examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and, deg q(x)= deg r(x)
Answers
Answer:
x³ + 1
Step-by-step explanation:
Solution:
The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x), there are polynomials q(x) and r(x) such that
p(x) = g(x) × q(x) + r(x),
where r(x) = 0 or degree r(x) < degree g(x)
The degree of a polynomial is the highest power of the variable in the polynomial.
(i) deg p(x) = deg q(x)
The degree of the quotient will be equal to the degree of dividend when the divisor is constant (i.e. when any polynomial is divided by a constant).
Let us assume the division of 6x² + 2x + 2 by 2
p(x) = 6x² + 2x + 2
g(x) = 2
q(x) = 3x² + x + 1
r(x) = 0
Degree of p(x) and q(x) is same that is 2.
Checking for division algorithm:
p(x) = g(x) × q(x) + r(x)
= 2(3x2 + x + 1) + 0
= 6x² + 2x + 2
Thus, the division algorithm is satisfied.
(ii) deg q(x) = deg r(x)
Let us assume the division of x³ + x by x²
p(x) = x³ + x
g(x) = x²
q(x) = x
r(x) = x
Clearly, degree of r(x) and q(x) is same i.e, 1.
Checking for division algorithm
p(x) = g(x) × q(x) + r(x)
= (x² × x ) + x
= x³ + x
Thus, the division algorithm is satisfied.
(iii) deg r(x) = 0
The degree of the remainder will be 0 when the remainder turns out to be a constant.
Let us assume the division of x³ + 1 by x².
p(x) = x³ + 1
g(x) = x²
q (x) = x
r(x) = 1
Clearly, the degree of r(x) is 0.
Checking for division algorithm
p(x) = g (x) × q (x) + r (x)
= (x² × x) + 1
= x³ + 1
Thus, the division algorithm is satisfied.
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