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Understanding Polynomials: Definitions, Types, and Operations

This resource provides a comprehensive overview of polynomials, including their definitions, classifications (monomial, binomial, trinomial, multinomial), and fundamental operations such as addition, subtraction, and multiplication. It details the characteristics that define a polynomial and provides examples to illustrate each type. Additionally, readers will learn how to perform polynomial operations effectively using column forms and the rules of exponents. This guide aims to enhance your understanding of polynomials as essential algebraic expressions.

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Understanding Polynomials: Definitions, Types, and Operations

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  1. POLYNOMIALS in ACTION by Lorence G. Villaceran Ateneo de Zamboanga University

  2. Polynomials in Action What is a Polynomial?

  3. 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"

  4. Polynomials in Action An expression is nota Polynomial if • It has a negativeexponent • It has a fractional exponent • It has a variable in the denominator • It has a variable inside the square root sign

  5. Polynomials in Action Determine the ff. if it is a polynomial or not Polynomial 6x2 1/x2 not a Polynomial not a Polynomial √x

  6. Polynomials in Action 4y6/3 Polynomial 9y3 Polynomial Z-4 not a Polynomial Polynomial √x2

  7. Polynomials in Action 6x2 TERM Parts of a TERM • It composes the polynomial • It composes of a numerical, literal coefficient and exponent Numerical Coefficient Exponent/Degree Literal Coefficient/Variable

  8. Polynomials in Action Similar Terms x2+xy-y2 2x2+3xy-2y2 • Terms that have the same degree or exponent of the same variable Similar Term

  9. Polynomials in Action Types of Polynomials Monomial • 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.

  10. Polynomials in Action Examples 6x2 Monomial x2+3x Binomial Trinomial 9y3+3y+4 Multinomial x3+y-x+3

  11. Polynomials in Action Trinomial x3+x2y+3y3 x3+x2y2+xy-y3 Multinomial Monomial x3yz2 x3y+wxy Binomial w3+wxy+x2z Trinomial

  12. Polynomials in Action Four Fundamental Operations in Polynomial

  13. Polynomials in Action Addition and Subtraction of Polynomials

  14. 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

  15. Polynomials in Action Example of adding polynomials in column form Add the following polynomials: 4x3+8x2-x-8; x2+6x+9; 9x3+5x-9 4x3+8x2-x-8 x2+6x+9 9x3+ 5x-9 13x3+9x2+10x-8

  16. 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

  17. Polynomials in Action Example of subtracting polynomials in Column form Subtract the following polynomials: 10y4-4y3-y2+y+20; 15y4-4y2-3y+7 10y4 - 4y3 - y2 + y + 20 15y4 4y2 3y 7 - - - - + + + - 5y4 – 4y3+ 3y2 + 4y + 13

  18. Polynomials in Action Multiplication of Polynomials

  19. Polynomials in Action Rules of Exponent Let a and b be the numerical coefficient or the literal coefficient and m, n and p be the exponent. • am x an  =  a(m+n) • (am)n  =  a(mxn) • (ab)m  =  ambm • (ambn)p =  a(m x p)b(mxp) • am/an  =  a(m-n)

  20. Polynomials in Action Rules of Exponent Let 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

  21. 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

  22. (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

  23. 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

  24. 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 Example

  25. am + am  =  2am a3 + a3  =  2a3 5a4 + 2a4  =  7a4 7a6 - 4a6  =  3a6 am + an  =  am + an a6 + a4 = a6 + a4 6a4 + 3a2 - 8a3 = 6a4 + 3a2 - 8a3 Polynomials in Action Example

  26. 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

  27. Polynomials in Action Example of multiplying monomial by a monomial Simplify (5x2)(–2x3) (5x2)(–2x3) = (5)(-2)(x2+3) = -10x5 Simplify (-3y5)(–9y0) (-3y5)(–9y0) = (-3)(-9)(y5+0) = 27y5

  28. Polynomials in Action Rules for multiplication of a polynomial by a monomials • Apply the distributive property of multiplication over addition or subtraction

  29. Polynomials in Action Example of Multiplying monomial by a polynomial Multiply 3x2 and 12x3-4x2 = 3x2(12x3-4x2) = 3x2(12x3) - 3x2(4x2) = 3(12)(x2+3) - 3(4)(x2+2) = 36x5-12x4

  30. Polynomials in Action Example of Multiplying monomial by a polynomial Multiply 7y4 and 5y4-9y3+8 = 7y4(5y4-9y3+8) = 7y4(5y4)-7y4(9y3)+7y4(8) = 7(5)(y4+4)-7(9)(y4+3)+7(8)(y4) = 35y8-63y7+56y4

  31. Polynomials in Action Rules for multiplication of a polynomial by another polynomial • 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

  32. Polynomials in Action Example of Multiplying polynomial by a polynomial Multiply (3x+5) and (3x-4) = (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

  33. Polynomials in Action Example of Multiplying polynomial by a polynomial Multiply(2x2+3x+5) and (x2-2x-3) 2x2+ 3x+ 5                      x2- 2x- 3                       -6x2- 9x-15              -2x3-6x2-10x 2x4+3x3+5x2    2x4+ x3 -7x2 -19x-15

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