Polynomials Calculator

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Polynomials Addition
Polynomials Subtraction
Polynomials Multiplication
Polynomials Division
Check for Polynomial
Degree of Polynomial
Polynomial Equation Solver Calculator
Leading Term of a Polynomial Calculator
Ascending of a Polynomial
Descending of a Polynomial
Factoring Over Multivariable Polynomials
GCF of Polynomials
Factor Out GCF from Polynomials
Determining if Polynomial is Prime
Finding the LCM using GCF
Factoring Binomials as Sum or Difference of cube
Factoring Binomials as Difference of Squares
Polynomial Root Calculator
Expand using the Binomial Theorem
Factoring over the Complex Numbers
Remainder Theorem Calculator

Ex: Polynomial Addition of x^5+3x^5+1+x^6+x^3+x

The given Expression is x^5+3x^5+1+x^6+x^3+x

After grouping similar terms we get x^6 + x^5 + 3 x^5 + x^3 + x + 1

After combining similar terms we get x + x^3 + x^6 + x^5 + 3 x^5 + 1

x^5 has similar terms x^5,3 x^5

x^3 has similar terms x^3

x has similar terms x

x^6 has similar terms x^6

By adding all the similar terms we get x^6 + 4 x^5 + x^3 + x + 1

Ex: Polynomials Subtraction of (x^2+2)-(x+8)

The given Expression is (x^2+2)-(x+8)

After removing all the brackets this expression can be written as x^2+2-x - 8.

After grouping similar terms we get x^2 - x - 8 + 2

After combining similar terms we get x^2 - x - 8 + 2

x^2 has similar terms x^2

x has similar terms - x

By adding all the similar terms we get x^2 - x - 6

Ex: Polynomials Multiplication of (x+2)(x+4)(x+6)

(x + 4)(x + 6)(x + 2)

=((x)((x + 6))+(4)((x + 6)))(x + 2)

=((x)(x)+(4)(x)+(x)(6)+(4)(6))(x + 2)

=((x^2)((x + 2))+(10x)((x + 2))+(24)((x + 2)))

=((x^2)(x)+(10x)(x)+(24)(x)+(x^2)(2)+(10x)(2)+(24)(2))

=(x^3)+(10x^2)+(24x)+(2x^2)+(20x)+(48)

=x^3 + 12x^2 + 44x + 48

∴ (x + 4)(x + 6)(x + 2) = x^3 + 12x^2 + 44x + 48

Ex: Polynomial Division of (x^4+2x+8)/(x+6)

The given expression is (x^4+2x+8)/(x+6)

The Divident is x^4 + 2 x + 8 and Divisor is x + 6

x + 6)x^4 + 2 x + 8(x^3 - 6 x^2 + 36 x - 214
     - x^4 - 6 x^3

     -----------------
      - 6 x^3 + 2 x + 8

      6 x^3 + 36 x^2

      ------------------
       36 x^2 + 2 x + 8

       - 36 x^2 - 216 x

       ------------------
        8 - 214 x

        214 x + 1284

        ------------
         1292

After the division the quotient is x^3 - 6 x^2 + 36 x - 214 and reminder is 1292

Ex: Determining x^5+3x^5+1+x^6+x^3+x Expression is a Polynomial

The expression can be written as x^6 + 4 x^5 + x^3 + x + 1

polynomial is a combination of terms separated using + or − signs. Polynomials cannot contain any of the following:

i)Variables raised to a negative or fractional exponent.

i)Variables in the denominator.

iii)Variables under a radical.

iv)Special features. (trig functions, absolute values, logarithms, … ).

x^5+3x^5+1+x^6+x^3+x is a polynomial.

Ex 1: Finding Polynomial x^5+x^5+1+x^5+x^3+x in Ascending Order

The Given Polynomial is x^5+x^5+1+x^5+x^3+x
The ascending Order of polynimial is 1+x+x^3+3 x^5

Ex 2: Find the Polynomial x^5+3x^5+1+x^6+x^3+x in Ascending Order

The Given Polynomial is x^5+3x^5+1+x^6+x^3+x
The ascending Order of polynimial is 1+x+x^3+4 x^5+x^6

Ex 3: Find the Polynomial x^3+x^5+1+x^3+x^3+x in Ascending Order

The Given Polynomial is x^3+x^5+1+x^3+x^3+x
The ascending Order of polynimial is 1+x+3 x^3+x^5

Ex 1: Determining Polynomial of Descending Order of x^5+x^5+1+x^5+x^3+x

The Given Polynomial is x^5+x^5+1+x^5+x^3+x
The descending Order of polynimial is 3 x^5 + x^3 + x + 1

Ex 2: Determining Polynomial of Descending Order of x^5+3x^5+1+x^6+x^3

The Given Polynomial is x^5+3x^5+1+x^6+x^3
The descending Order of polynimial is x^6 + 4 x^5 + x^3 + 1

