Polynomials: An expression of the form P(X) = a0 +a1x+a2x2 ……anxn where an ≠ 0 is called a polynomial in variable x of degree n. where; a0, a2 ….an are real numbers and each power of x is a non-negative integer.
Example: 3x2 + 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.
- (ax2+bx+c) is a polynomial of degree 2 called a quadratic polynomial.
- (ax3+ bx2+ 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, ax0 + 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) = ax2+ bx + c then:
- Sum of zeroes = α + β = -b/a = (-coefficient of x) / (coefficient of x2)
- Product of Zeroes = α.β = c/a = (coefficient of x) / (coefficient of x2)
- A quadratic polynomial whose zeroes are α and β is given by:
p(x) = k[x2 -(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) = ax3+bx2+cx+d then:
- α + β + γ = -b/a =(coefficient of x2) / (coefficient of x3)
- αβ + βγ + γα = c/a =(coefficient of x) / (coefficient of x3)
- a.β.γ = -d/a =(-coefficient term) / (coefficient of x3)
- A cubic polynomial whose zeroes are α, β and γ, is given by:
p(x) = k[x3 – (α + β + γ)x2 + (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?
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 $ (2x2 – 4y2 + 3xy – 5) and his expenditure is $ (-2y2 + 5x2 + 9). Use the concept of subtraction of polynomials to find his savings.
We all know that Savings = Income – Expenditure. Now, applying the same thing here, we will get:
Savings = 2x2 – 4y2 + 3xy – 5 – (9 – 2y2 + 5x2) = 2x2 – 4y2 + 3xy – 5 + 2y2 – 5x2 – 9 = -3x2 – 2y2 + 3xy – 14
Answer: Hence, his savings will be $(-3x2 – 2y2 + 3xy – 14).
Example 3: Add the following polynomials: (2x2 + 16x – 7) + (x3 + x2 – 9x + 1).
Solution: To add polynomials, we have to compile the like terms.
(2x2 + 16x – 7) + (x3 + x2 – 9x + 1) = x3 + (2 + 1)x2 + (16 – 9)x – 7 + 1 = x3 + 3x2 + 7x – 6
Answer: x3 + 3x2 + 7x – 6