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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.
The coefficient of x3 in the polynomial 5 + 2x + 3x2 – 7x3 is
The quadratic polynomial whose sum of zeroes is 3 and the product of zeroes is –2 is :
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.
When the polynomial x3 + 3x2 + 3x + 1 is divided by x + 1, the remainder is :-
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
If the polynomial 2x3 – 3x2 + 2x – 4 is divided by x – 2, then the remainder is :-
The value of k for which x – 1 is a factor of the polynomial 4x3+ 3x2 – 4x + k is :-
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
The value of k for which x + 1 is a factor of the polynomial x3 + x2 + x + k is :-
The value of m for which x – 2 is a factor of the polynomial x4 – x3 + 2x2 – mx + 4 is :-
2x2 – 3x – 2
2x2 - 4x + x - 2 = 0
2x2(x-2) +1(x-2) = 0
(2x+1) (x-2)
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 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
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