Students should go through these JAC Class 9 Maths Notes Chapter 2 Polynomials will seemingly help to get a clear insight into all the important concepts.

### JAC Board Class 9 Maths Notes Chapter 2 Polynomials

**Polynomials:**

An algebraic expression f(x) of the form f(x) = a_{0} + a_{1}x + a_{2}x^{2} + …… + a_{n}x^{n}, where a_{0}, a_{1}, a_{2} ……, a_{n} are real numbers and all the index of x’ are nonnegative integers is called a polynomial in x.

→ Degree of a Polynomial: Highest Index of x in algebraic expression is called the degree of the polynomial, here a_{0}, a_{1}x, a_{2}x^{2} ….. a_{n}x^{n}, are called the terms of the polynomial and a_{0}, a_{1}, a_{2}, …… a_{n} are called various coefficients of the polynomial f(x).

Note: A polynomial in x is said to be in standard form when the terms are written either in increasing order or decreasing order of the indices of x in various terms.

→ Different Types of Polynomials: Generally, we divide the polynomials in the following categories.

→ Based on degrees:

There are four types of polynomials based on degrees. These are listed below:

- Linear Polynomials: A polynomial of degree one is called a linear polynomial. The general form of linear polynomial is ax + b, where a and b are any real constant and a ≠ 0.
- Quadratic Polynomials: A polynomial of degree two is called a quadratic polynomial. The general form of a quadratic polynomial is ax
^{2}+ bx + c, where a ≠ 0, a, b, c ∈ R. - Cubic Polynomials: A polynomial of degree three is called a cubic polynomial. The general form of a cubic polynomial is ax
^{3}+ bx^{2}+ cx + d, where a ≠ 0 and a, b, c, d ∈ R. - Biquadratic (or quadric) Polynomials: A polynomial of degree four is called a biquadratic (quadric) polynomial. The general form of a biquadratic polynomial is ax
^{4}+ bx^{3}+ cx^{2}+ dx + e, where a ≠ 0 and a, b, c, d, e are real numbers.

Note: A polynomial of degree five or more than five does not have any particular name. Such a polynomial usually called a polynomial of degree five or six or ….etc.

→ Based on number of terms:

There are three types of polynomials based on number of terms. These are as follow:

- Monomial: A polynomial is said to be monomial if it has only one term. e.g. x, 9x
^{2}, 5x^{3}all are monomials. - Binomial: A polynomial is said to be binomial if it contains only two terms e.g. 2x
^{2}+ 3x, \(\sqrt{3}\)x + 5x^{3}, -8x^{3}+ 3, all are binomials. - Trinomial: A polynomial is said to be a trinomial if it contains only three terms.e.g. 3x
^{3}– 8x + \(\frac{1}{2}\), \(\sqrt{7}\) x^{10}+ 8x^{4}– 3x^{2}, 5 – 7x + 8x^{9}, are all trinomials.

Note: A polynomial having four or more than four terms does not have particular name. These are simply called polynomials.

→ Zero degree polynomial: Any non-zero number (constant) is regarded as polynomial of degree zero or zero degree polynomial. i.e. f(x) = a. where a ≠ 0 is a zero degree polynomial, since we can write f(x) = a, as f(x) = ax^{0}.

→ Zero polynomial: A polynomial whose all coefficients are zero is called as zero polynomial i.e. f(x) = 0, we cannot determine the degree of zero polynomial.

**Algebraic Identities:**

An identity is an equality which is true for all values of the variables.

