Iemh 102
Iemh 102
Iemh 102
CHAPTER 2
POLYNOMIALS
2.1 Introduction
You have studied algebraic expressions, their addition, subtraction, multiplication and
division in earlier classes. You also have studied how to factorise some algebraic
expressions. You may recall the algebraic identities :
(x + y)2 = x2 + 2xy + y2
(x – y)2 = x2 – 2xy + y2
and x2 – y2 = (x + y) (x – y)
and their use in factorisation. In this chapter, we shall start our study with a particular
type of algebraic expression, called polynomial, and the terminology related to it. We
shall also study the Remainder Theorem and Factor Theorem and their use in the
factorisation of polynomials. In addition to the above, we shall study some more algebraic
identities and their use in factorisation and in evaluating some given expressions.
Rationalised 2023-24
26 MATHEMATICS
x
1
know that you can write x + = x + x–1? Here, the exponent of the second term, i.e.,
x
x–1 is –1, which is not a whole number. So, this algebraic expression is not a polynomial.
1
1
Again, x + 3 can be written as x 2 + 3 . Here the exponent of x is
, which is
2
not a whole number. So, is x + 3 a polynomial? No, it is not. What about
3 y + y2? It is also not a polynomial (Why?).
Rationalised 2023-24
POLYNOMIALS 27
Rationalised 2023-24
28 MATHEMATICS
Rationalised 2023-24
POLYNOMIALS 29
EXERCISE 2.1
1. Which of the following expressions are polynomials in one variable and which are
not? State reasons for your answer.
2
(i) 4x2 – 3x + 7 (ii) y2 + 2 (iii) 3 t + t 2 (iv) y +
y
(v) x10 + y3 + t50
2. Write the coefficients of x2 in each of the following:
π 2
(i) 2 + x2 + x (ii) 2 – x2 + x3 (iii) x +x (iv) 2 x − 1
2
3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
4. Write the degree of each of the following polynomials:
(i) 5x3 + 4x2 + 7x (ii) 4 – y2
(iii) 5t – 7 (iv) 3
5. Classify the following as linear, quadratic and cubic polynomials:
(i) x2 + x (ii) x – x3 (iii) y + y2 + 4 (iv) 1 + x
2 3
(v) 3t (vi) r (vii) 7x
Example 2 : Find the value of each of the following polynomials at the indicated value
of variables:
(i) p(x) = 5x2 – 3x + 7 at x = 1.
(ii) q(y) = 3y3 – 4y + 11 at y = 2.
(iii) p(t) = 4t4 + 5t3 – t2 + 6 at t = a.
Rationalised 2023-24
30 MATHEMATICS
Rationalised 2023-24
POLYNOMIALS 31
1
Now, 2x + 1 = 0 gives us x = –
2
1
So, – is a zero of the polynomial 2x + 1.
2
Now, if p(x) = ax + b, a ≠ 0, is a linear polynomial, how can we find a zero of
p(x)? Example 4 may have given you some idea. Finding a zero of the polynomial p(x),
amounts to solving the polynomial equation p(x) = 0.
Now, p(x) = 0 means ax + b = 0, a ≠ 0
So, ax = –b
b.
i.e., x= –
a
b
So, x = − is the only zero of p(x), i.e., a linear polynomial has one and only one zero.
a
Now we can say that 1 is the zero of x – 1, and –2 is the zero of x + 2.
Example 5 : Verify whether 2 and 0 are zeroes of the polynomial x2 – 2x.
Solution : Let p(x) = x2 – 2x
Then p(2) = 22 – 4 = 4 – 4 = 0
and p(0) = 0 – 0 = 0
Hence, 2 and 0 are both zeroes of the polynomial x2 – 2x.
Let us now list our observations:
(i) A zero of a polynomial need not be 0.
(ii) 0 may be a zero of a polynomial.
(iii) Every linear polynomial has one and only one zero.
(iv) A polynomial can have more than one zero.
EXERCISE 2.2
1. Find the value of the polynomial 5x – 4x2 + 3 at
(i) x = 0 (ii) x = –1 (iii) x = 2
2. Find p(0), p(1) and p(2) for each of the following polynomials:
(i) p(y) = y2 – y + 1 (ii) p(t) = 2 + t + 2t2 – t3
(iii) p(x) = x3 (iv) p(x) = (x – 1) (x + 1)
Rationalised 2023-24
32 MATHEMATICS
3. Verify whether the following are zeroes of the polynomial, indicated against them.
1 4
(i) p(x) = 3x + 1, x = – (ii) p(x) = 5x – π, x =
3 5
(iii) p(x) = x2 – 1, x = 1, –1 (iv) p(x) = (x + 1) (x – 2), x = – 1, 2
m
(v) p(x) = x2, x = 0 (vi) p(x) = lx + m, x = –
l
1 , 2 1
(vii) p(x) = 3x2 – 1, x = − (viii) p(x) = 2x + 1, x =
3 3 2
4. Find the zero of the polynomial in each of the following cases:
(i) p(x) = x + 5 (ii) p(x) = x – 5 (iii) p(x) = 2x + 5
(iv) p(x) = 3x – 2 (v) p(x) = 3x (vi) p(x) = ax, a ≠ 0
(vii) p(x) = cx + d, c ≠ 0, c, d are real numbers.
Rationalised 2023-24
POLYNOMIALS 33
Example 8 : Factorise 6x2 + 17x + 5 by splitting the middle term, and by using the
Factor Theorem.
Solution 1 : (By splitting method) : If we can find two numbers p and q such that
p + q = 17 and pq = 6 × 5 = 30, then we can get the factors.