Ex 3: Determining Polynomial of Descending Order of x^3+x^5+1+x^3+x^3+x

The Given Polynomial is x^3+x^5+1+x^3+x^3+x
The descending Order of polynimial is x^5 + 3 x^3 + x + 1

Ex 1: Degree of a Polynomial x^3+x^5+1+x^3+x^3+x

The given expression is x^3+x^5+1+x^3+x^3+x

• The degree of x is 1
• The degree of x^3 is 3
• The degree of x^3 is 3
• The degree of x^5 is 5
• The degree of x^3 is 3
• The degree of 1 is 0

But the degree of expression will the highest degree of the indivisual expression of above i.e 5

Ex 2: Degree of a Polynomial x^5+3x^5+1+x^6+x^3+x

The given expression is x^5+3x^5+1+x^6+x^3+x

• The degree of x is 1
• The degree of x^3 is 3
• The degree of x^6 is 6
• The degree of x^5 is 5
• The degree of 3 x^5 is 5
• The degree of 1 is 0

But the degree of expression will the highest degree of the indivisual expression of above i.e 6

Ex 1: Determining the Leading Term of a Polynomial x^5+3x^5+1+x^6+x^3+x

The given input is x^5+3x^5+1+x^6+x^3+x

The term can be simplified as x^6 + 4 x^5 + x^3 + x + 1

-- 1 term has degree 0 .

-- x term has degree 1 .

-- x^3 term has degree 3 .

-- x^6 term has degree 6 .

-- 4 x^5 term has degree 5 .

--Here highest degree is maximum of all degrees of terms i.e 6 .

--Hence the leading term of the polynomial will be the terms having highest degree i.e x^6 .

--x^6 has coefficient 1 .

Ex 2: Determining the Leading Term of a Polynomial xx^3+x^5+1+x^3+x^3+x

The given input is x^3+x^5+1+x^3+x^3+x

The term can be simplified as x^5 + 3 x^3 + x + 1

-- 1 term has degree 0 .

-- x term has degree 1 .

-- x^5 term has degree 5 .

-- 3 x^3 term has degree 3 .

--Here highest degree is maximum of all degrees of terms i.e 5 .

--Hence the leading term of the polynomial will be the terms having highest degree i.e x^5 .

--x^5 has coefficient 1 .

Ex 1: How to Find Factoring Multi Variable Polynomials for a^2-b^2?

The given polynomial is a^2-b^2

The polynomial can be written as a^2 - b^2

=(a)(a)+(b)(a)+(a)(-b)+(b)(-b)

=(a)((a - b))+(b)((a - b))

=(a - b) (a + b)

Ex 2: How to Find Factoring Multi Variable Polynomials for a^3-b^3?

The given polynomial is a^3-b^3

The polynomial can be written as a^3 - b^3

=(a)(a^2)+(-b)(a^2)+(a)(ab)+(-b)(ab)+(a)(b^2)+(-b)(b^2)

=(a)((a2 + ab + b2))+(-b)((a2 + ab + b2))

=(a - b) (a^2 + a b + b^2)

Ex 3: How to Find Factoring Multi Variable Polynomials for abc+8ab+ac+8a+bc+8b+c+8?

The given polynomial is abc+8ab+ac+8a+bc+8b+c+8

The polynomial can be written as a b c + 8 a b + a c + 8 a + b c + 8 b + c + 8

=((ab)((c + 8))+(a)((c + 8))+(b)((c + 8))+(1)((c + 8)))

=((ab)((c + 8))+(a)((c + 8))+(b)((c + 8))+(1)((c + 8)))

=((a)(b)+(1)(b)+(a)(1)+(1)(1))(c + 8)

=((a)((b + 1))+(1)((b + 1)))(c + 8)

=(a + 1) (b + 1) (c + 8)

Ex 1: Find the GCF of Polynomials x^2+2x+1,x+1

The given input is x^2+2x+1,x+1

x^2+2x+1 has factors i.e (x + 1)^2

x+1 has factors i.e x + 1

By verifying each polynomial factor we get the GCF i.e common factor of the polynomial is x + 1 and simplified as x + 1

Ex 2: Find the GCF of Polynomials x^2-1,x-1

The given input is x^2-1,x-1

x^2-1 has factors i.e (x - 1) (x + 1)

x-1 has factors i.e x - 1

By verifying each polynomial factor we get the GCF i.e common factor of the polynomial is x - 1 and simplified as x - 1

Ex 3: Find the GCF of Polynomials x^3-1,x+1

The given input is x^3-1,x+1

x^3-1 has factors i.e (x - 1) (x^2 + x + 1)

x+1 has factors i.e x + 1

By verifying each polynomial factor we get the GCF i.e common factor of the polynomial is 1 and simplified as 1

