OPERATIONS WITH POLYNOMIALS

About "Operations with polynomials"

Operations with polynomials : 

polynomial is an expression composed of coefficients and variables under addition, subtraction and multiplication and exponents on those variables must be non-negative integers.

Before going to learn how to add, subtract, multiply or divide polynomial we must know the terms 

  • Coefficients
  • like terms

What is coefficients ?

A numerical or constant quantity placed before and multiplying the variable in an algebraic expression.

For example, in -4x the coefficient of x is -4.

What are like terms ?

Like terms are terms that contain the same variables raised to the same power. Only the numerical coefficients are different. In an expression, only like terms can be combined.

For example, 5a, -3a, 2ab

in the above three terms 5a  and -3a are like terms, because they are having same variable, but 2ab is not like term of 5a or -3a, because it has a different variable "ab" other than "a".

Operations with polynomials :

  • Adding polynomials
  • Subtracting polynomials
  • Multiplying polynomials
  • Dividing polynomials

Let us discuss the above topics one by one with detailed examples.Operations with polynomials

Adding polynomials

Adding polynomials is nothing but combining the like terms.We follow two different methods to add two or more polynomials 

  • Horizontal method
  • Vertical method

In both ways we will get the same answer.

Adding polynomials horizontal method

Example 1 :

Add : (3 x³- 5 x²+ 2 x - 7) and (4 x² + x - 8)

Solution :

Here we give step by step explanation for adding the above two polynomials using horizontal method.

Step 1 :

Before going to add two polynomials, first we have to arrange the given polynomials one by one from highest power to lowest power.

(3x³-5x²+ 2x-7) and (4x²+x-8)

The two given polynomials are already in the arranged form.So we can leave it as it is.

Step 2 :

Now we have to write the like terms together starting from the highest power to lowest power.

     = 3 x³-5 x²+ 4 x²+ 2 x + x  - 7 - 8

Step 3:

Combine the like terms (Add or subtract) based on the signs of those terms.

     = 3 x³- x²+ 3 x - 15

Adding polynomials Vertical method

We have to line up the polynomials from highest power to lowest power one by one. If we have any missing term in the given polynomials then we have to put 0 instead of the missing term.

Subtracting polynomials

Subtracting polynomials is also nothing but combining the like terms.We follow two different methods to subtract two polynomials 

  • Horizontal method
  • Vertical method

In both ways we will get the same answer.

Example 2 :

Subtract the following polynomials:

(2 x³ - 2 x² + 4 x - 3)- (x³ + x² - 5 x + 4)

Solution :

Step 1:

In the first step we are going to multiply the negative with inner terms.

                    = 2 x³ -2 x² + 4 x - 3 - x³-x²+ 5 x - 4 

Step 2:

In the second step we have to combine the like terms 

                   = 2 x³ - x³ - 2 x²- x² + 4 x + 5 x - 3 - 4

Step 3:

After combining the like terms we will get the answer

                  = x³ - 3 x² + 9x - 7

Subtracting polynomials in vertical method :

We have to write the polynomials from highest power to lowest power one by one. If we have any missing term then we have to put 0 instead of that.

Change the signs of second polynomial from positive to negative and negative to positive. Hereafter we should not consider the original sign, we have to consider the changed sign.

According to the signs we have to combine the terms.

For more examples please visit "Adding and subtracting polynomials".

Multiplying polynomials

We will multiply two or more polynomials in the following order.

(1) Symbol

(2) Number

(3) Variable

Let us see how it works

Multiply ( 5 x² ) and (-2 x³)

            =  ( 5 x² )  x (-2 x³)

        =  10 x

Example 3 :

Multiply (2 a² + 5 a - 1) x (8 a² - 3 a + 5)

Solution :

         = (2 a² + 5 a - 1) x (8 a² - 3 a + 5)

To multiply these trinomials we have to distribute 2 a² with  (8 a² - 3 a + 5), distribute 5 a with  (8 a² - 3 a + 5) and -1 with  (8 a² - 3 a + 5).

 = 16a⁴-6 a³+10 a²+40 a³-15 a²+25 a-8 a²+3 a-5

Now we have to combine the like terms

         = 16a⁴-6a³+40a³+10a²-15a²-8a²+25a+3a-5

         = 16a⁴ + 34 a³ -13 a²+ 28 a - 5

For more examples please visit "Multiplying polynomials".

Dividing polynomials

The division of polynomials can be expressed by division algorithm.

Example 4 :

Find the quotient and remainder when 10 - 4x + 3x² is divided by x - 2

Solution :

To divide the given polynomial by other polynomial, first we have to write each polynomial is descending order.

Let f(x) = 3x² - 4x + 10 and g(x) = x - 2

Quotient = 3x + 2

Remainder = 14

After having gone through the stuff given above, we hope that the students would have understood "Operations with polynomials". 

