CBSE Class 10 Answered
Q: 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
Q: 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
Q: 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
Q: 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
Asked by Ashish Parashar | 21 Sep, 2013, 09:03: AM
Expert 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) x q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).
(i) Degree of quotient will be equal to degree of dividend when divisor is constant.
Let us consider the division of by 3.
Here, p(x) = and g(x) = 3
q(x) = and r(x) = 0
Here, degree of p(x) and q(x) is the same which is 2.
Checking:
p(x) = g(x) x q(x) + r(x)
Thus, the division algorithm is satisfied.
(ii) Let us consider the division of 2x4 + 2x by 2x3,
Here, p(x) = 2x4 + 2x and g(x) = 2x3
q(x) = x and r(x) = 2x
Clearly, the degree of q(x) and r(x) is the same which is 1.
Checking,
p(x) = g(x) x q(x) + r(x)
2x4 + 2x = (2x3 ) x x + 2x
2x4 + 2x = 2x4 + 2x
Thus, the division algorithm is satisfied.
(iii) Degree of remainder will be 0 when remainder obtained on division is a constant.
Let us consider the division of 10x3 + 3 by 5x2.
Here, p(x) = 10x3 + 3 and g(x) = 5x2
q(x) = 2x and r(x) = 3
Clearly, the degree of r(x) is 0.
Checking:
p(x) = g(x) x q(x) + r(x)
10x3 + 3 = (5x2 ) x 2x + 3
10x3 + 3 = 10x3 + 3
Thus, the division algorithm is satisfied.
p(x) = g(x) x q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).
(i) Degree of quotient will be equal to degree of dividend when divisor is constant.
Let us consider the division of by 3.
Here, p(x) = and g(x) = 3
q(x) = and r(x) = 0
Here, degree of p(x) and q(x) is the same which is 2.
Checking:
p(x) = g(x) x q(x) + r(x)
Thus, the division algorithm is satisfied.
(ii) Let us consider the division of 2x4 + 2x by 2x3,
Here, p(x) = 2x4 + 2x and g(x) = 2x3
q(x) = x and r(x) = 2x
Clearly, the degree of q(x) and r(x) is the same which is 1.
Checking,
p(x) = g(x) x q(x) + r(x)
2x4 + 2x = (2x3 ) x x + 2x
2x4 + 2x = 2x4 + 2x
Thus, the division algorithm is satisfied.
(iii) Degree of remainder will be 0 when remainder obtained on division is a constant.
Let us consider the division of 10x3 + 3 by 5x2.
Here, p(x) = 10x3 + 3 and g(x) = 5x2
q(x) = 2x and r(x) = 3
Clearly, the degree of r(x) is 0.
Checking:
p(x) = g(x) x q(x) + r(x)
10x3 + 3 = (5x2 ) x 2x + 3
10x3 + 3 = 10x3 + 3
Thus, the division algorithm is satisfied.
Note: We have provided one example, for each case. You can find more on similar lines.
Answered by | 22 Sep, 2013, 03:51: PM
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