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**Now discuss the concepts of the polynomial for class 10.**

Contents

**Polynomials**

An expression of the form p(x) = a_{0}+a_{1}x+a_{2}x^{2}+……+a_{n}x^{n}, where a_{n} is not equal to zero, is called a polynomial in x of degree n.

**Some important identity or polynomial for class 10 formulas used in polynomial:**

^{ }

**Degree of polynomials**

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).

Ex: The degree of polynomial P(x) = 2x^{3} + 5x^{2} -7 is 3 because the degree of the polynomial is the highest power of polynomial.

**Types of Polynomials**

**Linear polynomial:**Polynomial having degree 1 is called a linear polynomial. For example, (4x+7), (3x-4), etc.

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**Quadratic polynomial:**Polynomial having degree 2 is called a quadratic polynomial. For example, 2x^{2}+5x+3, 4x^{2}-3x+4 etc.

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**Cubic polynomial:**Polynomial having degree 3 is called a cubic polynomial. For example, 3x^{3}+4x^{2}+5x+2, 9x^{3}+5x^{2}+3x+5 etc.

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**Biquadratic polynomial:**Polynomial having degree 4 is called a biquadratic polynomial. For example, 5x^{4}+3x^{3}+8x^{2}+x+2 etc.

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**Zero of polynomial **

If the value of P(x) at x = K is zero then K is called a zero of the polynomial P(x).

**Relationship between zeros and coefficients of a polynomial**

**For Quadratic equation:**If α and β are zeros of the**quadratic equation**ax^{2}+bx+c, a is not equal to zero, then

- α+ β = -b/a
- αx β = c/a

Equation of quadratic polynomial,

P(x) = αx^{2} – (α +β)x + xβ

**For Cubic equation:**If α, β and γ are zeros of cubic polynomial ax^{3}+bx^{2}+cx+d, a is not equal to zero, then

- α+ β + γ = -b/a
- αβ + βγ + γα = c/a
- αβγ = -d/a

** ** Equation of cubic polynomial,

P(x) = α x^{3} – (α +β+γ)x^{2} + (α β+βγ+γα )x – αβγ

**Division Algorithm for polynomials**

If P(x) and g(x) are any two polynomials with g(x) is not equal to zero, then we can find polynomials q(x) and r(x) such that

P(x) = g(x) X q(x) + r(x)

When r(x) = 0 or degree of r(x) < degree of g(x). This result is called the division algorithm for polynomials.

**Some important identities used in Trigonometry and polynomials chapter**

There are 8 important identities which is used in **Trigonometry** and Polynomial chapter:

- (a+b)
^{2}= a^{2}+ b^{2}+ 2ab - (a-b)
^{2}= a^{2}+ b^{2}– 2ab - a
^{2}– b^{2}= (a+b)(a-b) - (a+b+c)
^{2}= a^{2}+b^{2}+c^{2}+2ab+2bc+2ca - (a+b)
^{3}= a^{3}+b^{3}+3a^{2}b+3ab^{2} - (a-b)
^{3}= a^{3}+b^{3}-3a^{2}b+3ab^{2} - a
^{3}+b^{3}= (a+b)(a^{2}+b^{2}-2ab) - a
^{3}-b^{3}= (a+b)(a^{2}+b^{2}+2ab)

**Graphical representation of zero of a polynomial**

- The zeroes of 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 zeros and a cubic polynomial can have at most 3 zeros.
- For any quadratic polynomial ax
^{2}+bx+c, a is not equal to zero, its graph one of the two shapes either open upwards or open downwards on whether a>0 or a<0.

**Factorization is the important concept used in Polynomial **

**Factorization: **The process of finding two or more factors of a given expression is called factorization.

**EX: Find the factor of x ^{2}-4.**

Solution: We know that a^{2}-b^{2}= (a+b)(a-b)

X^{2}-4 = x^{2}-2^{2} = (x+2)(x-2)

Hence, (x+2) and (x-2) is the factor of x^{2}-4.

**Factorize x ^{2}+5x+6.**

Solution: By using middle splitting term method,

= x^{2}+5x+6

= x^{2}+2x+3x+6

= x(x+2) +3(x+2)

= (x+2)(x+3)

Hence, (x+2) and (x+3) is the factor of x^{2}+5x+6.

** I hope you enjoy this post polynomial for class 10 all concept notes.**