The Polynomial X3 + 8 is Equal to

The Polynomial X3 + 8 is Equal to


Polynomials
are algebraic expressions that consist of variables and coefficients. Variables are also sometimes called indeterminates. We can perform arithmetic operations such every bit addition, subtraction, multiplication and also positive integer exponents for polynomial expressions but non segmentation by variable. An example of a polynomial with one variable is x2

+x-12. In this example, in that location are three terms: x

2

, x and -12.

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The word polynomial is derived from the Greek words ‘poly’ means ‘many‘ and ‘nominal’ ways ‘terms‘, so birthday it said “many terms”. A polynomial tin accept any number of terms but not infinite. Learn about degree, terms, types, backdrop, polynomial functions in this article.

Table of Contents:

  • Definition
  • Note
  • Caste
  • Terms
  • Types
    • Monomial
    • Binomial
    • Trinomial
  • Backdrop
  • Equations
  • Function
  • Solving Polynomials
    • Linear Polynomial
    • Quadratic Polynomial
  • Operations
    • Add-on
    • Subtraction
    • Multiplication
    • Division
  • Examples
  • FAQs

What is a Polynomial?

Polynomial is fabricated upwardly of two terms, namely Poly (significant “many”) and Nominal (meaning “terms.”). A polynomial is defined equally an expression which is composed of variables, constants and exponents, that are combined using the mathematical operations such every bit addition, subtraction, multiplication and sectionalization (No division performance by a variable). Based on the numbers of terms present in the expression, information technology is classified as monomial, binomial, and trinomial. Examples of constants, variables and exponents are as follows:

  • Constants. Example: one, 2, iii, etc.
  • Variables. Example: g, h, 10, y, etc.
  • Exponents: Example: 5 in ten5
    etc.

Notation

The polynomial function is denoted by P(x) where ten represents the variable. For example,

P(x) = x

2

-5x+11

If the variable is denoted by a, then the function volition be P(a)

Degree of a Polynomial

The degree of a polynomial is defined as the highest degree of a monomial within a polynomial. Thus, a polynomial equation having one variable which has the largest exponent is called a degree of the polynomial.

Polynomial Degree Example
Abiding or Cipher Polynomial 6
Linear Polynomial one 3x+1
Quadratic Polynomial 2 4xii+1x+1
Cubic Polynomial 3 6x3+4xthree+3x+i
Quartic Polynomial 4 6x4+3x3+3x2+2x+1

Example:
Find the degree of the polynomial 6s4+ 3x2+ 5x +19

Solution:

The degree of the polynomial is 4.

Terms of a Polynomial

The terms of polynomials are the parts of the equation which are generally separated by “+” or “-” signs. And then, each function of a polynomial in an equation is a term. For instance, in a polynomial, say, 2x2
+ 5 +iv, the number of terms will be 3. The nomenclature of a polynomial is done based on the number of terms in it.

Polynomial Terms Degree
P(x) = xthree

-2x

2

+3x+4
x3

, -2xtwo

, 3x and 4
3

Types of Polynomials

Polynomials are of 3 different types and are classified based on the number of terms in it. The three types of polynomials are:

  • Monomial
  • Binomial
  • Trinomial

These polynomials can be combined using improver, subtraction, multiplication, and division only is never division past a variable. A few examples of
Non Polynomials
are: 1/x+2, x-3

Monomial

A monomial is an expression which contains only i term. For an expression to be a monomial, the single term should be a non-zero term. A few examples of monomials are:

  • 5x
  • 3
  • 6afour
  • -3xy

Binomial

A binomial is a polynomial expression which contains exactly 2 terms. A binomial can be considered as a sum or difference between two or more monomials. A few examples of binomials are:

  • – 5x+3,
  • 6aiv
    + 17x
  • xy2+xy

Trinomial

A trinomial is an expression which is equanimous of exactly three terms. A few examples of trinomial expressions are:

  • – 8a4+2x+7
  • 4x2
    + 9x + 7

Properties

Some of the important properties of polynomials forth with some important polynomial theorems are equally follows:

Property i: Partitioning Algorithm

If a polynomial P(x) is divided by a polynomial G(x) results in quotient Q(ten) with remainder R(x), then,

P(ten) = G(x)



Q(ten) + R(x)

Holding two: Bezout’s Theorem

Polynomial P(10) is divisible by binomial (x – a) if and merely if
P(a) = 0.

Property 3: Remainder Theorem

If P(x) is divided by (x – a) with remainder r, then
P(a) = r.

Property four: Gene Theorem

A polynomial P(x) divided by Q(x) results in R(x) with zero remainders if and only if Q(x) is a factor of P(x).

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Cistron Theorem

Belongings 5: Intermediate Value Theorem

If P(x) is a polynomial, and P(10) ≠ P(y) for (x < y), then P(ten) takes every value from P(ten) to P(y) in the airtight interval [x, y].

