POLYNOMIALS in

ACTIONby Lorence G. Villaceran

Ateneo de Zamboanga University

What is aPolynomial?

Polynomials in Action

A Polynomial is

• Is an algebraic expression which consist more than one summed term

• Is a finite sum of terms each of which is a real number or the product of a numerical factor and one or more factor raised to whole-number powers

• each part that is being added, is called a "term"

Polynomials in Action

An expression is not a Polynomial if

• It has a negative exponent

• It has a fractional exponent

• It has a variable in the denominator

• It has a variable inside the square root sign

Polynomials in Action

6x2

Determine the ff. if it is a polynomial or not

√x

Polynomial

1/x2 not a Polynomial

not a Polynomial

Polynomials in Action

4y6/3

9y3

Z-4

√x2

Polynomial

Polynomial

not a Polynomial

Polynomial

Polynomials in Action

TERM• It composes the polynomial

• It composes of a numerical, literal coefficient and exponent

Parts of a TERM

Numerical Coefficient

Literal Coefficient/Variable

6x2 Exponent/Degree

Polynomials in Action

Similar Terms• Terms that have the same degree

or exponent of the same variable

x2+xy-y2 2x2+3xy-2y2

Similar Term

Polynomials in Action

Types of PolynomialsMonomial

• If a polynomial contains only one term.

Binomial• If a polynomial contains two terms.

Trinomial• If a polynomial contains three terms.

Multinomial• If a polynomial contains more than three terms.

Polynomials in Action

Examples6x2

9y3+3y+4

x2+3x

x3+y-x+3

Binomial

Monomial

Trinomial

Multinomial

Polynomials in Action

Polynomials in Action

x3+x2y+3y3

x3y+wxy x3yz2

x3+x2y2+xy-y3

w3+wxy+x2z

Monomial

Multinomial

Binomial

Trinomial

Trinomial

Polynomials in Action

Four Fundamental Operations in Polynomial

Addition and Subtraction of Polynomials

Polynomials in Action

How to add polynomials in column form

• Arrange the polynomials in either descending or ascending order of the variable/s and place similar terms in same vertical column

• For addition, the similar terms by finding the sum of coefficients

• Apply rules in adding signed numbers and retain the common literal factor

Polynomials in Action

Add the following polynomials:

4x3+8x2-x-8; x2+6x+9; 9x3+5x-9

Example of adding polynomials in column form

4x3+8x2-x-8 x2+6x+9

9x3+ 5x-913x3+9x2+10x-8

Polynomials in Action

How to subtract polynomials in Column form

• Arrange the polynomials in either descending or ascending order of the variable/s and place similar terms in same vertical column

• For subtraction, set the subtrahend under the minuend so that similar terms fall in the same column

• Subtract the numerical coefficients of similar terms.

• Use the rule for subtraction for signed numbers and retain the common literal factor

Polynomials in Action

Subtract the following polynomials:

10y4-4y3-y2+y+20; 15y4-4y2-3y+7

Example of subtracting polynomials in Column form

10y4 - 4y3 - y2 + y + 20

15y4 4y2 3y 7- + + -+- -

- 5y4 – 4y3+ 3y2 + 4y + 13

Polynomials in Action

Multiplication of Polynomials

Polynomials in Action

Rules of ExponentLet a and b be the numerical coefficient or the literal coefficient and m, n and p be the exponent.

1. am x an  =  a(m+n)

2. (am)n  =  a(mxn)

3. (ab)m  =  ambm

4. (ambn)p =  a(m x p)b(mxp)

5. am/an  =  a(m-n)

Polynomials in Action

Rules of ExponentLet a and b be the numerical coefficient or the

literal coefficient and m, n and p be the exponent.

