What is a polynomial
What is a polynomial?
Table of contents
1.What is a polynomial
2.Degree of polynomial
3.Linear polynomial
4.Quadratic polynomial
5.Cubic polynomial
6.Value of polynomial
7.Zeroes of a polynomial
8.Relation between zeroes and coefficients of Polynomial
1.What is a polynomial?
Algebraic expressions involving addition, subtraction or multiplication of variables, constants, coefficients of variables and non-exponential exponents are called polynomials (POLYNOMIAL).
2. Degree of Polynomial
If p (x) p (x) is a polynomial (POLYNOMIAL), the polynomial p (x) p (x) of the variable xx, is called the degree of polynomial power (Power) polynomial.
Suppose that 4x + 24x + 2 is a polynomial.
The power of the variable xx of this polynomial is one (1). Hence, this polynomial is called an exponential polynomial or an exponential polynomial or linear polynomial.
Therefore, the polynomial of power 1 is called a polynomial or linear polynomial.
4. Quadratic Polynomial
Let x2 + x + 2x2 + x + 2 be a polynomial.
The highest power of Variable xx in this polynomial is 2 (two). Hence such a polynomial is called quadratic polynomial or quadratic polynomial.
Hence the polynomial of power 2 (two) is called quadratic polynomial.
5.Cubic Polynomial
The polynomial of power 3 (three) is called CUBIC POLYNOMIAL.
Examples:
x3 + 2x2 − x + 1x3 + 2x2-x + 1, 2 − x32-x3, √2x2x, etc.
Since the highest power of the variable xx in these polynomials is 3 (three), they are all cubic polynomials.
The most widespread form of Cubic Polynomial is:
ax3 + bx2 + cx + dax3 + bx2 + cx + d where a, b, c, da, b, c, d are real numbers and a ≠ 0a ≠ 0.
6. Value of Polynomial
If p (x) is a polynomial in p (x), xx and kk is a real number, p (x) is the real number p (x) p (x) obtained by replacing xx with kk in p (x). x = kx = k is called the value and represent p (k) with p (k).
Example:
Let p (x) = x2−3x − 4p (x) = x2-3x-4
Putting x = 2x = 2 in it we find that
p (2) = 223 × 2−4 = −6p (2) = 223 × 2-4 = -6
The value 22 obtained here is called x = 2x = 2 of p (x) p (x).
7. Zeroes of a Polynomial
A real number kk is called the zero of a polynomial of the polynomial p (x) p (x), if p (k) = 0p (k) = 0.
Examples
Suppose a polynomial p (x) = x2−3x − 4p (x) = x2-3x-4
Putting x = −1x = −1 in this polynomial we find that
p (−1) = (- 1) 2−3 (−1) −4p (-1) = (- 1) 2-3 (-1) -4
⇒p (−1) = 1 + 3−4 = 0⇒p (-1) = 1 + 3-4 = 0
Now by placing x = 4x = 4 in this polynomial, we get
p (4) = 42? (3 × 4) −4p (4) = 42? (3 × 4) -4
⇒p (4) = 16−12−4 = 0⇒p (4) = 16-12-4 = 0
Since here p (−1) = 0p (-1) = 0 and p (4) = 0p (4) = 0
Hence the polynomial given −1-1 and 44 is called the zeroes of x2−3x − 4x2-3x-4.
Broadly, if p (x) = ax + bp (x) = ax + b has a zero kk,
8.Relation between Zeroes and Coefficients of a Polynomial .
If αα and ββ are zeroes of the quadratic polynomial p (x) = ax2 + bx + cp (x) = ax2 + bx + c, a ≠ 0a ≠ 0, then x − αx-α and x − βx-β, p ( x) are factors of p (x).
Hence,
ax2 + bx + cax2 + bx + c = k (x − α) (x − β) = k (x − α) (x − β), where kk is a constant.
= k [x2 (α + β) x + αβ] = k [x2 (α + β) x + αβ]
= kx2 − k (α + β) x + kαβ = kx2-k (α + β) x + kαβ
Comparing the coefficients and constant terms of x2, xx2, x on both sides, we find that
a = ka = k ------- (i)
b = −k (α + β) b = −k (α + β) ---------- (ii)
c = kαβc = kαβ ------ (iii)
Now,
∵ b = −k (α + β) b = −k (α + β)
∴α + β = −bk∴α + β = −bk
Substituting k = ak = a from equation (i) in the above, we get
α + β = −baα + β = -ba ------- (iv)
Now,
∵ c = kαβc = kαβ
∴α + β = ck∴α + β = ck
Substituting k = ak = a from equation (i) in the above, we get
α + β = caα + β = ca --------- (v)
Hence, the sum of zeroes
= α + β == α + β
And the product of zeroes
= αβ = ca = αβ = ca
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