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Intermediate Value Theorem

Property vi

The add-on, subtraction and multiplication of polynomials P and Q result in a polynomial where,

Degree(P ± Q) ≤ Caste(P or Q)

Degree(P × Q) = Degree(P) + Caste(Q)

Holding vii

If a polynomial P is divisible past a polynomial Q, then every zero of Q is also a nix of P.

Property 8

If a polynomial P is divisible by two coprime polynomials Q and R, then it is divisible by (Q • R).

Property 9

If P(x) = a
+ a110 + a2ten2
+ …… + antenn
is a polynomial such that deg(P) = n ≥ 0 then, P has at most “north” distinct roots.

Holding 10: Descartes’ Rule of Sign

The number of positive real zeroes in a polynomial part P(x) is the aforementioned or less than by an fifty-fifty number as the number of changes in the sign of the coefficients. So, if at that place are “K” sign changes, the number of roots will be “chiliad” or “(chiliad – a)”, where “a” is some even number.

Property 11: Fundamental Theorem of Algebra

Every non-abiding single-variable polynomial with circuitous coefficients has at least one complex root.

Belongings 12

If P(x) is a polynomial with real coefficients and has 1 complex zip (x = a – bi), so ten = a + bi will besides exist a zero of P(x). Also, 102
– 2ax + a2
+ b2
will be a cistron of P(10).

Polynomial Equations

The polynomial equations are those expressions which are fabricated upwardly of multiple constants and variables. The standard form of writing a polynomial equation is to put the highest degree first then, at last, the constant term. An case of a polynomial equation is:

b = a4
+3athree
-2aii
+a +one

Polynomial Functions

A polynomial role is an expression constructed with one or more terms of variables with constant exponents. If there are real numbers denoted by a, and so function with one variable and of caste n tin exist written equally:

f(x) = a10n
+ aixn-1
+ atwoxnorth-two
+ ….. + an-iix2
+ an-1x + an

Solving Polynomials

Any polynomial can be easily solved using bones algebra and factorization concepts. While solving the polynomial equation, the outset step is to ready the right-hand side equally 0. The explanation of a polynomial solution is explained in two dissimilar ways:

  • Solving Linear Polynomials
  • Solving Quadratic Polynomials

Solving Linear Polynomials

Getting the solution of linear polynomials is easy and uncomplicated. First, isolate the variable term and brand the equation equally equal to zero. Then solve as basic algebra operation. An example of finding the solution of a linear equation is given below:

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Example:
Solve 3x – 9

Solution:

First, make the equation as 0. So,

3x – 9 = 0

⇒ 3x = 9

⇒ ten = 9/3

Or, x = iii.

Thus, the solution of 3x-9 is x = three.

Solving Quadratic Polynomials

To solve a quadratic polynomial, first, rewrite the expression in the descending lodge of degree. Then, equate the equation and perform polynomial factorization to get the solution of the equation. An case to find the solution of a quadratic polynomial is given below for better agreement.

Example:
Solve 3xii
– 6x + x3
– xviii

Solution:

First, accommodate the polynomial in the descending lodge of degree and equate to zilch.

⇒ xiii
+ 3x2
-6x – xviii = 0

Now, accept the common terms.

x2(10+3) – 6(x+3) =0

⇒ (xtwo-6)(ten+3)=0

Then, the solutions volition be x =-three and

xii
= 6

Or, ten = √half-dozen

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Polynomial Operations

There are four principal polynomial operations which are:

  • Addition of Polynomials
  • Subtraction of Polynomials
  • Multiplication of Polynomials
  • Division of Polynomials

Each of the operations on polynomials is explained below using solved examples.

Addition of Polynomials

To add together polynomials, always add the like terms, i.eastward. the terms having the aforementioned variable and power. The addition of polynomials e’er results in a polynomial of the aforementioned caste. For case,

Example:
Notice the sum of two polynomials: 5x3+3x2y+4xy−6y2, 3x2+7xtwoy−2xy+4xyii−5

Solution:

First, combine the like terms while leaving the different terms as they are. Hence,

(5x3+3xiiy+4xy−6ytwo)+(3xii+7x2y−2xy+4xyii−five)

= 5xthree+3xtwo+(3+seven)teniiy+(4−2)xy+4xy2−6yii−five

= 5x3+3xii+10xtwoy+2xy+4xy2−6yii−5

Subtraction of Polynomials

Subtracting polynomials is similar to addition, the only difference being the type of operation. So, subtract the like terms to obtain the solution. It should exist noted that subtraction of polynomials also results in a polynomial of the same degree.

