CBSE Class 10 Polynomials Important Formulas and concepts for exams

CBSE Class 10 Polynomials Important Formulas and concepts for exams. There are many more useful educational material which the students can download in pdf format and use them for studies. Study material like concept maps, important and sure shot question banks, quick to learn flash cards, flow charts, mind maps, teacher notes, important formulas, past examinations question bank, important concepts taught by teachers. Students can download these useful educational material free and use them to get better marks in examinations.  Also refer to other worksheets for the same chapter and other subjects too. Use them for better understanding of the subjects.

An algebraic expression of the form p(x) = a₀ + a₁x + a₂x²+ a₃x³+ …………….anxn, where a ≠  0, iscalled a polynomial in variable x of degree n.

Here, a₀, a₁, a₂, a₃, ………,an are real numbers and each power of x is a non-negative integer.e.g. 3x² – 5x + 2 is a polynomial of degree 2.

3√x+2 is not a polynomial.

* 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,

 * A polynomial of degree 0 is called a constant polynomial.

 *A polynomial p(x) = ax + b of degree 1 is called a linear polynomial.

 * A polynomial p(x) = ax² + bx + c of degree 2 is called a quadratic polynomial.

 *A polynomial p(x) = ax³+ bx² + cx + d of degree 3 is called a cubic polynomial.

 * A polynomial p(x) = ax⁴ + bx³+ cx²+ dx + e of degree 4 is called a bi-quadratic polynomial.

VALUE OF A POLYNOMIAL AT A GIVEN POINT x = k

If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k, and is denoted by p(k).

ZERO OF A POLYNOMIAL

A real number k is said to be a zero of a polynomial p(x), if p(k) = 0.

* Geometrically, 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.

 * In general, a polynomial of degree ‘n’ has at the most ‘n’ zeroes. 

♦ A quadratic polynomial whose zeroes are α and β is given by p(x) = x² - (α + β )x + αβ

i.e. x² – (Sum of zeroes)x + (Product of zeroes)

♦ A cubic polynomial whose zeroes are α,β and γ is given by
p(x) = x³ - (α + β + γ )x² + (αβ + βγ + γα )x - αβγ

The zeroes of a quadratic polynomial ax² + bx + c, a  0, are precisely the x-coordinates of the points where the parabola representing y = ax² + bx + c intersects the x-axis.

In fact, for any quadratic polynomial ax² + bx + c, a ≠ 0, the graph of the corresponding equation y = ax² + bx + c has one of the two shapes either open upwards like υ or open downwards like  depending on whether a > 0 or a < 0. (These curves are called parabolas.)

The following three cases can be happen about the graph of quadratic polynomial ax² + bx + c :

Case (i) : Here, the graph cuts x-axis at two distinct points A and A'. The x-coordinates of A and A' are the two zeroes of the quadratic polynomial ax² + bx + c in this case
 

Case (ii) : Here, the graph cuts the x-axis at exactly one point, i.e., at two coincident points. So, the two points A and A′ of Case (i) coincide here to become one point A. The x-coordinate of A is the only zero for the quadratic polynomial ax² + bx + c in this case.

Case (iii) : Here, the graph is either completely above the x-axis or completely below the x-axis. So, it does not cut the x-axis at any point. So, the quadratic polynomial ax² + bx + c has no zero in this case.

DIVISION ALGORITHM FOR POLYNOMIALS

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x),
where r(x) = 0 or degree of r(x) < degree of g(x).

♦ If r(x) = 0, then g(x) is a factor of p(x).
♦ Dividend = Divisor × Quotient + Remainder

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