`text(Polynomial :)`
An expression of the type `P_n(x) = a_0x^ + a_1 x^(n-1) + ...... + a_n` is called a polynomial of degree `'n'`, where all powers of `x` are non-negative integers and `a_0` which is called leading coefficient of the polynomial should not be equal to zero.
`=>` If co-efficients `a_0, a_1, a_2 ...... a`, are real then polynomial is called real polynomial and if co-efficients are in the form of `(a + ib)` then it is called complex polynomial.
e.g., : `(2 + 3i)x^3 + 5x^2 + 6x + 3` is called a complex polynomial.
If `n = 1` then `P(x) = a_0x + a_1` is called a linear polynomial.
If `n = 2` then `P(x) = a_0x^2 + a_1x + a_2` is called a quadratic polynomial.
If `n = 3` then` P(x) = a_0x^3 +a_1x^2 + a_2x +a_3` is called a cubic polynomial.
If `n = 4` then `P(x) = a_0x^4 + a_1x^3 + a_2x^2 + a_3x + a_4` is called a bi-quadratic polynomial.
`=> P_n (alpha)` means value of the polynomial `P_n(x)` at `x = a`.
lf `P_n(alpha) = 0`, then a is called as root or zero of the polynomial.
`text(Degree of Polynomial : )`
The highest power of variable (x) present in the polynomial is called the degree of the polynomial.
For example `P_n(x) = a_0x^ + a_1 x^(n-1) + ...... + a_n` is a polynomial in x of degree n.
`text(Important Results : )`
1.A polynomial equation has at.least one root.
2.A polynomial equation of degree n has n roots.
3. The real roots of an equation `f(x) = 0` are the values of x, where the curve `y = f(x)` crosses X-axis.
`text(Remainder Theorem :)`
The remainder theorem states that if a polynomial `P(x)` is divided by a linear function `x - k`, then the remainder is `P(k)`.
`(P(x))/(x-k)=Q(x)+R/(x-k)` where `Q(x)` is quotient and `R` is remamder.
`=> P(x) = Q(x) (x - k) + R` at `x = k , P(k) = R`
`text(Factor Theorem :)`
Let `P(x) = (x - k) Q(x) + R`
when `P(k) = 0, P(x) = (x - k) Q(x)`. Therefore, `P(x)` is exactly divisible by `x - k`.
`text(Note : )`
=>Every equation of an odd degree has atleast one real root, whose sign is opposite to that of its last term, provided that the coefficient of the first term is positive.
=> Every equation of an even degree has atleast two real roots, one positive and one negative, whose last term is negative, provided that the coefficient of the first term is positive.
=> If an equation has no odd powers of x, then all roots of the equation are complex provided all the coefficients of the equation have positive sign.
=>Let f(x) = 0 be a polynomial equation and `lambda,mu ` are two real numbers. Then, `f(x) = 0` will have atleast one real root or an odd number of roots between `lambda ` and `mu .` if `f(lambda)` and `f(mu)` are of opposite signs. But if `f(lambda)` and `f(mu)` are of same signs, then either f(x) = 0 has no real roots or an even number of roots between `f(lambda)` and `f(mu) .`
`text(Quadratic Expression and Quadratic Equation :)`
A second degree expression in one variable contains the variable with an exponent of `2` ; but not higher power. Such expressions are called as quadratic expression.
`=>` e.g., : `y = ax^2 + bx + c` ,
where `a =` leading coefficient & `c =` absolute term of quadratic polynomial.
`=>` If above is equated to zero called as quadratic equation.
e.g., : `ax^2 + bx + c = 0 ; a ne 0`
`=>` If leading coefficient is `I` then polynomial is called monic polynomial.
Solving a quadratic equation means finding the values of `x` for which `ax^2 + bx + c` vanishes and these values of `x` are also called the roots of quadratic equation.
`text(Identity :)`
If two expressions are equal for all values of x, then the statement of equality between the two expressions is called an identity.
e.g. For example, `(x + 1)^2 = x^2 + 2x + 1` is an identity in x.
Let `ax^2 + bx + c = 0` be a quadratic equation. Now, if this quadratic equation has more than two distinct roots then it becomes an identity and in this case `a= b = c = 0` .
Note: Identity is an equation which is true for all values of `x`.
Let us say `alpha , beta, gamma` are three distinct roots of the given quadratic equation. Then,
`ax^2 + bx + c = k (x - alpha) (x - beta ) (x - gamma)` , for some constant `k`.
`=> ax^2 +bx + c = k [ x^3 - (alpha + beta + gamma) x^2 + ( alpha beta + beta gamma + gamma alpha) x - alpha beta gamma]`
Comparing the co-efficient of `x^3` on both sides, we get `k = 0`
and `k = 0 => a = 0, b = 0` and `c = 0`
`=>` If a quadratic equation is satisfied by more than two distinct values of `x`, then all the co-efficients must be zero. And when all the coefficients are zero, quadratic equation is true for all `x in R` and hence, it becomes an identity.
