Maths portfolio manvi

Detailed ppt on the topic polynomials, which will help in the revision for exam. Also for the ideas of making mathematics portfolio.

1. MATHEMATICS PORTFOLIO TOPIC - POLYNOMIALS Class - 10th ‘A’

2. PORTFOLIO INFORMATION Assigned by : Tanfees Sir Submitted by : Manvi Gangwar Class : 10th ‘A’ Roll no. : 742017 Subject : Mathematics Topic : Polynomials

3. Table of contents Introduction to polynomials Polynomial classifications Degrees of a polynomial Terms of a polynomial Types of polynomials Graphical representation of equations Zeroes of a polynomial Graphical meaning of zeroes of polynomial Number of zeroes Factorization of polynomials Relations between zeroes and coefficient of a polynomial Zeroes of a polynomial examples Division algorithm Algebraic identities

4. Introduction to Polynomials The word polynomial is derived from the Greek words ‘poly’ means ‘many‘ and ‘nominal’ means ‘terms‘, so altogether it said “many terms”. A polynomial can have any number of terms but not infinite. It is defined as an expression which is composed of variables, constants and exponents, that are combined using the mathematical operations such as addition, subtraction, multiplication and division . Polynomials are algebraic expressions that consist of variables and coefficients. Variables are also sometimes called indeterminates. For example: x²+x-12.

5. Based on the numbers of terms present in the expression, it is classified as monomial, binomial, and trinomial. Examples of constants, variables and exponents are as follows Constants. Example: 1, 2, 3, etc. Variables. Example: g, h, x, y, etc. Exponents: Example: 5 in x5 etc. Polynomial Classifications The polynomial function is denoted by P(x) where x represents the variable. For example, P(x) = x ²+ 5x+11 If the variable is denoted by a, then the function will be P(a)

6. Degree of a polynomial The degree of a polynomial is defined as the highest degree of a monomial within a polynomial. Thus, a polynomial equation having one variable which has the largest exponent is called a degree of the polynomial. Example: Find the degree of the polynomial 6s4+ 3x2+ 5x +19 Solution: The degree of the polynomial is 4.

7. TERMS OF POLYNOMIAL The terms of polynomials are the parts of the equation which are generally separated by “+” or “-” signs. So, each part of a polynomial in an equation is a term. For example, in a polynomial, say, 2x2 + 5 +4, the number of terms will be 3. The classification of a polynomial is done based on the number of terms in it.

8. Polynomials are of 3 different types and are classified based on the number of terms in it. The three types of polynomials are: Monomial Binomial Trinomial TYPES OF POLYNOMIALS These polynomials can be combined using addition, subtraction, multiplication, and division but is never division by a variable. A few examples of Non Polynomials are: 1/x+2, x-3

9. TYPES OF POLYNOMIALS

10. GRAPHICAL REPRESENTATION OF EQUATIONS Any equation can be represented as a graph on the Cartesian plane, where each point on the graph represents the x and y coordinates of the point that satisfies the equation. An equation can be seen as a constraint placed on the x and y coordinates of a point, and any point that satisfies that constraint will lie on the curve For example, the equation y = x, on a graph, will be a straight line that joins all the points which have their x coordinate equal to their y coordinate. Example – (1,1), (2,2) and so on.

11. The graph of a linear polynomial is a straight line. It cuts the X-axis at exactly one point.

12. The graph of a quadratic polynomial is a parabola . It looks like a U which either opens upwards or opens downwards depending on the value of ‘a’ in ax²+bx+c . If ‘a’ is positive, then parabola opens upwards and if ‘a’ is negative then it opens downwards It can cut the x-axis at 0, 1 or two points

13. ZEROES OF A POLYNOMIAL A zero of a polynomial p(x) is the value of x for which the value of p(x) is 0. If k is a zero of p(x), then p(k)=0. For example, consider a polynomial p(x)=x2−3x+2. When x=1, the value of p(x) will be equal to p(1)=12−3×1+2 =1−3+2 =0 Since p(x)=0 at x=1, we say that 1 is a zero of the polynomial x2−3x+2

14. Graphical Meaning of zeroes of a Polynmomial G raphically , zeros of a polynomial are the points where its graph cuts the x-axis. (i) One zero (ii) Two zeros (iii) Three zeros Here A, B and C correspond to the zeros of the polynomial represented by the graphs.

15. In general, a polynomial of degree n has at most n zeros. A linear polynomial has one zero, A quadratic polynomial has at most two zeros. A cubic polynomial has at most 3 zeros. Number of zeroes

16. Factorization of Polynomials Quadratic polynomials can be factorized by splitting the middle term. For example, consider the polynomial 2x²−5x+3 Splitting the middle term : The middle term in the polynomial 2x²−5x+3 is -5x. This must be expressed as a sum of two terms such that the product of their coefficients is equal to the product of 2 and 3 (coefficient of x² and the constant term) −5 can be expressed as (−2)+(−3), as −2×−3=6=2×3 2x²−5x+3 = 2x²−2x−3x+3 Now, identify the common factors in individual groups 2x²−2x−3x+3=2x(x−1)−3(x−1) Taking (x−1) as the common factor, this can be expressed as: 2x(x−1)−3(x−1)= (x−1)(2x−3)

17. Relation Between Zeros and Coefficient of a Polynomial A real number say “a” is a zero of a polynomial P(x) if P(a) = 0. The zero of a polynomial is clearly explained using the Factor theorem. If “k” is a zero of a polynomial P(x), then (x-k) is a factor of a given polynomial.

18. The linear polynomial is an expression, in which the degree of the polynomial is 1. The linear polynomial should be in the form of ax+b. Here, “x” is a variable, “a” and “b” are constant. The Quadratic polynomial is defined as a polynomial with the highest degree of 2. The quadratic polynomial should be in the form of ax 2 + bx + c. In this case, a ≠ 0. The cubic polynomial is a polynomial with the highest degree of 3. The cubic polynomial should be in the form of ax 3 + bx 2 + cx + d, where a ≠ 0. Let say α, β, and γ are the three zeros of a polynomial, then Zeroes (α, β, γ) follow the rules of algebraic identities, i.e., (α + β)² = α² + β² + 2αβ ∴(α² + β²) = (α + β)² – 2αβ

19. Zeros of a Polynomial Solved Examples

20. Division Algorithm To divide one polynomial by another, follow the steps given below. Step 1: arrange the terms of the dividend and the divisor in the decreasing order of their degrees. Step 2: To obtain the first term of the quotient, divide the highest degree term of the dividend by the highest degree term of the divisor Then carry out the division process. Step 3: The remainder from the previous division becomes the dividend for the next step. Repeat this process until the degree of the remainder is less than the degree of the divisor. If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then p(x) = g(x) × q(x) + r(x) Dividend = Divisor x Quotient + Remainder If r (x) = 0, then g (x) is a factor of p (x). If r (x) ≠ 0, then we can subtract r (x) from p (x) and then the new polynomial formed is a factor of g(x) and q(x).

21. Algebraic Identities

22. mathematics portfolio ends here.

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