So, let us look for the pairs of factors of 30. Some are 1 and 30, 2 and 15, 3 and 10, 5
and 6. Of these pairs, 2 and 15 will give us p + q = 17.
Rationalised 2023-24
34 MATHEMATICS
2 17 5
6x2 + 17x + 5 = 6 x + x + = 6 p(x), say. If a and b are the zeroes of p(x), then
6 6
5
6x2 + 17x + 5 = 6(x – a) (x – b). So, ab = . Let us look at some possibilities for a and
6
1 1 5 5 1 1 17 1 5
b. They could be ± , ± , ± , ± , ± 1 . Now, p = + + ≠ 0. But
2 3 3 2 2 4 6 2 6
−1 1
p = 0. So, x + is a factor of p(x). Similarly, by trial, you can find that
3 3
5
x + is a factor of p(x).
2
1 5
Therefore, 6x2 + 17x + 5 = 6 x + x +
3 2
3x + 1 2 x + 5
= 6
3 2
= (3x + 1) (2x + 5)
For the example above, the use of the splitting method appears more efficient. However,
let us consider another example.
Rationalised 2023-24
POLYNOMIALS 35
Also, p(3) = 32 – (5 × 3) + 6 = 0
So, y – 3 is also a factor of y2 – 5y + 6.
Therefore, y2 – 5y + 6 = (y – 2)(y – 3)
Note that y2 – 5y + 6 can also be factorised by splitting the middle term –5y.
Now, let us consider factorising cubic polynomials. Here, the splitting method will not
be appropriate to start with. We need to find at least one factor first, as you will see in
the following example.
EXERCISE 2.3
1. Determine which of the following polynomials has (x + 1) a factor :
(i) x3 + x2 + x + 1 (ii) x4 + x3 + x2 + x + 1
Rationalised 2023-24
36 MATHEMATICS
Rationalised 2023-24
POLYNOMIALS 37
5 y5 y
= x+ x−
2 32 3
So far, all our identities involved products of binomials. Let us now extend the Identity
I to a trinomial x + y + z. We shall compute (x + y + z)2 by using Identity I.
Let x + y = t. Then,
(x + y + z)2 = (t + z)2
= t2 + 2tz + t2 (Using Identity I)
= (x + y)2 + 2(x + y)z + z2 (Substituting the value of t)
Rationalised 2023-24
38 MATHEMATICS
Rationalised 2023-24
POLYNOMIALS 39
Rationalised 2023-24
40 MATHEMATICS
EXERCISE 2.4
1. Use suitable identities to find the following products:
(i) (x + 4) (x + 10) (ii) (x + 8) (x – 10) (iii) (3x + 4) (3x – 5)
3 3
(iv) (y2 + ) (y2 – ) (v) (3 – 2x) (3 + 2x)
2 2
2. Evaluate the following products without multiplying directly:
(i) 103 × 107 (ii) 95 × 96 (iii) 104 × 96
3. Factorise the following using appropriate identities:
y2
(i) 9x2 + 6xy + y2 (ii) 4y2 – 4y + 1 (iii) x2 –
100
Rationalised 2023-24
POLYNOMIALS 41
1 9 1
(v) 27p3 – – p2 + p
216 2 4
9. Verify : (i) x3 + y3 = (x + y) (x2 – xy + y2) (ii) x3 – y3 = (x – y) (x2 + xy + y2)
10. Factorise each of the following:
(i) 27y3 + 125z3 (ii) 64m3 – 343n3
[Hint : See Question 9.]
11. Factorise : 27x3 + y3 + z3 – 9xyz
1
12. Verify that x3 + y3 + z3 – 3xyz = ( x + y + z ) ( x − y )2 + ( y − z )2 + ( z − x)2
2
13. If x + y + z = 0, show that x3 + y3 + z3 = 3xyz.
14. Without actually calculating the cubes, find the value of each of the following:
(i) (–12)3 + (7)3 + (5)3
(ii) (28)3 + (–15)3 + (–13)3
15. Give possible expressions for the length and breadth of each of the following
rectangles, in which their areas are given:
(i) (ii)
Rationalised 2023-24
42 MATHEMATICS
16. What are the possible expressions for the dimensions of the cuboids whose volumes
are given below?
(i) (ii)
2.6 Summary
In this chapter, you have studied the following points:
1. A polynomial p(x) in one variable x is an algebraic expression in x of the form
p(x) = anxn + an–1xn – 1 + . . . + a2x2 + a1x + a0,
where a0, a1, a2, . . ., an are constants and an ≠ 0.
a0, a1, a2, . . ., an are respectively the coefficients of x0, x, x2, . . ., xn, and n is called the degree
of the polynomial. Each of anxn, an–1 xn–1, ..., a0, with an ≠ 0, is called a term of the polynomial
p(x).
2. A polynomial of one term is called a monomial.
3. A polynomial of two terms is called a binomial.
4. A polynomial of three terms is called a trinomial.
5. A polynomial of degree one is called a linear polynomial.
6. A polynomial of degree two is called a quadratic polynomial.
7. A polynomial of degree three is called a cubic polynomial.
8. A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0. In this case, a is also called a
root of the equation p(x) = 0.
9. Every linear polynomial in one variable has a unique zero, a non-zero constant polynomial
has no zero, and every real number is a zero of the zero polynomial.
10. Factor Theorem : x – a is a factor of the polynomial p(x), if p(a) = 0. Also, if x – a is a factor
of p(x), then p(a) = 0.
11. (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
12. (x + y)3 = x3 + y3 + 3xy(x + y)
13. (x – y)3 = x3 – y3 – 3xy(x – y)
14. x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)
Rationalised 2023-24