`text(Polynomial :)`
An expression of the type `P_n(x) = a_0x^ + a_1 x^(n-1) + ...... + a_n` is called a polynomial of degree `'n'`, where all powers of `x` are non-negative integers and `a_0` which is called leading coefficient of the polynomial should not be equal to zero.
`=>` If co-efficients `a_0, a_1, a_2 ...... a`, are real then polynomial is called real polynomial and if co-efficients are in the form of `(a + ib)` then it is called complex polynomial.
e.g., : `(2 + 3i)x^3 + 5x^2 + 6x + 3` is called a complex polynomial.
If `n = 1` then `P(x) = a_0x + a_1` is called a linear polynomial.
If `n = 2` then `P(x) = a_0x^2 + a_1x + a_2` is called a quadratic polynomial.
If `n = 3` then` P(x) = a_0x^3 +a_1x^2 + a_2x +a_3` is called a cubic polynomial.
If `n = 4` then `P(x) = a_0x^4 + a_1x^3 + a_2x^2 + a_3x + a_4` is called a bi-quadratic polynomial.
`=> P_n (alpha)` means value of the polynomial `P_n(x)` at `x = a`.
lf `P_n(alpha) = 0`, then a is called as root or zero of the polynomial.
`text(Degree of Polynomial : )`
The highest power of variable (x) present in the polynomial is called the degree of the polynomial.
For example `P_n(x) = a_0x^ + a_1 x^(n-1) + ...... + a_n` is a polynomial in x of degree n.
`text(Important Results : )`
1.A polynomial equation has at.least one root.
2.A polynomial equation of degree n has n roots.
3. The real roots of an equation `f(x) = 0` are the values of x, where the curve `y = f(x)` crosses X-axis.
`text(Remainder Theorem :)`
The remainder theorem states that if a polynomial `P(x)` is divided by a linear function `x - k`, then the remainder is `P(k)`.
`(P(x))/(x-k)=Q(x)+R/(x-k)` where `Q(x)` is quotient and `R` is remamder.
`=> P(x) = Q(x) (x - k) + R` at `x = k , P(k) = R`
`text(Factor Theorem :)`
Let `P(x) = (x - k) Q(x) + R`
when `P(k) = 0, P(x) = (x - k) Q(x)`. Therefore, `P(x)` is exactly divisible by `x - k`.
`text(Note : )`
=>Every equation of an odd degree has atleast one real root, whose sign is opposite to that of its last term, provided that the coefficient of the first term is positive.
=> Every equation of an even degree has atleast two real roots, one positive and one negative, whose last term is negative, provided that the coefficient of the first term is positive.
=> If an equation has no odd powers of x, then all roots of the equation are complex provided all the coefficients of the equation have positive sign.
=>Let f(x) = 0 be a polynomial equation and `lambda,mu ` are two real numbers. Then, `f(x) = 0` will have atleast one real root or an odd number of roots between `lambda ` and `mu .` if `f(lambda)` and `f(mu)` are of opposite signs. But if `f(lambda)` and `f(mu)` are of same signs, then either f(x) = 0 has no real roots or an even number of roots between `f(lambda)` and `f(mu) .`
`text(Quadratic Expression and Quadratic Equation :)`
A second degree expression in one variable contains the variable with an exponent of `2` ; but not higher power. Such expressions are called as quadratic expression.
`=>` e.g., : `y = ax^2 + bx + c` ,
where `a =` leading coefficient & `c =` absolute term of quadratic polynomial.
`=>` If above is equated to zero called as quadratic equation.
e.g., : `ax^2 + bx + c = 0 ; a ne 0`
`=>` If leading coefficient is `I` then polynomial is called monic polynomial.
Solving a quadratic equation means finding the values of `x` for which `ax^2 + bx + c` vanishes and these values of `x` are also called the roots of quadratic equation.
`text(Identity :)`
If two expressions are equal for all values of x, then the statement of equality between the two expressions is called an identity.
e.g. For example, `(x + 1)^2 = x^2 + 2x + 1` is an identity in x.
Let `ax^2 + bx + c = 0` be a quadratic equation. Now, if this quadratic equation has more than two distinct roots then it becomes an identity and in this case `a= b = c = 0` .
Note: Identity is an equation which is true for all values of `x`.
Let us say `alpha , beta, gamma` are three distinct roots of the given quadratic equation. Then,
`ax^2 + bx + c = k (x - alpha) (x - beta ) (x - gamma)` , for some constant `k`.
`=> ax^2 +bx + c = k [ x^3 - (alpha + beta + gamma) x^2 + ( alpha beta + beta gamma + gamma alpha) x - alpha beta gamma]`
Comparing the co-efficient of `x^3` on both sides, we get `k = 0`
and `k = 0 => a = 0, b = 0` and `c = 0`
`=>` If a quadratic equation is satisfied by more than two distinct values of `x`, then all the co-efficients must be zero. And when all the coefficients are zero, quadratic equation is true for all `x in R` and hence, it becomes an identity.