.

(a+b)(a-b)

The resulting expression follows this pattern $a^{2} - b^{2}$

Ex. (x+3)(x-3) = $x^{2} - 3^{2}$ or $x^{2} -9$

You can use FOIL to check your

F: (x)(x) = $x^{2}$

O: (x)(-3) = -3x

I: (3)(x) =

L: (3)(-3) =

Arrange the expression:

$x^{2} - 3x + 3x - 9$ Combining -3x+3x = 0

$x^{2} -9$

The next special case is squaring a binomial and there are two scenarios that you can consider.

$(a+b)^{2}$ and $(a-b)^{2}$

The resulting expression follows a certain pattern for each:

$(a+b)^{2}$ = $a^{2} + 2ab + b^{2}$

$(a-b)^{2}$ = $a^{2} - 2ab + b^{2}$

Let's use an example of each to demonstrate this and check with FOIL:

$(a+b)^{2}$

$(2x+4)^{2}$

a = 2x b = +4

Let's insert that into our pattern:

$a^{2} + 2ab + b^{2}$

$2x^{2} + 2(2x)(4) + 4^{2}$

Simplify the expression:

$2x^{2} + 16x + 4^{2}$

$4x^{2} + 16x + 16$

Let's check with FOIL

$(2x+4)^{2}$ = (2x+4)(2x+4)

F: (2x)(2x) = $4x^{2}$

O: (2x)(4) =

I: (4)(x) =

L: (4)(4) =

Let's arrange the terms:

$4x^{2} + 8x + 8x + 16$ Combine the like terms
$4x^{2} + 16x + 16$ It's the same.

Now let's use the second scenario:

$(a-b)^{2}$
$(2x-4)^{2}$

a = 2x b = -4

Let's insert that into our pattern:

$a^{2} - 2ab + b^{2}$

$2x^{2} - 2(2x)(-4) + (-4)^{2}$

Simplify the expression:

$2x^{2} - 16x + (-4)^{2}$

$4x^{2} - 16x + 16$

Let's check with FOIL

$(2x+4)^{2}$ = (2x-4)(2x-4)

F: (2x)(2x) = $4x^{2}$

O: (2x)(-4) = -8x

I: (-4)(x) = -8x

L: (-4)(-4) =

Let's arrange the terms:

$4x^{2} - 8x - 8x + 16$ Combine the like terms
$4x^{2} - 16x + 16$ It's the same.

Mathematics

3x+12 classify the following polynomials by agree and number terms

Step-by-step answer

P Answered by PhD

2

The degree of the polynomial is 1 and the number of terms is 2.

Step-by-step explanation:

The degree of the polynomial 1 and the number of terms is 2.

The degree of a term is the sum of the exponents of the variable factors.

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

For example:

$3x+12\\$

The degree is 1 and there are two terms , and +12

Why degree of 1, there is no exponent though?

In math there is an invisible variable and exponent. Pretend x is 3.

$3x=3(3)=9\\\\3x^1=3(3)^1=3(3)=9$

Both expressions above have the same value.

12 can be written as 12x^0 .

You need to solve exponents first so any number multiplied by the power of 0 is 1. x*0 is 1. Therefore, 1 times 12 is still 12.

$12=12\\12x^0=12(3)^0=12(1)=12$

Mathematics

3x+12 classify the following polynomials by agree and number terms

Step-by-step answer

P Answered by PhD

2

The degree of the polynomial is 1 and the number of terms is 2.

Step-by-step explanation:

The degree of the polynomial 1 and the number of terms is 2.

The degree of a term is the sum of the exponents of the variable factors.

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

For example:

$3x+12\\$

The degree is 1 and there are two terms , and +12

Why degree of 1, there is no exponent though?

In math there is an invisible variable and exponent. Pretend x is 3.

$3x=3(3)=9\\\\3x^1=3(3)^1=3(3)=9$

Both expressions above have the same value.

12 can be written as 12x^0 .

You need to solve exponents first so any number multiplied by the power of 0 is 1. x*0 is 1. Therefore, 1 times 12 is still 12.

$12=12\\12x^0=12(3)^0=12(1)=12$

Mathematics

Which of the following represents the polynomial 2x − 5x3 − 10x5 + 9 in standard form and identifies the degree of the polynomial and the number of terms? 2x + 9 − 5x3 − 10x5 degree: 9 number of terms: 3 9 + 2x − 10x5 − 5x3 degree: 9 number of terms: 3 −10x5 − 5x3 + 2x + 9 degree: 4 number of terms: 5 −10x5 − 5x3 + 2x + 9 degree: 5 number of terms:4

Step-by-step answer

P Answered by Specialist

$-10x^{5} -5x^{3} +2x+9$ is the standard form

5 is the degree

Step-by-step explanation:

Your correct equation is $2x - 5x^{3} -10x^{5} +9$

Mathematics

Which of the following represents the polynomial 2x − 5x3 − 10x5 + 9 in standard form and identifies the degree of the polynomial and the number of terms? 2x + 9 − 5x3 − 10x5 degree: 9 number of terms: 3 9 + 2x − 10x5 − 5x3 degree: 9 number of terms: 3 −10x5 − 5x3 + 2x + 9 degree: 4 number of terms: 5 −10x5 − 5x3 + 2x + 9 degree: 5 number of terms:4

Step-by-step answer

P Answered by Specialist

$-10x^{5} -5x^{3} +2x+9$ is the standard form

5 is the degree

Step-by-step explanation:

Your correct equation is $2x - 5x^{3} -10x^{5} +9$

Mathematics

Match the vocabulary terms to the correct definitions. 1. a two-term polynomial a.monomial 2. a one-term polynomial b.binomial 3. a term or a sum of terms whose variables have whole number exponents c. trinomial 4. a three-term polynomial d.polynomial

Step-by-step answer

P Answered by Master

19

Two term polynomial is a binomial

One term polynomial is a monomial

A term or a sum of terms whose variables have whole number exponents is a polynomial

Three term polynomial is a trinomial

Mathematics

Match the vocabulary terms to the correct definitions. 1. a two-term polynomial a.monomial 2. a one-term polynomial b.binomial 3. a term or a sum of terms whose variables have whole number exponents c. trinomial 4. a three-term polynomial d.polynomial

Step-by-step answer

P Answered by Specialist

19

Two term polynomial is a binomial

One term polynomial is a monomial

A term or a sum of terms whose variables have whole number exponents is a polynomial

Three term polynomial is a trinomial

Classify the following polynomials by the number of terms and degrees
-2x^3 +5x

Step-by-step answer

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