Polynomials

Class 9 - NCERT

Intro:

• Polynomial in one variable
• value of variable keeps changing but that of constant remains the same
• terms, coefficients of terms
• constant polynomials & zero polynomial
• monomials, binomials, trinomials, quadrinomials
• Degree of the polynomial - highest power
  • degree of non zero constant polynomial is zero
  • linear, quadratic, cubic
  • degree of zero polynomial is not defined

Ex. 2.1

• find the polynomial in one variable
• write the coefficients
• binomials
• find the degree of the polynomials
• types of polynomials based on degree

Zeroes of the polynomials

• values of p(x)
• a zero or root of the polynomial p(x) = 0 is a number c, such that p(c) = 0
• a non zero constant polynomial has no zero
• every real number is a zero of the zero polynomial
• zero of the polynomial need not be zero
• zero may be a zero of the polynomial
• every linear polynomial has one and only one zero
• the degree of the polynomial decides the number of zeroes of the polynomial

Ex. 2.2

• find the value using the given zero
• find the zero of the linear polynomial

Factorisation of the polynomial

• Factor theorem
  • "if p(x) is a polynomial of degree n>=1, and a is any real number, then
  • x-a is a factor of p(x), if p(a) = 0
  • p(a) = 0, if x-a is the factor of p(x)"
• splitting the middle term method
• for cubic polynomial, first find one factor by substituting 1,-1,2,-2, etc.. then divide to get a quadratic polynomial, then use the splitting method

Ex. 2.3

• determine the polynomial with given factor
• use factor theorem
• find the value of k if the factor is given
• find the factor using splitting the middle term or factorise method for quadratic and cubic polynomial

Algebraic identities:

Ex. 2.4

RSA - Class 9th

Class 10 - NCERT

Zero of the polynomial

• value of p(k) = 0 when x = k
• zero of linear polynomial ax+b is - b / a = - constant / coefficient of x
• a polynomial of degree n has n zeroes

Geometrical meaning of zeroes of polynomial

• intersects the graph at x or y axis correspondingly
• linear polynomial intersects exactly at one point
• quadratic polynomial forms parabola which intersects at 2 points

Ex.2.1

1. find the number of zeroes from the graph given

Relationship between zero and coefficients of polynomial

• Quadratic polynomial
  • sum of zeroes = - b / a = - coefficient of x / coefficient of x²
  • product of zeroes = c / a = constant / coefficient of x²
• Cubic polynomial
  • sum of zeroes = - b / a = - coefficient of x² / coefficient of x³
  • product of zeroes = - d / a = - constant / coefficient of x³
  • sum of products of zeroes taken in pair = c / a = coefficient of x / coefficient of x^3

Ex.2.2

1. find the zeroes of the quadratic polynomial and verify the relationship between coefficients and zeroes
2. find the quadratic polynomial from the sum and zeroes given

RD Sharma new - Class 10th

1. Find the roots of the polynomial and find the value of alpha / beta + beta / alpha
2. If alpha and beta are the roots of the polynomial, then show that alpha + beta = 1-c
3. If the alpha + beta = 24, alpha - beta = 8 ,then find the quadratic polynomial.
   1. Find the value of alpha and beta using elimination method
   2. Then find sum and product of alpha and beta
   3. Construct the polynomial
4. If alpha and beta are the roots of the polynomial, then construct the polynomial with roots 2alpha and 2beta
5. If squared differences of the zeroes of the polynomial f(x) = x² + px + 45 is 144, find the value of p
   1. Find sum and product
   2. Then (alpha - beta)² = 144, expand the LHS
   3. p = +/- 18

In case of coefficients are alphabets, then find the sum and product, before proceeding If the coefficients are numbers, then find the roots either by factorisation or by formula, and then proceed ahead.