Students can go through AP 9th Class Maths Notes Chapter 2 Polynomials to understand and remember the concepts easily.
Class 9 Maths Chapter 2 Notes Polynomials
What we are going to learn in this chapter are:
- What is polynomial ? What are coefficients?
- Degree of a polynomial
- Types of polynomials
- Zeroes of a polynomial and finding them
- Remainder theorem, factor theorem
- Certain useful identities
- Applications of polynomials, factorisation methods.
→ Constant : Constant is a term (or) coefficient which does not change overtime and has a fixed value.
In general, we use alphabets a, b, c etc. to represent constants.
Ex-1 : 2, -5, 6, a, b, c,
Ex-2: Size of your room (is always fixed)
→ Variables : A variable is denoted by symbols like x, y, z, ………. that can take any
real value.
Ex-1 : x, y, z, ………. etc.
Ex-2 : Size of your pencil (Using), Weight of your daily school bag.
→ Expression : A term or group of terms together is called expression.
Terms are in a form of (constant × variable)
Ex : 2x, 2x5, 4y3, -5z2, ……. etc.
ax, by, cz, ……..
- These are called algebraic expressions.
- In a term (axp)
‘a’ is called coefficient
‘x’ is called base
‘p’ is called exponent. - Exponent of a variable present in a term should be a whole number only.
- It means exponent of a variable in a term cannot be a negative or fraction.
→ Polynomial: A polynomial is an expression consisting of terms in which exponents of variables are whole numbers.
Ex : 2x, 3x2 + 4x, 5x3 – 7x2 + 10x + 6
→ Zero polynomial : Constant ‘zero’ is called zero polynomial, because zero can be expressed as following
0 = 0 (x) or 0 (x2) or 0 (x10) or 0 (x100)
So, ‘0’ is called zero polynomial.
→ Constant polynomial : All non-zero integers are called as constant polynomials.
Ex : 5, 8, -6, 4, ….. etc.
because we can express
5 = 5 (x0); 8 = 8 (y0); -6 = -6 (x0)
So, we call all non-zero integers as con-stant polynomials.
- Polynomials can be written in any (n) variables.
For example,
x + 2 is a polynomial in 1 variable
2x + 3y is a polynomial in 2 variables
4x + 5y – 6z is a polynomial in 3 variables. - Polynomials are denoted by P(x) if the expressions have only (x) as variable.
Ex : P(x) = 2x2 – 3x + 4 - If (y) is the variable in expressions, then that polynomial will be denoted by P(y) or Q(y) etc.
Ex : Q(y) = y3 – 3y2 + 4y - If x, y are two variables in those expressions we denote it P(x, y) or R(x, y) ……… etc.
Ex : R(x, y) = 2x + 3y – 4 - Monomial: If a polynomial has only one term, then that polynomial is called monomial.
Ex : P(x) = 2x; Q(y) = 3y10; R(x) = 2 (∵ 2 = 2x0) - Binomial : If the polynomial has only, two terms, then it is called binomial.
Ex:- 2x + 4 = P(x) (Binomial in one variable)
- Q(y) = 5y10 – 6y (Binomial in one variable)
- R(z) = Z9 + Z2 (Binomial in one variable)
Note : 2 + 3 is not a binomial because 2 + 3 gives us 5 which will be monomial.
→ Trinomial : If a polynomial has three terms in it, then it is called trinomial.
Ex:
P(x) = x2 + 5x + 6 (Trinomial in one variable)
Q(y) = -7y3 + 6y2 + 3y (Trinomial in one variable)
R(z) = z6 – z3 + z100 (Trinomial in one variable)
- All expressions having more than 3 terms are called polynomials (simply).
- Even though monomials, binomials, tri-nomials are polynomials, they have specific meaning.
→ Degree of a polynomial : The highest power (exponent) of all variables in a (non-monomial) polynomial is called degree of that polynomial.
Ex : 5x8 – 6x7 – 4x2 + 6 here powers of variables in each term are 8, 7, 2, 0 (∵ 6 = 6x0)
So, highest power = 8; then degree = 8
Ex : 3x9 – 8x7 – 6x5 – 4x3 – 2x2 + x + 6
In this polynomial exponents (powers) of variable are 9, 7, 5, 3, 2, 1, 0; then ‘9’ is degree of given polynomial.
