Ex 1. : Consider a polynomial f(x) = 3x2 - 4x + 2, find the value at x = 3.
Sol. replace x by 3 everywhere
So, the value of f(x) = 3x2 - 4x + 2, at x = 3 is
f(3) = 3 x 32 - 4 x 3 + 2 = 27 -12 + 2 = 17
Similarly the value of polynomiak f(x) = 3x2 - 4x + 2
at x = -2 is,
at x = 0 is,
at x = 1/2 is,
DO YOUR SELF
Find the value of polynomial 5x - 4x2 + 3 at :
(i) x = 0
(ii) x = -1
(iii) x = 2
(iv) ZEROES OF A POLYNOMIAL
A real number α is a zero of a polynomial p(x) if the value of the polynomial p(x) is zero at x = α. i.e. p(α) = 0
OR
The value of the variable x, for which the polynomial p(x) becomes zero is called zero of the polynomial.
Ex : consider, a polynomial p(x) = x2 - 5x + 6 ; replace x by 2 and 3.
p(2) = (2)2 - 5 × 2 + 6 = 4 - 10 + 6 = 0,
p(3) = (3)2 - 5 × 3 + 6 = 9 - 15 + 6 = 0
∴ 2 and 3 are the zeroes of the polynomial p(x).
REMARK
1. The constant polynomial has no zero.
2. Every linear polynomial has one and only one zero or root.
consider a linear polynomial p(x) = ax + b, a ≠ 0 ⇒ p(x) = 0 ⇒ ax + b = 0 ⇒ ax = -b ⇒ x = -b/a is a zero of the polynomial.
3. A given polynomial can have more than one zero or root.
4. If the degree of a polynomial is n, the maximum number of zeroes it can have is also n.
Ex. : If the degree of a polynomial is 5, the polynomial can have at the most 5 zeroes, if the degree of
polynomial is 8, maximum number of zeroes it can have is 8.
5. A zero of a polynomial need not be 0
Ex . : Consider a polynomial f(x) = x2 - 4, then f(2) = (2)2 - 4 = 4 - 4 = 0 here, zero of the polynomial
f(x) = x2 - 4 is 2 which itself is not 0.
6. 0 may be zero of the polynomial.
Ex. Consider a polynomial f(x) = x2 - x, then f(0) = 02 - 0 = 0 here, 0 is the zero of the polynomial
f(x) = x2 - x
Ex 2. Find which of the following algebraic expression is a polynomial.
(i)
(iv)
Sol. (i) Is a polynomial in one variable.
(ii) = x1 + x-1 Is not a polynomial as the second term 1/x has degree (-1).
(iii) = y1/2 - 8 Is not a polynomial since, the power of the first term √y is 1/2, which is not a whole number.
(iv)
Since, the exponent of the second term is 1/3, which in not a whole number. Therefore, the given expression is not a polynomial.
Ex 3. Find the degree of the polynomial :
(i) 5x2 - 6x3 + 8x7 + 6x2
(ii) 2y12 + 3y10 - y15 + y + 3
(iii) x
(iv) 8
Sol. (i) Since the term with highest exponent (power) is 8x7 and its power is 7.
∴ The degree of given polynomial is 7.
(ii) The highest power of the variable is 15. ⇒ degree = 15
(iii) x = x1 ⇒ degree is 1.
(iv) 8 = 8x0 ⇒ degree is 0.
Ex 4. Write the coefficient of x2 in each of the following :
(i)
(ii)
(iii)
(iv)
Sol. (i) In the polynomial the coefficient of x2 is 1.
(ii) In the polynomial the coefficient of x2 is - 1.
(iii) In the polynomial the coefficient of x2 is 1/2.
(iv) In the polynomial the coefficient of x2 = 0. [We may write = ax2 +. So the
coefficient of x2 is 0]
Ex 5. Classify the following as linear, quadratic and cubic polynomials :
(i) x2 + x
(ii) x - x3
(iii) 1 + x
(iv) 3t
(v) r2
(vi) 7x3
(vii) y + y2 + 5
(viii) 3xyz
Sol. (i) The polynomial x2 + x is a quadratic polynomial as its degree is 2.
(ii) Degree of the polynomial x - x3 is 3. It is a cubic polynomial
(iii) Degree of the polynomial 1 + x is 1. It is a linear polynomial.
