a. Linear Polynomial: Degree 1
b. Quadratic Polynomial: Degree 2
c. Cubic Polynomial: Degree 3
The next section describes the geometrical meaning of the zeroes of a polynomial. Various graphs are shown to explain how plotting of equations is done.
In general, for a linear polynomial ax + b, a 0, the graph of y = ax + b is a straight line which intersects the x-axis at exactly one point.
In quadratic polynomials, Parabolas are curves which have one of the two shapes either open upwards or open downwards.
After that, solved examples are given to show how the number of zeroes can be determined by using the graphical method.
Zeroes of a quadratic polynomial are precisely the x- coordinates of the point where parabola intersects the x-axis.
Exercise 2.1 is based on the same concept. In the next section, the relationship between zeroes and coefficients of a polynomial is explained.
Exercise 2.2 consists of questions in which students need to find the sum and product of zeroes of the given polynomial.
Moving further, students will learn about the division algorithm for polynomials. In this section, the method of dividing one polynomial by another is discussed.
If p(x) and g(x) are any two polynomials with g(x) not equal to 0,then we can find polynomial q(x) and r(x) such that:
p(x)=g(x) q(x)+ r(x), where r(x) = 0 or degree of r(x)< degree of g(x).
Exercise 2.3 is based on the same.
Additionally, an optional exercise is given but that exercise is just for practice and not given from the examination point of view.
For a perfect finish, a few important points are given to give a summary of the chapter.
Page No 28:
Question 1:
The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case.
(i)
(ii)
(iii)
(iv)
(v)
(v)
Answer:
(i) The number of zeroes is 0 as the graph does not cut the x-axis at any point.
(ii) The number of zeroes is 1 as the graph intersects the x-axis at only 1 point.
(iii) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.
(iv) The number of zeroes is 2 as the graph intersects the x-axis at 2 points.
(v) The number of zeroes is 4 as the graph intersects the x-axis at 4 points.
(vi) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.
Video Solution for polynomials (Page: 28 , Q.No.: 1)
NCERT Solution for Class 10 math - polynomials 28 , Question 1
Page No 33:
Question 1:
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
Answer:
The value of is zero when x − 4 = 0 or x + 2 = 0, i.e., when x = 4 or x = −2
Therefore, the zeroes of are 4 and −2.
Sum of zeroes =
Product of zeroes
The value of 4s2 − 4s + 1 is zero when 2s − 1 = 0, i.e.,
Therefore, the zeroes of 4s2 − 4s + 1 areand.
Sum of zeroes =
Product of zeroes
The value of 6x2 − 3 − 7x is zero when 3x + 1 = 0 or 2x − 3 = 0, i.e., or
Therefore, the zeroes of 6x2 − 3 − 7x are.
Sum of zeroes =
Product of zeroes =
The value of 4u2 + 8u is zero when 4u = 0 or u + 2 = 0, i.e., u = 0 or u = −2
Therefore, the zeroes of 4u2 + 8u are 0 and −2.
Sum of zeroes =
Product of zeroes =
The value of t2 − 15 is zero when or , i.e., when
Therefore, the zeroes of t2 − 15 are and.
Sum of zeroes =
Product of zeroes =
The value of 3x2 − x − 4 is zero when 3x − 4 = 0 or x + 1 = 0, i.e., when or x = −1
Therefore, the zeroes of 3x2 − x − 4 are and −1.
Sum of zeroes =
Product of zeroes
Video Solution for polynomials (Page: 33 , Q.No.: 1)
NCERT Solution for Class 10 math - polynomials 33 , Question 1
Page No 33:
Question 2:
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
Answer:
Let the polynomial be , and its zeroes be and .
Therefore, the quadratic polynomial is 4x2 − x − 4.
Let the polynomial be , and its zeroes be and .
Therefore, the quadratic polynomial is 3x2 − x + 1.
Let the polynomial be , and its zeroes be and .
Therefore, the quadratic polynomial is .
Let the polynomial be , and its zeroes be and .
Therefore, the quadratic polynomial is .
Let the polynomial be , and its zeroes be and .
Therefore, the quadratic polynomial is .
Let the polynomial be .
Therefore, the quadratic polynomial is.
Video Solution for polynomials (Page: 33 , Q.No.: 2)
NCERT Solution for Class 10 math - polynomials 33 , Question 2
Page No 36:
Question 1:
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:
(i)
(ii)
(iii)
Answer:
Quotient = x − 3
Remainder = 7x − 9
Quotient = x2 + x − 3
Remainder = 8
Quotient = −x2 − 2
Remainder = −5x +10
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NCERT Solution for Class 10 math - polynomials 36 , Question 1
Page No 36:
Question 2:
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
Answer:
=
Since the remainder is 0,
Hence, is a factor of .
Since the remainder is 0,
Hence, is a factor of .
Since the remainder ,
Hence, is not a factor of .
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NCERT Solution for Class 10 math - polynomials 36 , Question 2
Page No 36:
Question 3:
Obtain all other zeroes of , if two of its zeroes are .
Answer:
Since the two zeroes are ,
is a factor of .
Therefore, we divide the given polynomial by .
We factorize
Therefore, its zero is given by x + 1 = 0
x = −1
As it has the term , therefore, there will be 2 zeroes at x = −1.
Hence, the zeroes of the given polynomial are, −1 and −1.
Video Solution for polynomials (Page: 36 , Q.No.: 3)
NCERT Solution for Class 10 math - polynomials 36 , Question 3
Page No 36:
Question 4:
On dividing by a polynomial g(x), the quotient and remainder were x − 2 and − 2x + 4, respectively. Find g(x).
