**Polynomials: **An expression of the form P(X) = a_{0} +a_{1}x+a_{2}x^{2} ……a_{n}x^{n} where a_{n} ≠ 0 is called a polynomial in variable x of degree n. where; a_{0}, a_{2} ….an are real numbers and each power of x is a non-negative integer.**Example:** 3x^{2} + 5x + 3 is a polynomial of degree two which is a non-negative integer.

√x + 5 is not a polynomial because the degree of x is not a non-negative integer.

**Polynomials** of degrees 1, 2, and 3 are called linear, quadratic, and cubic polynomials respectively.

- (ax + b) is a polynomial of degree 1 called a linear polynomial.
- (ax
^{2}+bx+c) is a polynomial of degree 2 called a quadratic polynomial. - (ax
^{3}+ bx^{2}+ cx + d) is a polynomial of degree 3 cubic polynomial.

**Zero Polynomial:** A polynomial of degree zero is called zero polynomial. Or, A polynomial that contains the only constant term, is called a zero polynomial.**Example:** 5, ax^{0} + 3.

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

- A linear polynomial has at most one zero.
- A Quadratic polynomial has at most two zeroes.
- A Cubic polynomial has at most three zeroes.
- A polynomial of degree n has at most n zeroes.

**For quadratic polynomial: **If α,β are zeroes of polynomial p(x) = ax^{2}+ bx + c then:

**Sum of zeroes =**α + β = -b/a = (-coefficient of x) / (coefficient of x^{2})**Product of Zeroes =**α.β = c/a = (coefficient of x) / (coefficient of x^{2})- A quadratic polynomial whose zeroes are α and β is given by:

p(x) = k[x^{2}-(α + β)x (α.β)] where k is any real number.

**For cubic polynomial: **If α,β and γ are zeroes of polynomial p(x) = ax^{3}+bx^{2}+cx+d then:

- α + β + γ = -b/a =(coefficient of x
^{2}) / (coefficient of x^{3}) - αβ + βγ + γα = c/a =(coefficient of x) / (coefficient of x
^{3}) - a.β.γ = -d/a =(-coefficient term) / (coefficient of x
^{3}) - 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 – a.β.γ] where k is any real number.

**Division Algorithm:** 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:

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

**Solved Examples of Polynomials:**

**Example 1:** Mr. Stark wants to plant a few rose bushes on the borders of his triangular-shaped garden. If the sides of the garden are given by the polynomials (4x – 2) feet, (5x + 3) feet, and (x + 9) feet, what is the perimeter of the garden?**Solution:**

Perimeter of garden = (4x – 2) + (5x + 3) + (x + 9) = 4x + 5x + x – 2 + 3 + 9 = 10x + 10**Answer:** ∴ The perimeter is (10x + 10) feet.

**Example 2:** The income of Mr. Smith is $ (2x^{2} – 4y^{2} + 3xy – 5) and his expenditure is $ (-2y^{2} + 5x^{2} + 9). Use the concept of subtraction of polynomials to find his savings.**Solution:**

We all know that Savings = Income – Expenditure. Now, applying the same thing here, we will get:

Savings = 2x^{2} – 4y^{2} + 3xy – 5 – (9 – 2y^{2} + 5x^{2}) = 2x^{2} – 4y^{2} + 3xy – 5 + 2y^{2} – 5x^{2} – 9 = -3x^{2} – 2y^{2} + 3xy – 14**Answer:** Hence, his savings will be $(-3x^{2} – 2y^{2} + 3xy – 14).

**Example 3:** Add the following polynomials: (2x^{2} + 16x – 7) + (x^{3} + x^{2} – 9x + 1).**Solution:** To add polynomials, we have to compile the like terms.

(2x^{2} + 16x – 7) + (x^{3} + x^{2} – 9x + 1) = x^{3} + (2 + 1)x^{2} + (16 – 9)x – 7 + 1 = x^{3} + 3x^{2} + 7x – 6**Answer:** x^{3} + 3x^{2} + 7x – 6