Polynomial
A polynomial in which the power of the variable is a whole number.
Examples: π₯+2π₯, π₯3+π₯2+3π₯+4, π₯3+3,6 etc.
Types of Polynomials
1. On the basis of power:
- Linear Polynomial (eg- 2x+y+4)
- Quadratic Polynomial (eg- 3x2-8x+9)
- Cubic Polynomial (eg- 4x3-x2+6x+3)
2. On the basis of terms:
- Monomial
- Binomial
- Trinomial
Fundamentals
An algebraic equation of the form:
p(x) = a₀xβΏ + a₁xβΏ⁻¹ + ... + an-1x + an
is called a polynomial, provided it has no negative exponent for any variable.
Where, a₀, a₁, ..., an are constants (real numbers); a₀ ≠ 0.
Degree of Polynomial
n is called the degree (highest power of variable x):
- Linear polynomial: ax + b (where, a ≠ 0)
- Quadratic polynomial: ax² + bx + c (where, a ≠ 0)
- Cubic polynomial: ax³ + bx² + cx + d (where, a ≠ 0)
Zeroes of Polynomial
For polynomial p(x), the value of x for which p(x) = 0, is called zero(es) of the polynomial.
- Linear polynomial can have at most 1 root (zero).
- Quadratic polynomial can have at most 2 roots (zeroes).
- Cubic polynomial can have at most 3 roots.
Relationship between Zeroes and Coefficients
- Zero of linear polynomial ax + b is given by x = -π / π
- If Ξ± and Ξ² are zeroes of the quadratic polynomial ax² + bx + c:
- Sum of zeroes, Ξ± + Ξ² = -π / π
- Product of zeroes, Ξ±Ξ² = π / π
- If Ξ±, Ξ², and Ξ³ are zeroes of the cubic polynomial ax³ + bx² + cx + d:
- Sum of zeroes, Ξ± + Ξ² + Ξ³ = -π / π
- Sum of product of pairs, Ξ±Ξ² + Ξ²Ξ³ + Ξ³Ξ± = π / π
- Product of zeroes, πΌπ½πΎ = -π / π
Division Algorithm for Polynomials
If p(x) and g(x) are two polynomials with g(x) ≠ 0, then:
p(x) = g(x) x q(x) + r(x)
where, Dividend = Divisor x Quotient + Remainder.
Tips
- Graph of linear equation is a straight line, while graph of quadratic equation is a parabola.
- Degree of polynomial = maximum number of zeroes of polynomial.
- If remainder r(x) = 0, then g(x) is a factor of p(x).
- To form a quadratic polynomial, if sum and product of zeroes are given: P(x) = x² - (sum of zeroes)x + (product of zeroes)
Graphs
Graph of Linear Polynomial
Example: Y = x + 2
![Graph of Linear Polynomial](linear_polynomial_graph.png)
Graph of a linear polynomial is a straight line which intersects the x-axis at one point, indicating a degree of 1.
Graph of Quadratic Polynomial
Examples:
- Case 1: Graph cuts the x-axis at 2 points, indicating 2 zeroes.
- Case 2: Graph touches the x-axis at 1 point, indicating a perfect square.
- Case 3: Graph does not intersect the x-axis, indicating no real zeroes.
![Graph of Quadratic Polynomial](quadratic_polynomial_graph.png)
NCERT Exercises
Exercise 2.1
Question 1: Find the zeroes of the quadratic polynomial ( x^2 + 7x + 10 ), and verify the relationship between the zeroes and the coefficients.
To find the zeroes of ( x^2 + 7x + 10 ):
- Factorize the polynomial: ( x^2 + 7x + 10 = (x+5)(x+2) ).
- Set each factor to zero: ( x+5=0 ) or ( x+2=0 ).
- Solve for ( x ): ( x=-5 ) or ( x=-2 ).
The zeroes of ( x^2 + 7x + 10 ) are -5 and -2.
Verification:
- Sum of zeroes: ( -5 + (-2) = -7 ), which is equal to ( -b/a ) (coefficient of ( x ) divided by coefficient of ( x^2 )).
- Product of zeroes: ( -5 * (-2) = 10 ), which is equal to ( c/a ) (constant term divided by coefficient of ( x^2 )).
Previous Year's Question
1. If the sum of zeroes of the quadratic polynomial 3x² - Kx + 6 is 3, then find the value of K.