Answer:
g(x) = ? (Divisor)
Quotient = (x − 2)
Remainder = (− 2x + 4)
Dividend = Divisor × Quotient + Remainder
g(x) is the quotient when we divide by
Video Solution for polynomials (Page: 36 , Q.No.: 4)
NCERT Solution for Class 10 math - polynomials 36 , Question 4
Page No 36:
Question 5:
Give examples of polynomial p(x), g(x), q(x) and r(x), which satisfy the division algorithm and
(i) deg p(x) = deg q(x)
(ii) deg q(x) = deg r(x)
(iii) deg r(x) = 0
Answer:
According to the division algorithm, if p(x) and g(x) are two polynomials with
g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that
p(x) = g(x) × q(x) + r(x),
where r(x) = 0 or degree of r(x) < degree of g(x)
Degree of a polynomial is the highest power of the variable in the polynomial.
(i) deg p(x) = deg q(x)
Degree of quotient will be equal to degree of dividend when divisor is constant ( i.e., when any polynomial is divided by a constant).
Let us assume the division of by 2.
Here, p(x) =
g(x) = 2
q(x) = and r(x) = 0
Degree of p(x) and q(x) is the same i.e., 2.
Checking for division algorithm,
p(x) = g(x) × q(x) + r(x)
= 2()
=
Thus, the division algorithm is satisfied.
(ii) deg q(x) = deg r(x)
Let us assume the division of x3 + x by x2,
Here, p(x) = x3 + x
g(x) = x2
q(x) = x and r(x) = x
Clearly, the degree of q(x) and r(x) is the same i.e., 1.
Checking for division algorithm,
p(x) = g(x) × q(x) + r(x)
x3 + x = (x2 ) × x + x
x3 + x = x3 + x
Thus, the division algorithm is satisfied.
(iii)deg r(x) = 0
Degree of remainder will be 0 when remainder comes to a constant.
Let us assume the division of x3 + 1by x2.
Here, p(x) = x3 + 1
g(x) = x2
q(x) = x and r(x) = 1
Clearly, the degree of r(x) is 0.
Checking for division algorithm,
p(x) = g(x) × q(x) + r(x)
x3 + 1 = (x2 ) × x + 1
x3 + 1 = x3 + 1
Thus, the division algorithm is satisfied.
Video Solution for polynomials (Page: 36 , Q.No.: 5)
NCERT Solution for Class 10 math - polynomials 36 , Question 5
Page No 36:
Question 1:
Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
Answer:
(i)
Therefore, , 1, and −2 are the zeroes of the given polynomial.
Comparing the given polynomial with , we obtain a = 2, b = 1, c = −5, d = 2
Therefore, the relationship between the zeroes and the coefficients is verified.
(ii)
Therefore, 2, 1, 1 are the zeroes of the given polynomial.
Comparing the given polynomial with , we obtain a = 1, b = −4, c = 5, d = −2.
Verification of the relationship between zeroes and coefficient of the given polynomial
Multiplication of zeroes taking two at a time = (2)(1) + (1)(1) + (2)(1) =2 + 1 + 2 = 5
Multiplication of zeroes = 2 × 1 × 1 = 2
Hence, the relationship between the zeroes and the coefficients is verified.
Video Solution for polynomials (Page: 36 , Q.No.: 1)
NCERT Solution for Class 10 math - polynomials 36 , Question 1
Page No 36:
Question 2:
Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, − 7, − 14 respectively.
Answer:
Let the polynomial be and the zeroes be .
It is given that
If a = 1, then b = −2, c = −7, d = 14
Hence, the polynomial is .
Video Solution for polynomials (Page: 36 , Q.No.: 2)
NCERT Solution for Class 10 math - polynomials 36 , Question 2
Page No 37:
Question 3:
If the zeroes of polynomial are, find a and b.
Answer:
Zeroes are a − b, a + a + b
Comparing the given polynomial with , we obtain
p = 1, q = −3, r = 1, t = 1
The zeroes are .
Hence, a = 1 and b = or .
Video Solution for polynomials (Page: 37 , Q.No.: 3)
NCERT Solution for Class 10 math - polynomials 37 , Question 3
Page No 37:
Question 4:
]It two zeroes of the polynomial are, find other zeroes.
Answer:
Given that 2 + and 2 are zeroes of the given polynomial.
Therefore, = x2 + 4 − 4x − 3
= x2 − 4x + 1 is a factor of the given polynomial
For finding the remaining zeroes of the given polynomial, we will find the quotient by dividing by x2 − 4x + 1.
Clearly, =
It can be observed that is also a factor of the given polynomial.
And =
Therefore, the value of the polynomial is also zero when or
Or x = 7 or −5
Hence, 7 and −5 are also zeroes of this polynomial.
Video Solution for polynomials (Page: 37 , Q.No.: 4)
NCERT Solution for Class 10 math - polynomials 37 , Question 4
Page No 37:
Question 5:
If the polynomial is divided by another polynomial, the remainder comes out to be x + a, find k and a.
Answer:
By division algorithm,
Dividend = Divisor × Quotient + Remainder
Dividend − Remainder = Divisor × Quotient
will be perfectly divisible by .
Let us divide by
It can be observed that will be 0.
Therefore, = 0 and = 0
For = 0,
2 k =10
And thus, k = 5
For = 0
10 − a − 8 × 5 + 25 = 0
10 − a − 40 + 25 = 0
− 5 − a = 0
Therefore, a = −5
Hence, k = 5 and a = −5
Video Solution for polynomials (Page: 37 , Q.No.: 5)
NCERT Solution for Class 10 math - polynomials 37 , Question 5
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