Polynomial

 Polynomial 

An algebraic expression in which the variables involved have only non-negative integral powers is called Polynomial. 

Let a₀, a₁, a₂.........aₙ are real numbers, then

f(x) = a₀ + a₁x + a₂x² +.........aₙxⁿ is known as polynomial of degree 'n'.

When f(x) = 0, then it is known as Polynomial equation. 

• When f(x) is a polynomial of degree 1, then f(x) = 0 is known as Linear Equation.
• When f(x) is a polynomial of degree 2, then f(x) = 0 is known as Quadratic Equation.

So, general quadratic equation is ax² + bx + c = 0, where a, b, c are real numbers.

If b = 0, then quadratic equation is ax² + c = 0 is known as Pure Quadratic Equation.

Linear Equation 

1. ax + b = 0 ﹣linear equation with one variable
2. ax + by + c = 0 ﹣linear equation with two variable

Simultaneous Linear Equations or System of Equations

Let equations are:- 

                            a₁x + b₁y + c₁ = 0

                            a₂x + b₂y + c₂ = 0

Then, we have

1. a Unique Solution, if   a₁/a₂ ≠ b₁/b₂ (Consistent & independent)
2. an Infinite no. of Solutions, if   a₁/a₂ =  b₁/ b₂ = c₁/c₂  (Consistent & dependent)
3. No Solution, if  a₁/a₂ =  b₁/ b₂ ≠ c₁/c₂  (Inconsistent)

Example:- Find value of 'p'  for which the system of linear equations x + 2y − 3 = 0 & 5x + py +7 = 0 have no solution?

Solution: 

              For no solution, a₁/a₂ = b₁/ b₂ ≠ c₁/c₂ 

                         ⟹ 1/5 = 2/p ≠ ﹣3/7 

                         ⟹ p = 10.

How to find roots or zeroes of a quadratic equation?

1. By factorisation method
If quadratic equation ax² + bx + c = 0 can be expressed in the form (x - ɑ)(x - β) = 0, then the roots or zeroes are ɑ and β.
  where a, b are real numbers & a ≠ 0.

2. By using formula 
The roots are x = - b ± √(b² - 4ac) / 2a
Then, ɑ = - b + √(b² - 4ac) / 2a 
       β = - b - √(b² - 4ac) / 2a

Formation of a quadratic equation:-
x² - x(sum of the roots) + product of roots = 0

Sum & Products of a quadratic equation ax² + bx + c = 0 :-
1. Sum of the roots =  ɑ + β = -b/a 
2. Product of the roots = ɑβ = c/a

Sum & Products of a cubic equation ax³ + bx² + cx + d = 0 , a ≠0 :-
1. ɑ + β +  γ= -b/a 
2. ɑβγ = -d/a
3. ɑβ + βγ + γɑ = c/a

Factor Theorem 
Let f(x) be a polynomial of degree greater than equal to one and 'a' be any real number; such that f(a) = 0, then (x -a) is a factor of f(x).
In other words,  if (x - a) is a factor of f(x), then f(a) = 0.

Remainder Theorem 
Let f(x) be a polynomial of degree greater than equal to one and 'a' be any real number; when f(x) is divided by (x - a), then the remainder is f(a).

Nature of the roots or zeroes

The discriminate 'D' is given by b² - 4ac, determines nature of the roots of quadratic equation ax² + bx + c = 0, 
1. If D = b² - 4ac > 0 and a perfect square,  then the roots are real, rational and unequal. 
2. If D = b² - 4ac > 0 and is not a perfect square,  then the roots are real, irrational and unequal. 
3. If D = b² - 4ac = 0, then the roots are real, rational and equal.
4. If D = b² - 4ac < 0, then the roots are distinct conjugate complex number or imaginary. 

Do it Yourself

1. Find the value of “p” from the polynomial x² + 3x + p, if one of the zeroes of the polynomial is 2.

2. Does the polynomial a⁴ + 4a² + 5 have real zeroes?

3. Compute the zeroes of the polynomial 4x² – 4x – 8. Also, establish a relationship between the zeroes and coefficients. (sum and products)

4. Find the quadratic polynomial if its zeroes are 0, √5. (ans. x² – √5x)

5. Find the value of “x” in the polynomial 2a² + 2xa + 5a + 10 if (a + x) is one of its factors. (Ans. x = 2)

6. How many zeros does the polynomial (x – 3)² – 4 have? Also, find its zeroes. (ans.2)

7. α and β are zeroes of the quadratic polynomial x² – 6x + y. Find the value of ‘y’ if 3α + 2β = 20. (Ans. y= -16)

8. If the zeroes of the polynomial x³ – 3x² + x + 1 are a – b, a, a + b, then find the value of a and b. (by taking sum and product, Ans. a = 1, b = √2)

9. Find a quadratic polynomial whose zeroes are reciprocals of the zeroes of the polynomial f(x) = ax² + bx + c, a ≠ 0, c ≠ 0.

10. If p(x) = x² + 5x + 2, then find p(3) + p(2) + p(0).

11. What is the value of p(x) = x² – 3x – 4 at x = -1 ?

12. Find the value of p for which the polynomial x³ + 4x² – px + 8 is exactly divisible by (x – 2). (Ans.16)

13. If α and β are zereos of the polynomial 2x² – 5x + 7, then find the value of α⁻¹ + β⁻¹. (Ans. 5/7)

14. If p and q are the roots of ax² – bx + c = 0, a ≠ 0, then find the value of p + q.

15. If – 1 is a zero of quadratic polynomial, p(x) = kx² – 5x – 4, then find the value of k. (k = – 1)

16. If one of the zeroes of a quadratic polynomial (k - 1)x² + kx + 1 is -3, then k = ?

17. If one of the zeroes of a cubic polynomial x³ + ax² + bx + c is -1, then product of other two zeroes is:-

18. If m and n are the zeroes of a quadratic polynomial x² + x -2, then the value of (1/m - 1/n) is :-