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

2. What is a Polynomial? Polynomials in Action

3. 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. 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

5. 6x 2 Determine the ff. if it is a polynomial or not √x Polynomial 1/x 2 not a Polynomial Polynomials in Action

6. 4y 6/3 9y 3 Z -4 √x 2 Polynomial not a Polynomial Polynomial Polynomials in Action

7. TERM It composes the polynomial It composes of a numerical, literal coefficient and exponent Parts of a TERM Numerical Coefficient Literal Coefficient/Variable 6x 2 Exponent/Degree Polynomials in Action

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

9. 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. Polynomials in Action

10. Examples 6x 2 9y 3 +3y+4 x 2 +3x x 3 +y-x+3 Binomial Monomial Trinomial Multinomial Polynomials in Action

11. x 3 +x 2 y+3y 3 x 3 y+wxy x 3 yz 2 x 3 +x 2 y 2 +xy-y 3 w 3 +wxy+x 2 z Monomial Multinomial Binomial Trinomial

12. Polynomials in Action Four Fundamental Operations in Polynomial

13. Addition and Subtraction of Polynomials Polynomials in Action

14. 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

15. Add the following polynomials: 4x 3 +8x 2 -x-8; x 2 +6x+9; 9x 3 +5x-9 Example of adding polynomials in column form 4x 3 +8x 2 -x-8 x 2 +6x+9 9x 3 + 5x-9 13x 3 +9x 2 +10x-8 Polynomials in Action

16. 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

17. Subtract the following polynomials: 10y 4 -4y 3 -y 2 +y+20; 15y 4 -4y 2 -3y+7 Example of subtracting polynomials in Column form 10y 4 - 4y 3 - y 2 + y + 20 15y 4 4y 2 3y 7 - + + - +- - - 5y 4 – 4y 3 + 3y 2 + 4y + 13 Polynomials in Action

18. Multiplication of Polynomials Polynomials in Action

19. Rules of Exponent Let a and b be the numerical coefficient or the literal coefficient and m, n and p be the exponent. 1.a m x a n = a (m+n) 2.(a m ) n = a (mxn) 3.(ab) m = a m b m 4.(a m b n ) p = a (m x p) b (mxp) 5.a m /a n = a (m-n) Polynomials in Action

20. Rules of Exponent Let a and b be the numerical coefficient or the literal coefficient and m, n and p be the exponent. 6. a 0 = 1 7. a 1 = a 8. a -m = 1/a m or 1/a m = a m /1 = a m 9. a m + a m = 2a m 10. a m + a n = a m + a n Polynomials in Action

21. Example a m x a n = a (m+n)  a 3 x a 2 = a 3+2 = a 5  (3 4 )(3 7 ) = 3 11  (2 3 a 4 )(2 5 a 6 ) = 2 8 a 10 (a m ) n = a (mxn)  (a 5 ) 3 = a 5x3 = a 15  (4 2 ) 3 = 4 6  [(x 2 ) 2 ] 2 = x 2x2x2 = x 8 Polynomials in Action

22. Example (ab) m = a m b m (ab) 4 = a 4 b 4 (5x) 3 = 5 3 x 3 (4xy) 5 = 4 5 x 5 y 5 (a m b n ) p = a (mxp) b (mxp)  (a 2 b 3 ) 4 = a 2x4 b 3x4 = a 8 b 12  (4 3 x 7 y 4 ) 5 = 4 15 x 35 y 20 Polynomials in Action

23. Example a m /a n = a (m-n) aa 5 /a 3 = a 5-3 = a 2 aa 7 /a 10 = a 7-10 = a 3 or 1/a 3 aa 3 b 8 c 12 /a 5 b 8 c 7 = a -2 c 5 a 0 = 1 ®8®8 0 = 1 ®5®5a 0 = 5(1) = 5 Polynomials in Action

24. Example a 1 = a ⌂8 1 = 8 ⌂5a 1 = 5(a) = 5a a -m = 1/a m or 1/a m = a m /1 = a m ∂a -3 = 1/a 3 ∂a -5 /b -2 = b 2 /a 5 ∂6a -2 b 5 /7c - 6 d 3 = 6b 5 c 6 /7a 2 d 3 Polynomials in Action

25. Example a m + a m = 2a m oa 3 + a 3 = 2a 3 o5a 4 + 2a 4 = 7a 4 o7a 6 - 4a 6 = 3a 6 a m + a n = a m + a n a 6 + a 4 = a 6 + a 4 6a 4 + 3a 2 - 8a 3 = 6a 4 + 3a 2 - 8a 3 Polynomials in Action

26. 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

27. Example of multiplying monomial by a monomial Simplify (5x 2 )(–2x 3 ) Polynomials in Action (5x 2 )(–2x 3 ) = (5)(-2)(x 2+3 ) = -10x 5 Simplify (-3y 5 )(–9y 0 ) (-3y 5 )(–9y 0 ) = (-3)(-9)(y 5+0 ) = 27y 5

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

29. Example of Multiplying monomial by a polynomial Multiply 3x 2 and 12x 3 -4x 2 Polynomials in Action = 3x 2 (12x 3 -4x 2 ) = 3x 2 (12x 3 ) - 3x 2 (4x 2 ) = 3(12)(x 2+3 ) - 3(4)(x 2+2 ) = 36x 5 -12x 4

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

31. 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

32. 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)(x 1+1 )+(3)(-4)(x)+(5)(3)(x)+(5)(-4) =9x 2 -12x+15x-20 =9x 2 +3x-20

33. Example of Multiplying polynomial by a polynomial Multiply(2x 2 +3x+5) and (x 2 -2x-3) Polynomials in Action 2x 2 + 3x+ 5 x 2 - 2x- 3 -6x 2 - 9x-15 -2x 3 -6x 2 -10x 2x 4 +3x 3 +5x 2 2x 4 + x 3 -7x 2 -19x-15