→ Degree of a constant polynomial: ‘6’ is a constant polynomial. It can be expressed as 6x0. Hence exponent of variable in 6x0 is zero.
Hence degree of all (non-zero) constant polynomial is zero.
→ Degree of a zero polynomial : The constant zero is called zero polynomial. We can express zero as following ways :
0 = 0 (x0) or 0 (x10) or 0 (x100) or 0 (x1000……)
So, we cannot define it.
So, degree of a zero polynomial is not defined.
→ Linear polynomial : If the degree of a polynomial is one. then it is called linear polynomial.
Ex : P(x) = 5x + 6 (Linear polynomial in one variable)
Q(x, y) = 3x + 2y (Linear polynomial in two variables)
R(x, y, z) = 5x – 3y + 2z (Linear polynomial in three variables)
→ Quadratic polynomial: If the degree of a polynomial is two, then it is called quadratic polynomial.
Ex : P(x) = 3x2 – 5x + 6 (Quadratic polynomial in 1 variable)
Q(x, y) = 2x2 – 3y2 + 4x – 5y (Quadratic polynomial in 2 variables)
R(s, t, u) = \(\frac{2}{5}\)s2 – \(\frac{3}{4}\)t + \(\frac{4}{7}\)u (Quadratic polynomial in 3 variables)
No. of possible terms in a polynomial in one variable is 1 more than its degree.
It means highest possible terms in a monomial (one variable) is 1 + 1 = 2.
For a binomial, highest possible terms are 2 + 1 = 3 and
For cubic, highest possible terms are 3 + 1 = 4
For ‘n’ degree polynomial (in one variable) the maximum possible terms are (n + 1).
General form of n-degree polynomial in one variable is
P(x) = a0xn + a1xn – 1 + a2xn – 2 + ………. + an – 1x + an (a0 ≠ 0)
(or)
P(x) = anxn + an – 1xn – 1 + an – 2xn – 2 + ………… + a1x + a0 (an ≠ 0)
where a0, a1 a2, …… an are coefficients and ’n’ is an positive integer.
General form of linear polynomial in one variable is
P(x) = ax + b (a ≠ 0) (or)
= ay + b (a ≠ 0) (or)
= az + b (a ≠ 0)
What happens if a = 0 in above polynomial?
If the coefficient of a term having ‘ highest power becomes zero, then the degree of the polynomial varies (changes to downside). Hence, the coefficient of highest power of any polynomial should not be zero.
General form of linear polynomial in 2 variables P(x) = ax + by + c is
P(x, y) = 2x + 3y + 4
P(r, s) = 5r – 10s + 100
Q(l, m) = 10l + 20m + 30, ……… etc.
General form of
i) a quadratic polynomial in one variable is P(x) = ax2 + bx + c (a ≠ 0).
Ex: Q(x) = 5x2 – 4x + 9
ii) a quadratic polynomial in two variables is P(x, y) = ax2 + bxy + cy2 + dx + ey + f (a ≠ 0)
Ex: P(x, y) = -2x2 + 3xy + 4y2 + 5x – 6y + 7
General form of cubic polynomial in one variable is
P(x, y) = ax3 + bx2 + cx + d (a ≠ 0)
General form of a cubic polynomial in two variables is
P(x, y) = ax3 + bx2y + cxy3 + dy3 + ex2 + fy2 + gx + hy + i.
→ Value of a polynomial at given value of variable: Value of a polynomial at given value can be obtained by substituting given value in the place of variable.
Ex : Let P(x) = 4x2 + 10x + 3
Value of P(x) at x = 5 will be (by replacing)
P(5) = 4(5)2 + 10(5) + 3
= 4(25) + 10(5) + 3
= 100 + 50 + 3
P(5) = 153
∴ If P(x) = 4x2 + 10x + 3, then its value at x = 5 is P(5) = 153
→ Zeroes of a polynomial : ‘a’ is said to be zero of a polynomial P(x) if P(a) = 0 that means if the value of polynomial is zero at a particular value, then this particular value is called zero of given polynomial.
