Question

Question 5 (i)
Give examples of polynomial p(x), g(x), q(x) and r(x), which satisfy the division algorithm and

deg p(x) = deg q(x)

Answer 1

Division algorithm for polynomials states that, suppose p(x) and g(x) are the two polynomials, where $g\left(x\right)\ne 0$, we can write:

$f\left(x\right)=q\left(x\right)g\left(x\right)+r\left(x\right)$

which is same as the Dividend = Divisor * Quotient + Remainder and where r(x) is the remainder polynomial and is equal to 0 or $degreer\left(x\right)<degreeg\left(x\right)$.

Let us assume the division of $6x^2+2x+2$ by 2

Here, p(x) = $6x^2+2x+2$

g(x) = 2

q(x) = $3x^2+x+1$

r(x) = 0

Degree of p(x) and q(x) is same i.e. 2. $\left(degreeofquotientwillbeequaltodegreeofdividendwhendivisorisconstant\right)$

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

Or, $6x^2+2x+2=2(3x^2+x+1)+0$

Hence, division algorithm is satisfied.