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Notes for polynomials chapter of class 10 Mathematics. Dronstudy provides free comprehensive chapterwise class 10 Mathematics notes with proper images & diagram.

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**(1) Polynomial :Â The expression which contains one or more terms with non-zero coefficient is called a polynomial. A polynomial can have any number of terms.
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**(2) Degree of polynomial : The highest power of the variable in a polynomial is called as the degree of the polynomial.
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**(3) Linear polynomial : A polynomial of degree one is called a linear polynomial.
**

**(4) Quadratic polynomial : A polynomial having highest degree of two is called a quadratic polynomial. The term â€˜quadraticâ€™ is derived from word â€˜quadrateâ€™ which means square. In general, a quadratic polynomial can be expressed in the form ax ^{2} + bx + c, where aâ‰ 0 and a, b, c are constants.
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**(5) Cubic Polynomial : A polynomial having highest degree of three is called a cubic polynomial. In general, a quadratic polynomial can be expressed in the form ax ^{3} + bx^{2} + cx + d, where aâ‰ 0 and a, b, c, d are constants.**

**(6) Zeroes of a Polynomial : The value of variable for which the polynomial becomes zero is called as the zeroes of the polynomial. In general, if k is a zero of p(x) = ax + b, then p(k) = ak + b = 0, i.e., k = -b/a. Hence, the zero of the linear polynomial ax + b is â€“b/a = -(Constant term)/(coefficient of x)
**

If we put x = -2 in p(x), we get,

p(-2) = -2 + 2 = 0.

Thus, -2 is a zero of the polynomial p(x).

**(7) Geometrical Meaning of the Zeroes of a Polynomial:
**

** For Example:Â **The graph of y = 2x - 3 is a straight line passing through points (0, -3) and (3/2, 0).

x |
0 |
3/2 |

y = 2x - 3 |
6 |
0 |

Here, the graph of y = 2x - 3 is a straight line which intersects the x-axis at exactly one point, namely, (3/2 , 0).

**(ii) For Quadratic Polynomial:
**In general, for any quadratic polynomial ax

** Case 2:** The Graph cuts x-axis at exactly one point.The x-coordinates of the quadratic polynomial ax

** Case 3:** The Graph is completely above x-axis or below x-axis.The quadratic polynomial ax

** For Example:Â **For the given graph, find the number of zeroes of p(x).From the figure, we can see that the graph intersects the x-axis at four points.

Therefore, the number of zeroes is 4.

**(8) Relationship between Zeroes and Coefficients of a Polynomial:
**

Moreover, Î± + Î² = -b/a and Î± Î² = c/a.

In general, sum of zeros = -(Coefficient of x)/(Coefficient of x

Product of zeros = (Constant term)/ (Coefficient of x

** For Example:Â **Find the zeroes of the quadratic polynomial x

On finding the factors of x

Thus, value of x

Hence, zeros of x

Now, sum of zeros = -2 + (-5) = -7 = -7/1 = -(Coefficient of x)/(Coefficient of x

** For Example:Â **Find the zeroes of the quadratic polynomial t

On finding the factors of t

Thus, value of t

Hence, zeros of t

Now, sum of zeros = âˆš15 + (-âˆš15) = 0 = -0/1 = -(Coefficient of t)/(Coefficient of t

** For Example:Â **Find a quadratic polynomial for the given numbers as the sum and product of its zeroes respectively 4, 1.

Let the quadratic polynomial be ax

Given, Î± + Î² = 4 = 4/1 = -b/a.

Î± Î² = 1 = 1/1 = c/a.

Thus, a = 1, b = -4 and c = 1.

Therefore, the quadratic polynomial is x

** (ii) Cubic Polynomial:Â **In general, it can be proved that if Î±, Î², Î³ are the zeroes of the cubic polynomial ax^{3} + bx^{2} + cx + d, then,

Î± + Î² + Î³ = â€“b/a ,

Î±Î² + Î²Î³ + Î³Î± = c/a andÂ Î± Î² Î³ = â€“ d/a .

**(9) Division Algorithm for Polynomials : If p(x) and g(x) are any two polynomials with g(x) â‰ 0, then we can find polynomials q(x) and r(x) such that p(x) = g(x) Ã— q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).**

** For Example:Â **Divide 3x

On dividing 3x

Here, quotient is (x â€“ 2) and remainder is 3.

Now, as per the division algorithm, Divisor x Quotient + Remainder = Dividend

LHS = (-x^{2} + x + 1)(x â€“ 2) + 3

= (â€“x^{3} + x^{2} â€“ x + 2x^{2} â€“ 2x + 2 + 3)

= (â€“x^{3} + 3x^{2} â€“ 3x + 5)

RHS = (â€“x^{3} + 3x^{2} â€“ 3x + 5)

Thus, division algorithm is verified.

** For Example:Â **On dividing x

Given, dividend = p(x) = (x

Let divisor be denoted by g(x).

Now, as per the division algorithm,

Divisor x Quotient + Remainder = Dividend

(x

(x

(x

Hence, g(x) is the quotient when we divide (x

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