Apart from the stuff given above, if you want to know more about "Operations with polynomials", please click here

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

ALGEBRA

Variables and constants

Writing and evaluating expressions

Solving linear equations using elimination method

Solving linear equations using substitution method

Solving linear equations using cross multiplication method

Solving one step equations

Solving quadratic equations by factoring

Solving quadratic equations by quadratic formula

Solving quadratic equations by completing square

Nature of the roots of a quadratic equations

Sum and product of the roots of a quadratic equations 

Algebraic identities

Solving absolute value equations 

Solving Absolute value inequalities

Graphing absolute value equations  

Combining like terms

Square root of polynomials 

HCF and LCM 

Remainder theorem

Synthetic division

Logarithmic problems

Simplifying radical expression

Comparing surds

Simplifying logarithmic expressions

Negative exponents rules

Scientific notations

Exponents and power

COMPETITIVE EXAMS

Quantitative aptitude

Multiplication tricks

APTITUDE TESTS ONLINE

Aptitude test online

ACT MATH ONLINE TEST

Test - I

Test - II

TRANSFORMATIONS OF FUNCTIONS

Horizontal translation

Vertical translation

Reflection through x -axis

Reflection through y -axis

Horizontal expansion and compression

Vertical  expansion and compression

Rotation transformation

Geometry transformation

Translation transformation

Dilation transformation matrix

Transformations using matrices

ORDER OF OPERATIONS

BODMAS Rule

PEMDAS Rule

WORKSHEETS

Converting customary units worksheet

Converting metric units worksheet

Decimal representation worksheets

Double facts worksheets

Missing addend worksheets

Mensuration worksheets

Geometry worksheets

Comparing  rates worksheet

Customary units worksheet

Metric units worksheet

Complementary and supplementary worksheet

Complementary and supplementary word problems worksheet

Area and perimeter worksheets

Sum of the angles in a triangle is 180 degree worksheet

Types of angles worksheet

Properties of parallelogram worksheet

Proving triangle congruence worksheet

Special line segments in triangles worksheet

Proving trigonometric identities worksheet

Properties of triangle worksheet

Estimating percent worksheets

Quadratic equations word problems worksheet

Integers and absolute value worksheets

Decimal place value worksheets

Distributive property of multiplication worksheet - I

Distributive property of multiplication worksheet - II

Writing and evaluating expressions worksheet

Nature of the roots of a quadratic equation worksheets

Determine if the relationship is proportional worksheet

TRIGONOMETRY

SOHCAHTOA

Trigonometric ratio table

Problems on trigonometric ratios

Trigonometric ratios of some specific angles

ASTC formula

All silver tea cups

All students take calculus 

All sin tan cos rule

Trigonometric ratios of some negative angles

Trigonometric ratios of 90 degree minus theta

Trigonometric ratios of 90 degree plus theta

Trigonometric ratios of 180 degree plus theta

Trigonometric ratios of 180 degree minus theta

Trigonometric ratios of 180 degree plus theta

Trigonometric ratios of 270 degree minus theta

Trigonometric ratios of 270 degree plus theta

Trigonometric ratios of angles greater than or equal to 360 degree

Trigonometric ratios of complementary angles

Trigonometric ratios of supplementary angles 

Trigonometric identities 

Problems on trigonometric identities 

Trigonometry heights and distances

Domain and range of trigonometric functions 

Domain and range of inverse  trigonometric functions

Solving word problems in trigonometry

Pythagorean theorem

MENSURATION

Mensuration formulas

Area and perimeter

Volume

GEOMETRY

Types of angles 

Types of triangles

Properties of triangle

Sum of the angle in a triangle is 180 degree

Properties of parallelogram

Construction of triangles - I 

Construction of triangles - II

Construction of triangles - III

Construction of angles - I 

Construction of angles - II

Construction angle bisector

Construction of perpendicular

Construction of perpendicular bisector

Geometry dictionary

Geometry questions 

Angle bisector theorem

Basic proportionality theorem

ANALYTICAL GEOMETRY

Analytical geometry formulas

Distance between two points

Different forms equations of straight lines

Point of intersection

Slope of the line 

Perpendicular distance

Midpoint

Area of triangle

Area of quadrilateral

Parabola

CALCULATORS

Matrix Calculators

Analytical geometry calculators

Statistics calculators

Mensuration calculators

Algebra calculators

Chemistry periodic calculator

MATH FOR KIDS

Missing addend 

Double facts 

Doubles word problems

LIFE MATHEMATICS

Direct proportion and inverse proportion

Constant of proportionality 

Unitary method direct variation

Unitary method inverse variation

Unitary method time and work

SYMMETRY

Order of rotational symmetry

Order of rotational symmetry of a circle

Order of rotational symmetry of a square

Lines of symmetry

CONVERSIONS

Converting metric units

Converting customary units

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

Word problems on comparing rates

Converting customary units word problems 

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles 

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed 

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6