Test: Polynomials - 1

A cubic polynomial is a polynomial of degree 3. A univariate cubic polynomial has the form f(x) = a₃ x³ + a₂ x² + a₁ x + a₀. An equation involving a cubic polynomial is called a cubic equation. A closed-form solution known as the cubic formula exists for the solutions of an arbitrary cubic equation.

A polynomial of degree 5 is of the form p(x) =, where a, b, c, d, e, and f are real numbers and a ≠ 0.

Thus, p(x) can have at most 6 terms and at least one term containing. 

Sum of zeros = 3/1 
-b/a = 3/1 .....................(1)

Product of zeros = -2/1
c/a = -2/1 ...................(2)

From equation (1) and (2)
a = 1
-b = 3, b = -3
c = -2

The required quadratic equation is 
ax^2+by+c
= x²-3x-2

A number of zeroes of an n -degree polynomial = n. 

First, a linear polynomial is in the form of ax + b, a≠0, a,b ∈R 

The degree of the polynomial = highest degree of the terms 
So here the highest degree is 1. 

Hence, Linear polynomial has only one zero.

The zero of x + 1 is –1

                And by remainder theorem, when

                p(x) = x³ + 3x² + 3x + 1 is divided by x + 1, then remainder is p(–1).

                ∴ p(–1) = (–1)³ + 3 (–1)² + 3(–1) + 1

                = –1 + (3 × 1) + (–3) + 1

                = –1 + 3 – 3 + 1

                = 0

                Thus, the required = 0

X - 1 is a factor of 4x³ + 3x² -4x +k
then x=1 is one root of 4x³ + 3x² -4x +k

put x= 1
4x³ +3x² -4x +k = 0

=> 4 (1)³ +3 (1)²-4 (1) +k =0

=> 4 + 3 - 4 + k = 0

=> k = -3 

We know that x³ + y³ + z³ – 3xyz = (x + y + z) (x² + y² + z² – xy – yz – zx).
If x + y + z = 0, then x³ + y³ + z³ – 3xyz = 0 or x³ + y³ + z³ = 3xyz.

Let x = (a – b), y = (b – c) and z = (c – a)
Consider, x + y + z = (a – b) + (b – c) + (c – a) = 0
⇒ x³ + y³ + z³ = 3xyz
That is (a – b)³ + (b – c)³ + (c – a)³ = 3(a – b)(b – c)(c – a)

The highest power of the variable is known as the degree of the polynomial.

√2x^0 = √2
hence the degree of the polynomial is zero.

The degree of the polynomial 4x⁴+0x³+0x⁵+5x+74x⁴+0x³+0x⁵+5x+7 is

The degree of zero polynomial is not defined, because, in zero polynomial, the coefficient of any variable is zero i.e., Ox² or Ox⁵, etc. Hence, we cannot exactly determine the degree of the variable.  

 Let p (x) = 5x – 4x2 + 3 …(i)
On putting x = -1 in Eq. (i), we get
p(-1) = 5(-1) -4(-1)2 + 3= - 5 - 4 + 3 = -6

 p(x)=x+3

 p(-x)=-x+3

p(x)+p(-x)=x+3-x+3=6