Ex (1) : P(x) = 4x – 20, then
P(2) = 4(2) – 20 = 8 – 20 = -12
P(4) = 4(4) – 20 = 16 – 20 = -4
P(5) = 4(5) – 20 = 20 – 20 = 0
For x = 5, P(x) = 4x – 20 becomes zero.
Hence ’51 is zero of given P(x) = 4x – 20
Ex (2) : P(x) = 4x – 80 we notice P(20)
P(20) = 4(20) – 80 = 80 – 80 = 0
Hence for x = 20, P(x) = 4x – 80 = 0
∴ ’20’ is zero of given P(x) = 4x – 80.
- A non-zero constant polynomial has no zero.
- Every real number is a zero of the zero polynomial.
→ How to find zero of a polynomial ?
Solution:
Let P(x) = ax + b is given polynomial, then ax + b = 0
⇒ ax = 0 – b (shifting) ⇒ x = –\(\frac{b}{a}\)
∴ –\(\frac{b}{a}\) will be zero of given polynomial.
→ How to verify the given is zero of a polynomial or not ?
Solution:
First replace the given value in the place of variable in given polynomial.
If its value becomes zero, then that value is zero of that polynomial, other wise it won’t be a zero.
→ An algebraic expression ¡n which the variables involved have only non-negative integral powers is called a polynomial.
E.g.: 5x3 – 2x + 8
→ Polynomials can be classified as monomials. binomials, trinomials and polynomials based on the number of terms it contains.
→ A polynomial with a single term is a monomial.
E.g.: 2x, -5x2, \(\frac{6}{7}\)x3 etc.
→ A polynomial with two terms is a binomial.
E.g.: 2x + 5y; -3x2 + 5x etc.
→ A polynomial with three terms is a trinomial.
E.g.: 3x2 + 5x – 8; 3x + 2y – 5z etc.
→ In general a polynomial may contain two or more than two terms.
E.g.: 5x + 8x2 – 3x3 + 7
→ Degree of a polynomial ¡s the heighest degree of its variable terms.
E.g.: Degree of 3x2 + 4xy3 + y2 is 4.
→ Degree of a constant term is considered as zero.
E.g.: Degree of 4 is zero.
→ The general form of a polynomial is a0xn + a1xn-1 + a2xn-2 …….. + an-1x + an where a0, a1, a2,…… an-1, an are constants and a0 ≠ 0. Its degree is ‘n’.
→ Polynomials are again classified based on their degrees.
→ The zero of a polynomial p(x) is the value of x at which p(x) becomes zero (i.e.) p(x) = 0.
E.g.: Zero of 3x – 5 is x = \(\frac{5}{3}\)
→ To find the zero of a polynomial we equate the polynomial to zero and solve for the value of the variable.
E.g.: To find zero of 7x + 8.
7x + 8 = 0
7x = – 8
x = \(\frac{-8}{7}\)
→ Let p(x) be any polynomial of degree greater than or equal to one and let ‘a’ be any real number. If p(x) is divided by the linear polynomial (x – a), then the remainder is p(a). This is called the Remainder theorem.
E.g.: If p(x) = 4x3 + 3x + 8 then the remainder when it is divided by x – 1 is p(1).
i.e., p(1) = 4 + 3 + 8 = 15.
→ If p(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number, then
(i) (x – a) is a factor of p(x), if p(a) = 0
(ii) and its converse “if (x – a) is a factor of a polynomial p(x) then p(a) = 0. This is called Factor theorem”.
E.g.: Let p(x) = x2 – 5x + 6 and p(2) = 22 – 5(2)+ 6 = 0 and hence (x – 2) is a factor of p(x) conversely; p(x) = x2 + 7x + 10 and (x + 2) is a factor, then p(-2) = 0.
→ Algebraic identities
- (x + y)2 = x2 + 2xy + y2
- (x – y)2 = x2 – 2xy + y2
- (x + y) (x-y) = x2 – y2
- (x + a) (x + b) = x2 + (a + b) x + ab
- (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
- (x + y)3 = x3 + y3 + 3xy (x + y)
- (x – y)3 = x3 – y3 – 3xy (x – y)
- (x + y + z) (x2 + y2 + z2 – xy – yz – zx) = x3 + y3 + z3 – 3xyz