#### Study Materials and Revision Notes for Ch 5 Arithmetic Progression Class 10th Maths

**Arithmetic Progression**

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__: An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number d to the preceding term, except the first term.__

*Arithmetic Progression*__•__

__: The fixed number d is called the common difference of the A.P.__

*Term*•

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*Common Difference**Each number in the list of arithmetic progression is called term. It can be positive, negative or zero.*

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(iii) ax

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We can find the required term by putting the numeric value of n(no. of term) which we have to find from starting.

nth term from the end: l - (n-1)d where, l is the last term

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Example: -3/2 is called zero of a polynomial p(x) = 2x + 3 because p(-3/2) = 2x + 3.

(i) A linear polynomial has at most one zero.

(ii) A Quadratic polynomial has at most two zeroes.

(iii) A Cubic polynomial has at most three zeroes.

(iv) A polynomial of degree n has at most n zeroes.

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(i) Sum of zeroes = α + β = -b/a = (-coefficient of x)/(coefficient of x

(ii) Product of zeroes = α.β = c/a = (constant term)/(coefficient of x

(iii) A quadratic polynomial whose zeroes are α and β, is given by:

p(x) = k[x

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(i) α + β + γ = -b/a = (-coefficient of x

(ii) αβ + βγ + γα = c/a = (constant term of x)/(coefficient of x

(iii) α.β.γ = -d/a = (-constant term)/(coefficient of x

(iv) A cubic polynomial whose zeroes are α, β and γ, is given by:

p(x) = k[x

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p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).

*: If a is first term and d is common difference of an A.P. The general form of an AP is: a, a+d, a+2d, a+3d*__General form of an AP__(iii) ax

^{3 }+ bx^{2 }+ cx + d is a polynomial of degree 3 cubic polynomial.•

__: We can find the nth term of an A.P by using T__*Finding General Term of an AP*_{n}= a + (n-1)dWe can find the required term by putting the numeric value of n(no. of term) which we have to find from starting.

nth term from the end: l - (n-1)d where, l is the last term

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__A real number k is said to be zero of a polynomial p(x) if p(k) = 0.__*Zero of a polynomial:*Example: -3/2 is called zero of a polynomial p(x) = 2x + 3 because p(-3/2) = 2x + 3.

(i) A linear polynomial has at most one zero.

(ii) A Quadratic polynomial has at most two zeroes.

(iii) A Cubic polynomial has at most three zeroes.

(iv) A polynomial of degree n has at most n zeroes.

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__If α,β are zeroes of polynomial p(x) = ax__*For quadratic polynomial:*^{2 }+ bx + c then:(i) Sum of zeroes = α + β = -b/a = (-coefficient of x)/(coefficient of x

^{2})(ii) Product of zeroes = α.β = c/a = (constant term)/(coefficient of x

^{2})(iii) A quadratic polynomial whose zeroes are α and β, is given by:

p(x) = k[x

^{2}- (α+β)x + αβ] where k is any real number.•

__If α,β and γ are zeroes of polynomial p(x) = ax__*For cubic polynomial:*^{3 }+ bx^{2 }+ cx + d then:(i) α + β + γ = -b/a = (-coefficient of x

^{2})/(coefficient of x^{3})(ii) αβ + βγ + γα = c/a = (constant term of x)/(coefficient of x

^{3})(iii) α.β.γ = -d/a = (-constant term)/(coefficient of x

^{3})(iv) A cubic polynomial whose zeroes are α, β and γ, is given by:

p(x) = k[x

^{3}- (α+β+γ)x^{2}+ (αβ+βγ+γα)x - αβγ] where k is any real number.•

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