Polynomials

Polynomial (Mathematics)

1. polynomials

2. Table Of Contents - Introduction History 1. History Of Notation Terms Types Of Polynomials Uses Zeroes Of Polynomial Degree Graphs Of Polynomial Function Table Algebraic Identities Arithmetic Of Polynomials Think Tanker ? ?

3. Introduction….. ~ What Is a Polynomial ?? In mathematics , a polynomial is an expression consisting of variables and coefficients , that involves only the operations of addition , subtraction , multiplication , and non-negative integer-exponents. Example : x 2 − 3 x + 6, which is a quadratic polynomial .

4. History Of Notation He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables. As can be seen above, in the general formula for a polynomial in one variable, where the a 's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.

5. Binomial Monomial Trinomial Polynomials NEXT

6. Types Of Polynomials... Monomial In mathematics, A monomial is a polynomial with just one term. For Example: 3x,4xy is a monomial. Binomial In algebra, A binomial is a polynomial, which is the sum of two monomials . For Example: 2x+5 is a Binomial. Trinomial In elementary algebra, A trinomial is a polynomial consisting of three terms or monomials. For Example : 3x+5y+7z is a Trinomial.

7. Binomial Trinomial Monomial Pictorial View

8. Uses... Polynomials appear in a wide variety of areas of mathematics and science. ~ For example, they are used to form “ Polynomial ” equations , which encode a wide range of problems, from elementary word problems to complicated problems in the sciences. They are used to define “ Polynomial Functions ” , which appear in settings ranging from basic chemistry and physics to economics and social science. They are used in calculus and numerical analysis to approximate other functions.

9. Zeroes Of Polynomial Consider the polynomial p(x) = 5 x 3 – 2 x 3 + 3x – 2. If we replace x by 1 everywhere in p(x), we get p(1) = 5 × (1)3 – 2 × (1)2 + 3 × (1) – 2 = 5 – 2 + 3 –2 = 4 So, we say that the value of p(x) at x = 1 is 4. Similarly, p(0) = 5(0)3 – 2(0)2 + 3(0) –2 = –2

10. Degree The degree of a polynomial is the highest degree of its terms, when the polynomial is expressed in canonical form (i.e., as a linear combination of monomials). The degree of a term is the sum of the exponents of the variables that appear in it.

11. DEGREE OF POLYNOMIAL Degree 0 – constant Degree 1 – linear Degree 2 – quadratic Degree 3 – cubic Degree 4 – quartic (or, less commonly, biquadratic )

12. Look at each term, whoever has the most letters wins! x 2 – 4x 4 + x 6 This is a 8 th degree polynomial: xy 4 + x 4 y 4 + 12 This guy has 6 letters… The degree is 6. This guy has 8 letters… The degree is 8 Here’s how you find the degree of a polynomial :

13. The graph of the zero polynomial f ( x ) = 0 is the x -axis. Graphs Of Polynomial Functions ..

14. The graph of the polynomial of degree 2 Graphs Of Polynomial Functions ..

15. Table Polynomial Degree Name Using Degree Nos. Of Terms Name Using Nos Of Terms 4 Constant 1 Monomial 3x+6 1 Linear 2 Binomial 3x 2 +2x+1 2 Quadratic 3 Trinomial 2x 3 3 Cubic 1 Monomial 6x 4 + 3x 4 Biquadratic 2 Binomial

16. Algebraic Identities (a + b ) 2 = a 2 + b 2 + 2 ab (a - b ) 2 = a 2 + b 2 - 2 ab (a 2 - b 2 )= (a + b)(a - b) (x - a)(x - b )= x 2 +( a+b )x - ab

17. Arithmetic Of Polynomials Addition ( + ) Subtraction( - ) Division ( / )

18. Addition Of Polynomials….. Polynomials can be added using the associative law of addition (grouping all their terms together into a single sum), possibly followed by reordering, and combining of like terms. For example, if Method 1: Line up like terms. Then add the coefficients. P = 3x + 7 Q = 2x + 3 P + Q = 5x + 10

19. Addition Of Polynomials….. Method 2 : Group like terms. Then add the coefficients. 4x 2 + 6x + 7 + 2x 2 – 9x + 1 = (4x 2 + 2x 2 )+(6x – 9x)+ (7+1) = 6x 2 – 3x + 8 » The sum of two polynomials is also a polynomial.

20. Subtraction Of Polynomials Earlier you learned that subtraction means to add the opposite. So when you subtract a polynomial, change the signs of each of the terms to its opposite. Then add the coefficients. Method 1: Line up like terms. Change the signs of the second polynomial, then add. For Example: 4x - 7 4x - 7 -(2x + 3)  -2x – 3 2x - 10

21. Subtraction Of Polynomials Method 2: Simplify : (5x 2 – 3x) – (-8x 2 + 11) Write the opposite of each term : 5x 2 – 3x + 8x 2 – 11 Group like terms : (5x 2 + 8x 2 ) + (3x + 0) + (-11 + 0) = 13x 2 + 3x – 11 » The difference of two polynomials is also a polynomial

22. Division Of Polynomials The Methods Used For Finding Divison Of Polynomials Are: 1. Long Division Method 2. Factor Theorem

23. Long- Division Method In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit numbers that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. Dividend = (Divisor × Quotient) + Remainder

24. Long- Division Method Dividend = (Divisor × Quotient) + Remainder

25. Factorisation Factor Theorem : If p(x) is a polynomial of degree n > 1 and a is any real number, Then : ( i ) x – a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x – a is a factor of p(x).

26. Think Tanker ? ?

27. Q.1 What is the simplified form of : 2x-3x+2?? A. -x+2 B. -5x+2 C. -10x+2 D. -2x+2

28. Q.2 What is the value of x when x+3=10 ?? A. 7 B. 4 C. 2 D. 9

29. Q 3. Solve 2x+4=108 ?? A. 32 B. 56 C. 52 D. 23

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