Some important identities are:

(i) (a + b)^{2} = a^{2} + 2ab + b^{2}

(ii) (a – b)^{2} = a^{2} – 2ab + b^{2}

(iii) a^{2} – b^{2} = (a + b)(a – b)

(iv) a^{3} + b^{3} = (a + b)(a^{2} – ab + b^{2})

(v) a^{3} – b^{3} = (a – b)(a^{2} + ab + b^{2})

(vi) (a + b)^{3} = a^{3} + b^{3} + 3ab (a + b)

(vii) (a – b)^{3} = a^{3} – b^{3} – 3ab (a – b)

(viii) a^{4} + a^{2}b^{2} + b^{4} = (a^{2} + ab + b^{2})(a^{2} – ab + b^{2})

(ix) a^{3} + b^{3} + c^{3} – 3abc = (a + b + c)(a^{2} + b^{2} + c^{2} – ab – bc – ac)

Special case: if a + b + c = 0 then a^{3} + b^{3} + c^{3} = 3abc.

Other Important Identities

(i) a^{2} + b^{2} = (a + b)^{2} – 2ab,

if a + b and ab are given

(ii) a^{2} + b^{2} = (a – b)^{2} + 2ab

if a – b and ab are given

(iii) a + b = \(\sqrt{(a-b)^2+4 a b}\)

if a – b and ab are given

(iv) a – b = \(\sqrt{(a+b)^2-4 a b}\)

if a + b and ab are given

**Factors Of A Polynomial:**

→ If a polynomial f(x) can be written as a product of two or more other polynomials f_{1}(x), f_{2}(x), f_{3}(x)…. then each of the polynomials f_{1}(x), f_{2}(x), f_{3}(x)….. is called a factor of polynomial f(x). The method of finding the factors of a polynomial is called factorisation.

**Zeroes Of A Polynomial:**

→ A real number α is a zero of polynomial f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + a_{n-2}x^{n-2 }+ ….. +a_{1}x + a_{0}, if f(α) = 0. i.e. a_{n}α^{n} + a_{n-1}α^{n-1} + a_{n-2}α^{n-2}+ ….. + a_{1}α + a_{0} = 0.

For example x = 3 is a zero of the polynomial f(x) = x^{3} – 6x^{2} + 11x – 6, because f(3) = (3)^{3} – 6(3)^{2} + 11(3) – 6 = 27 – 54 + 33 – 6 = 0.

but x = -2 is not a zero of the above mentioned polynomial,

∵ f(-2) = (-2)^{3} – 6(-2)^{2} + 11(-2) – 6

f(-2) = -8 – 24 – 22 – 6

f(-2) = -60 ≠ 0.

→ Value of a Polynomial: The value of a polynomial f(x) at x = a is obtained by substituting x a in the given polynomial and is denoted by f(a). Eg if f(x) = 2x^{3} – 13x^{2} + 17x + 12 then its value at x = 1 is

f(1) = 2(1)^{3} – 13(1)^{2} + 17(1) + 12

= 2 – 13 + 17 + 12 = 18.

**Remainder Theorem:**

Let ‘p(x)’ be any polynomial of degree greater than or equal to one and ‘a’ be any real number and if p(x) is divided by (x – a). then the remainder is equal to p(a). Let q(x) be the quotient and r(x) be the remainder when p(x) is divided by (x – a), then

Dividend = Divisor × Quotient + Remainder

∴ p(x) = (x – a) × q(x) + [r(x) or r], where r(x) = 0 or degree of r(x) < degree of (x – a). But (x – 2) is a polynomial of degree 1 and a polynomial of degree less than 1 is a constant. Therefore, either r(x) = 0 or r(x) = Constant. Let r(x) = r, then p(x) = (x – a)q(x) + r.

Putting x = a in above equation, p(a)

p(a) = (a – a)q(a) + r = 0 × q(a) + r

p(a) = 0 + r

⇒ p(a) = r

This shows that the remainder is p(a) when p(x) is divided by (x – a).

Remark: If a polynomial p(x) is divided by (x + a),(ax – b), (ax + b), (b – ax) then the remainder is the value of p(x) at x.

= \(-a, \frac{b}{a},-\frac{b}{a}, \frac{b}{a} \text { i.e. } p(-a)\)

\(p\left(\frac{b}{a}\right), p\left(-\frac{b}{a}\right), p\left(\frac{b}{a}\right)\) respectively.

