The chapter "Polynomials" is really important for learning the basics of algebra and how to use them to solve different math problems. It's like the foundation for solving all kinds of math questions in the future.
In this document, you'll find Class 10 Math Formulas related to Polynomials that can help you do well in your board exams and other important competitive exams.
An algebraic expression of the form p(x) = a₀ + a₁ x + a₂ x^{2} + a₃ x^{3} + ...... + a_{n} x^{n}, in which the variables involved have only nonnegative integral exponents, is called a polynomial in x of degree n.
Polynomial
Note:
In the polynomial a₀ + a₁ x + a₂ x^{2} + a₃ x^{3} + ...... + a_{n} x^{n}
a₀, a₁ x, a₂ x^{2}, a₃ x^{3} ..... a_{n  1} x^{n  1}, a_{n}x^{n} are terms.
a₀, a₁, a₂, ..... a_{n  1}, a_{n} are the coefficients of x^{0}, x^{1}, x^{2}, ....., x^{n1}, x^{n} respectively.
The highest power of the variable in a polynomial is called its degree.
Example:
5x + 3 is a polynomial in x of degree 1.
p(y) = 3y^{2} + 4y  4 is a polynomial in y of degree 2.
Examples:
If p (x) is a polynomial in x and ‘a’ is a real number. Then the value obtained by putting x = a in p (x) is called the value of p (x) at x = a.
Example: Let p(x) = 5x^{2}  4x + 2 then its value at x = 2 is given by
p(2) = 5 (2)^{2}  4 (2) + 2 = 5 (4)  8 + 2 = 20  8 + 2 = 14
Thus, the value of p(x) at x = 2 is 14.
A real number ‘a’ is said to be a zero of the polynomial p (x), if p (a) = 0.
Example: Let p (x) = x^{2}  x  2 Then p (2) = (2)^{2}  (2)  2 = 4  4 = 0,
and p (1) = (1)^{2}  (1)  2 = 2  2 = 0
∴ (1) and (2) are the zeroes of the polynomial x^{2 } x  2.
Note:
I. A linear polynomial has at the most one zero.
II. A quadratic polynomial has at the most two zeroes.III. In general a polynomial of degree n has at the most n zeroes.
The graph of a linear polynomial is always a straight line. It may or may not pass through the xaxis. In case the graph line is passing through a point on the xaxis, then the ycoordinate of that point must be zero. In general, for a linear polynomial ax + b = 0, (a ≠ 0), the graph is a straight line that can intersect the xaxis at exactly one point, namely, is the zero of the polynomial ax + b.
In the given figure, CD is meeting xaxis at x = 1.
∴ Zero of ax + b is 1.
Note:
A zero of a linear polynomial is the xcoordinate of the point, where the graph intersects the xaxis.
The graph of ax^{2} + bx + c, (a ≠0) is a curve of ∪ shape, called a parabola.
In the given figure, the graph of a quadratic polynomial x^{2}  3x  4 is shown. It intersects xaxis at (1, 0) and (4, 0). Therefore, its zeroes are 1 and 4. Here, a > 0, so the graph opens upwards.
Whereas the following figure is a graph of the polynomial  x^{2} + x + 6. Since it intersects the xaxis at (3, 0) and (2, 0). Therefore, the zeroes of  x^{2} + x + 6 are 2 and 3.
Here a < 0, so the parabola opens downwards.
Note: In the case of Quadratic polynomial  at most 2 zeroes, Cubic polynomial  at most 3 zeroes, Biquadratic polynomial  at most 4 zeroes.
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.
116 videos420 docs77 tests

1. What is the degree of a polynomial? 
2. How can we find the zeroes of a polynomial? 
3. How is the graph of a quadratic polynomial represented? 
4. What is the relationship between the zeroes and coefficients of a polynomial? 
5. Why was the Division Algorithm deleted from the NCERT textbook? 

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