(iv) Degree of the polynomial 3t is 1. It is a linear polynomial.
(v) Degree of the polynomial r2 is 2. It is a quadratic polynomial.
(vi) Degree of the polynomial 7x3 is 3. It is a cubic polynomial.
(vii)Degree of the polynomial y + y2 + 5 is 2. It is a quadratic polynomial.
(viii) 3xyz is a polynomial in 3 variables x, y and z. Its degree is 1 + 1 + 1 = 3. It is a cubic polynomial.
Ex 6. Find q(0), q(1) and q(2) for each of the following polynomials :
(i) q(x) = x2 + 3x
(ii) q(y) = 2 + y + 2y2 - 5y3
(iii) q(t) = t3
Sol.
q (x) = x2 + 3x
q(0) - (0)2 + 3 x 0 = 0
q(l) = (l)2 + 3 x 1 = 4
(i) q(2) = (2)2+ 3 x 2 = 4 + 6 = 10
q(y) = 2 + y + 2y2 - 5y3
q(0) = 2 + 0 + 2 (0)2 - 5 (0)3 = 2
(ii) q(1) = 2 + 1 + 2 (1)2 - 5 (1)3 = 2 + 1 + 2 - 5 = 0
q(t) = t3
q(0) =0
q(1) = (1)3 = 1
(iii) q(2) = (2)3 = 8.
E7. Check whether 0 and 3 are zeroes of the polynomial x2 - 3x.
Sol.
let p (x) = x1 - 3x
Then p(0) = (0)" - 3*0 = 0
∴ 0 is a zero of the given polymcinial
Again, p(3) = (3)2 -3 x 3 = 9 - 9 = 0
∴ 3 is alari a zero cf the given polynomial.
Hence 0 and 3 are both zeroes of the polynomial x2 - 3x
Ex 8. Show that 3 is a zero of the polynomial x3 - 8x2 + 8x + 21.
Sol . Let p(x) = x3 - 8x2 + 8x + 21.
Now p(3) = (3)3 - 8 (3)2 + 8.3 + 21 = 27 - 72 + 24 + 21 = 0
∴ 3 is zero of the polynomial - x3 + 8x2 + 8x + 21.
Ex 9. Which of the number 1, -1, and -3 are zeroes of the polynomial 2x4 + 9x3 + 11x2 + 4x - 6.
Sol.
Let f(x) = 2x4 + 9x3 + 11x2 + 4x - 6
f(1) = 2(1)4 + 9(1)3 + 11(1)2 + 4(1) - 6 = 2 + 9 + 11 + 4 - 6 = 20 0
∴ 1 is not a zero of the polynomial f(x)
Again f(-1) = 2(-1)4 + 9(-1)3 + 11(-1)2 + 4(-1) - 6 = 2 - 9 + 11 - 4 - 6 = - 6 0
∴ -1 is not a zero the polynomial f(x)
Also f(-3) = 2(-3)4 + 9(-3)3 + 11(-3)2 + 4(-3) - 6 = 162 - 243 + 99 - 12 - 6 = 0
∴ -3 is a zero of the polynomial f(x).
Thus 1 and -1 are not zeroes of f(x) whereas -3 is a zero of f(x).
Ex 10. Verify whether the indicated numbers are zeroes (roots) of the polynomial corresponding to them in the following cases:
(i)
(ii) p(x) = (x + 1) (x - 2), x = -1, 2
(iii) p(x) = x2, x = 0
(iv)
(v)
Sol.
(i) p(x) = 3x + 1
∴ is a zero of the polynomial.
(ii) p(x) - (x + 1) (x- 2)
⇒ P(-1) = (-1 + 1) (-1- 2) = 0 x (-3) = 0 and, p (2) = (2 + 1) [2 - 2) = 3 x 0 = 0
∴' x = -1 and x = 2 are zeroes of the given polynomial
(iii) p(x) = x2 ⇒ p(0) = (0)2 = 0
∴ x = 0 is a zero of the given polynomial
(iv) p (x) - ℓx + m
is a zero of the given polynomial.
(v) p(x) = 2x + 1
∴ x = 1/2 is not a zero of the given polynomial.
1. What are polynomials? |
2. Can you give an example of a polynomial? |
3. How do you add polynomials? |
4. What is the degree of a polynomial? |
5. How do you factorize a polynomial? |
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