Ex 1: Find the Factor out GCF of Polynomials x^2+2x+1,x+1

The given input is x^2+2x+1,x+1

x^2+2x+1 has factors i.e (x + 1)^2

x+1 has factors i.e x + 1

By verifying each polynomial factor we get the GCF i.e common factor of the polynomial is x + 1 and simplified as x + 1

Factor form of GCF is x + 1

Ex 2: Find the Factor out GCF of Polynomials x^2-1,x-1

The given input is x^2-1,x-1

x^2-1 has factors i.e (x - 1) (x + 1)

x-1 has factors i.e x - 1

By verifying each polynomial factor we get the GCF i.e common factor of the polynomial is x - 1 and simplified as x - 1

Factor form of GCF is x - 1

Ex 3: Find the Factor out GCF of Polynomials x^3-1,x+1

The given input is x^3-1,x+1

x^3-1 has factors i.e (x - 1) (x^2 + x + 1)

x+1 has factors i.e x + 1

By verifying each polynomial factor we get the GCF i.e common factor of the polynomial is 1 and simplified as 1

Factor form of GCF is 1

Ex 1: Finding Is x^5+x^5+1+x^5+x^3+x Prime Polynomial

The given polynomial is x^5 + x^5 + x^5 + x^3 + x + 1

So x^5 + x^5 + x^5 + x^3 + x + 1 = 3 x^5 + x^3 + x + 1

It is a prime polynomial because it has only two factors i.e 1 and 3 x^5 + x^3 + x + 1

Ex 2: Finding Is x^5+3x^5+1+x^6+x^3+x Prime Polynomial

The given polynomial is x^6 + x^5 + 3 x^5 + x^3 + x + 1

So x^6 + x^5 + 3 x^5 + x^3 + x + 1 = x^6 + 4 x^5 + x^3 + x + 1

It is a prime polynomial because it has only two factors i.e 1 and x^6 + 4 x^5 + x^3 + x + 1

Ex 3: Finding Is x^3+x^5+1+x^3+x^3+x Prime Polynomial

The given polynomial is x^5 + x^3 + x^3 + x^3 + x + 1

So x^5 + x^3 + x^3 + x^3 + x + 1 = x^5 + 3 x^3 + x + 1

It is a prime polynomial because it has only two factors i.e 1 and x^5 + 3 x^3 + x + 1

Ex 1: Finding LCM of Polynomials x^2+2x+1, x+1 Using GCF

The given Expressions are x^2+2x+1,x+1

x^2+2x+1 has factors i.e (x + 1)^2

x+1 has factors i.e x + 1

By finding the GCF of given expressions we get that the gcf is x + 1

There are 2 number of expressions are given.

To find the LCM we have to first multiply all the expressions (x^2+2x+1)(x+1) = x^3 + 3x^2 + 3x + 1

To find the LCM we have devide 2-1 power of gcf from x^3 + 3x^2 + 3x + 1

So by dividing x + 1 from x^3 + 3x^2 + 3x + 1 = (x^3 + 3x^2 + 3x + 1)/(x + 1) = x^2 + 2x + 1

So the LCM of x^2+2x+1,x+1 is x^2 + 2x + 1

Ex 2: Finding LCM of Polynomials x^2-1, x-1 Using GCF

The given Expressions are x^2-1,x-1

x^2-1 has factors i.e (x - 1)(x + 1)

x-1 has factors i.e x - 1

By finding the GCF of given expressions we get that the gcf is x - 1

There are 2 number of expressions are given.

To find the LCM we have to first multiply all the expressions (x^2-1)(x-1) = x^3 - x^2 - x + 1

To find the LCM we have devide 2-1 power of gcf from x^3 - x^2 - x + 1

So by dividing x - 1 from x^3 - x^2 - x + 1 = (x^3 - x^2 - x + 1)/(x - 1) = x^2 - 1

So the LCM of x^2-1,x-1 is x^2 - 1

Ex 3: Finding LCM of Polynomials x^3-1, x+1 Using GCF

The given Expressions are x^3-1,x+1

x^3-1 has factors i.e (x - 1)(x^2 + x + 1)

x+1 has factors i.e x + 1

By finding the GCF of given expressions we get that the gcf is 1

There are 2 number of expressions are given.

To find the LCM we have to first multiply all the expressions (x^3-1)(x+1) = x^4 + x^3 - x - 1

To find the LCM we have devide 2-1 power of gcf from x^4 + x^3 - x - 1

So by dividing 1 from x^4 + x^3 - x - 1 = (x^4 + x^3 - x - 1)/(1) = x^4 + x^3 - x - 1

So the LCM of x^3-1,x+1 is x^4 + x^3 - x - 1