Example:
Find the divergence of 2 polynomials: 5x3+3x2y+4xy−6y2, 3x2+7xtwoy−2xy+4xy2−5

Solution:

First, combine the like terms while leaving the unlike terms as they are. Hence,

(5x3+3x2y+4xy−6yii)-(3xtwo+7x2y−2xy+4xy2−5)

= 5x3-3x2+(3-7)x2y+(iv+2)xy-4xy2−6y2+5

= 5x3-3x2-4x2y+6xy-4xy2−6y2+v

Multiplication of Polynomials

Two or more polynomial when multiplied ever result in a polynomial of higher degree (unless one of them is a constant polynomial). An example of multiplying polynomials is given below:

Case:
Solve (6x−3y)×(2x+5y)

Solution:

⇒ 6x ×(2x+5y)–3y × (2x+5y) ———- Using distributive police of multiplication

⇒ (12xtwo+30xy) – (6yx+15yii) ———- Using distributive law of multiplication

⇒12x2+30xy–6xy–15y2
—————– as xy = yx

Thus, (6x−3y)×(2x+5y)=12x2+24xy−15y2

Division of Polynomials

Segmentation of two polynomial may or may not consequence in a polynomial. Let usa study below the partition of polynomials in details. To divide polynomials, follow the given steps:

Polynomial Division Steps:

If a polynomial has more than than ane term, we utilize long division method for the same. Following are the steps for information technology.

  1. Write the polynomial in descending order.
  2. Check the highest power and divide the terms past the same.
  3. Employ the answer in stride 2 as the division symbol.
  4. At present subtract it and bring downward the next term.
  5. Echo step 2 to 4 until you lot have no more than terms to conduct down.
  6. Note the last respond, including remainder, volition exist in the fraction form (final subtract term).
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Polynomial Examples

Example:

Given 2 polynomial 7siii+2s2+3s+nine and 5sii+2s+i.

Solve these using mathematical operation.

Solution:

Given polynomial:

7sthree+2sii+3s+9 and 5s2+2s+1

Polynomial Addition:
(7s3+2s2+3s+9) + (5s2+2s+1)

= 7sthree+(2s2+5s2)+(3s+2s)+(9+1)

= 7s3+7s2+5s+ten

Hence, addition issue in a polynomial.

Polynomial Subtraction:
(7s3+2s2+3s+9) – (5s2+2s+1)

= 7s3+(2s2-5stwo)+(3s-2s)+(9-1)

= 7siii-3stwo+due south+8

Hence addition consequence in a polynomial.

Polynomial Multiplication:(7s3+2s2+3s+9) × (5s2+2s+ane)

= 7s3
(5s2+2s+1)+2s2
(5s2+2s+1)+3s (5s2+2s+1)+9 (5s2+2s+1))

= (35s5+14s4+7s3)+ (10sfour+4siii+2s2)+ (15s3+6stwo+3s)+(45stwo+18s+9)

= 35s5+(14s4+10s4)+(7s3+4s3+15s3)+ (2stwo+6s2+45s2)+ (3s+18s)+9

= 35s5+24s4+26s3+ 53s2+ 21s +9

Polynomial Division: (7sthree+2s2+3s+nine) ÷ (5s2+2s+1)

(7s3+2s2+3s+9)/(5stwo+2s+1)

This cannot be simplified. Therefore, partition of these polynomial do not event in a Polynomial.

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Frequently Asked Questions – FAQs

What is a Polynomial?

A polynomial is an expression that consists of variables (or indeterminate), terms, exponents and constants. For example, 3xii
-2x-10 is a polynomial.

What are terms, degrees and exponents in a polynomial?

If 2xii
– 3x +nineteen is a polynomial, and so;
Terms: 2x2,-3x & 19
Degree: 2 (the highest exponent of variable x)
Exponents: Ability raised to variable x, i.e. ii and 1.

What is the standard form of the polynomial?

A standard polynomial is the one where the highest degree is the first term, and subsequently, the other terms come. For case, x3
– 3x2
+ x -12 is a standard polynomial. And then the highest degree here is 3, then comes ii and so 1.

What are the types of polynomials?

In general, at that place are 3 types of polynomials. They are Monomial, Binomial and Trinomial.
Monomial: Information technology is an expression that has one term. Ex: x, y, z, 23, etc.
Binomial: It is an expression that has ii terms. Ex: 2x+y, ten2
– x, etc.
Trinomial: It is an expression that has iii terms. Ex: x3
– 3x + 10.
A polynomial tin can have any number of terms but not infinite.

Is viii a polynomial?

8 can be written as 8x
or 0x2+0x+viii, which represents the polynomial expression. Therefore, we can consider 8 every bit a polynomial.

How to add and decrease polynomials?

To add together polynomials, e’er add the like terms, i.e. the terms having the same variable and power. The addition of polynomials always results in a polynomial of the same caste.
For example if we add x2+3x and 2x2
+ 2x + 9, then we get:
ten2+3x+2x2+2x+9 = 3xii+5x+9. Subtracting polynomials is similar to addition, the only departure beingness the blazon of operation. So, subtract the similar terms to obtain the solution. It should be noted that subtraction of polynomials likewise results in a polynomial of the same degree.
And then,
x2+3x-(2xii+2x+nine) = ten2+3x-2x2-2x-9 = -xii+x-9

The Polynomial X3 + 8 is Equal to

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