6. a0   =   1

7. a1   =   a   

8. a-m   =   1/am or 1/am = am/1 = am

9. am + am  =  2am 

10. am + an  =  am + an  

Polynomials in Action

Example

am x an  =  a(m+n)

a3 x a2 =  a3+2  =  a5

(34)(37)  =  311

(23a4)(25a6)  =  28a10

(am)n  =  a(mxn)

(a5)3 = a5x3 = a15

(42)3 = 46

[(x2)2]2 = x2x2x2 = x8

Polynomials in Action

Example(ab)m  =  ambm

(ab)4   =  a4b4

(5x)3  =  53x3

(4xy)5  =  45x5y5

(ambn)p =  a(mxp)b(mxp)

(a2b3)4 = a2x4b3x4 = a8b12

(43x7y4)5 = 415x35y20

Polynomials in Action

Example

am/an  =  a(m-n)

a5/a3 =  a5-3  =  a2

a7/a10 = a7-10 = a3 or 1/a3

a3b8c12/a5b8c7 = a-2c5

a0   =   1

® 80 =  1

® 5a0  =  5(1)  =  5

Polynomials in Action

Example

a1   =   a   

⌂ 81  =  8  

⌂ 5a1  =  5(a)  =  5a

a-m = 1/am or 1/am = am/1 = am

∂ a-3 =  1/a3

∂ a-5/b-2 =  b2/a5

∂ 6a-2b5/7c- 6d3 = 6b5c6/7a2d3 

Polynomials in Action

Exampleam + am  =  2am 

o a3 + a3  =  2a3 

o 5a4 + 2a4  =  7a4  

o 7a6 - 4a6  =  3a6

am + an  =  am + an 

• a6 + a4 = a6 + a4

• 6a4 + 3a2 - 8a3 = 6a4 + 3a2 - 8a3  

Polynomials in Action

Rules for multiplication of monomials

• Multiplying the coefficients by following the rule for multiplication of signed numbers to get the coefficient of the product

• Multiply the literal coefficients by following the laws of exponents to obtain the literal coefficient of the product

Polynomials in Action

Example of multiplying monomial by a monomial

Simplify (5x2)(–2x3)

Polynomials in Action

(5x2)(–2x3) = (5)(-2)(x2+3) = -10x5

Simplify (-3y5)(–9y0)

(-3y5)(–9y0) = (-3)(-9)(y5+0) = 27y5

Rules for multiplication of a polynomial by a monomials

• Apply the distributive property of multiplication over addition or subtraction

Polynomials in Action

Example of Multiplying monomial by a polynomial

Multiply 3x2 and 12x3-4x2

Polynomials in Action

= 3x2(12x3-4x2)

= 3x2(12x3) - 3x2(4x2)

= 3(12)(x2+3) - 3(4)(x2+2)

= 36x5-12x4

Example of Multiplying monomial by a polynomial

Multiply 7y4 and 5y4-9y3+8

Polynomials in Action

= 7y4(5y4-9y3+8)

= 7y4(5y4)-7y4(9y3)+7y4(8)

= 7(5)(y4+4)-7(9)(y4+3)+7(8)(y4)

= 35y8-63y7+56y4

• Take one term of the multiplier at a time and multiply the multiplicand

• Combine similar terms to get the required product

• Arrange the terms in descending order

Rules for multiplication of a polynomial by another polynomial

Polynomials in Action

Example of Multiplying polynomial by a polynomial

Multiply (3x+5) and (3x-4)

Polynomials in Action

= (3x)(3x)+(3x)(-4)+(5)(3x)+(5)(-4)

= 3(3)(x1+1)+(3)(-4)(x)+(5)(3)(x)+(5)(-4)

=9x2-12x+15x-20

=9x2+3x-20

Example of Multiplying polynomial by a polynomial

Multiply(2x2+3x+5) and (x2-2x-3)

Polynomials in Action

2x2+ 3x+ 5                      x2- 2x- 3

                      -6x2- 9x-15              -2x3-6x2-10x

2x4+3x3+5x2

   2x4+ x3 -7x2 -19x-15