A polynomial p(x) in one variable x is an algebraic expression in x of the form
p(x) = anxn + an-1xn-1 + . . . + a2x2+ a1x + a0,
where a0, a1, a2, . . ., an are constants and an ≠ 0.
A polynomial of one term is called a monomial. A polynomial of two terms is called a binomial. A polynomial of three terms is called a trinomial.
A polynomial of degree one is called a linear polynomial. A polynomial of degree two is called a quadratic polynomial. A polynomial of degree three is called a cubic polynomial.
A real number a is a zero of a polynomial p(x) if p(a) = 0. In this case, a is also called a root of the equation p(x) = 0. Every linear polynomial in one variable has a unique zero, a non-zero constant polynomial has no zero, and every real number is a zero of the zero polynomial.
Remainder Theorem: If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by the linear polynomial x - a, then the remainder is p(a).
Factor Theorem: x - a is a factor of the polynomial p(x), if p(a) = 0. Also, if x - a is a factor of p(x), then p(a) = 0.
Important Formulas
- (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
- (x + y)3 = x3 + y3 + 3xy(x + y)
- (x - y)3 = x3 - y3 - 3xy(x - y)
- x3 + y3 + z3 - 3xyz = (x + y + z)(x2 + y2 + z2 - xy - yz - zx)
Class 10
If p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the polynomial p(x). For example, 4x + 2 is a polynomial in the variable x of degree 1, 2y2 - 3y + 4 is a polynomial in the variable y of degree 2.
Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.
A quadratic polynomial in x with real coefficients is of the form ax2 + bx + c, where a, b, c are real numbers with a ≠ 0.
The zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the graph of y = p(x) intersects the x -axis. A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.
If α and β are the zeroes of the quadratic polynomial ax2 + bx + c, then
- α + β = -b/a
- αβ = c/a
If α, β, γ are the zeroes of the cubic polynomial ax3 + bx2 + cx + d, then
- α + β + γ = -b/a
- αβ + βγ + γα = c/a
- αβγ = -d/a
The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x), there are polynomials q(x) and r(x) such that
p(x) = g(x) q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).