**Factor Theorem:**

Let ‘p(x)’ be a polynomial of degree greater than or equal to 1 and ‘a’ be a real number such that p(a) = 0, then (x – a) is a factor of p(x). Conversely, if(x – a) is a factor of p(x). then p(a) = 0.

**Factorisation Of A Quadratic Polynomial:**

→ For factorisation of a quadratic expression ax^{2} + bx + c where a ≠ 0, there are two methods.

→ By Method of Completion of Square:

In the form ax^{2} + bx + c where a ≠ 0, firstly we take ‘a’ common in the whole expression then factorise by converting the expression \(a\left\{x^2+\frac{b}{a} x+\frac{c}{a}\right\}\) as the difference of two squares, which is

→ By Splitting the Middle Term:

→ x^{2} + lx + m = x^{2} + (a + b)x + ab

Where l = a + b and m = ab, such that a and b are real numbers

= x^{2} + ax + bx + ab

= x (x + a) + b (x + a)

= (x + a) (x + b)

Method: We express l as the sum of two such numbers whose product is m.

→ ax^{2} + bx + c = prx^{2} + (ps + qr)x + qs

where b = ps + qr, a = pr, c = qs

so that (ps) (gr) (pr) (qs) = ac

∴ prx^{2} + (ps + qr)x + qs

= prx^{2} + psx + qrx + qs

= px (rx + s) + q(rx + s)

= (px + q) (rx + x)

Method: We express b as the sum of two such numbers whose product is ac.

→ Integral Root Theorem:

If f(x) is a polynomial with integral coefficient and the leading coefficient is 1, then any integral root of f(x) is a factor of the constant term. Thus if f(x) = x^{3} – 6x^{2} + 11x – 6 has an Integral root, then it is one of the factors of 6 which are ±1, ±2, ±3, ±6.

Now in fact,

f(1) = (1)^{3} – 6(1)^{2} + 11(1) – 6 = 1 – 6 + 11 – 6 = 0

f(2) = (2)^{3} – 6(2)^{2} + 11(2) – 6

= 8 – 24 + 22 – 6 = 0

f(3) = (3)^{3} – 6(3)^{2} + 11(3) – 6

27 – 54 + 33 – 6 = 0

Therefore Integral roots of f(x) are 1, 2, 3.

→ Rational Root Theorem:

Let \(\frac{b}{c}\) be a rational fraction in lowest terms. If \(\frac{b}{c}\) is a rational root of the polynomial f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} +…+ a_{1}x + a_{0}, an ≠ 0 with integral coefficients, then b is a factor of constant term a_{0}, and C is a factor of the leading coefficient a_{n}.

For example: If \(\frac{b}{c}\) is a rational root of the polynomial f(x) = 6x^{3} + 5x^{2} – 3x – 2, then the values of b are limited to the factors of -2, which are ±1, ±2 and the values of care limited to the factors of 6, which are ±1, ±2, ±3, ±6. Hence, the possible rational roots of f(x) are ±1, ±2, \(\pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm \frac{2}{3}\). In fact -1 is an integral root and \(\frac{2}{3}\), –\(\frac{1}{2}\) are the rational roots of f(x) = 6x^{3} + 5x^{2} – 3x – 2.

Note: (i) n^{th} degree polynomial can have at most n real roots.

→ Finding a zero of polynomial f(x) means solving the polynomial equation f(x) = 0. It follows from the above discussion that if f(x) = ax + b, a ≠ 0 is a linear polynomial, then it has only one zero given by

f(x) = 0 i.e. f(x) = ax + b = 0

⇒ ax = -b

⇒ x = –\(\frac{b}{a}\)

Thus, x = –\(\frac{b}{a}\) is the only zero of f(x) = ax + b.

→ If a polynomial of degree n has more than n zeros then all the coefficients of powers of x including constant term of polynomial are zero.