In this section, let us consider the relationships between \(p(x)\), \(g(x)\), \(q(x)\) and \(r(x)\).
By Euclid's division algorithm, \(p(x) = (g(x) \times q(x)) + r(x)\).
For \(p(x)\), \(g(x)\), \(q(x)\) and \(r(x)\) to hold true, it should satisfy Euclid's division algorithm.
In other words, both the degrees of LHS and RHS should be equal.
Deg (\(r(x)\)) will never have the highest degree among \(p(x)\), \(g(x)\), \(q(x)\) and \(r(x)\).
Hence, for \(p(x)\), \(g(x)\), \(q(x)\) and \(r(x)\) to hold true, deg(\(p(x)\) \(=\) deg(\(g(x)\))\(+\) deg(\(q(x)\)) \(+\) Any (remainder).
Let us consider the example given below and check whether the given quotient could be a factor of the given dividend or not.
Dividend \(p(x)\) \(=\) \(x^7+ 5x^4 + x - 17\)
Quotient \(q(x)\) \(=\) \(x^2 + 5\)
Let the remainder be the \(r(x)\) and the divisor be the \(g(x)\).
Degree of the divisor is \(2\).
By Euclid's division algorithm, \(p(x) = g(x) \times q(x) + r(x)\).
deg\((p(x))\) \(=\) \(7\)
deg\((g(x))\) \(=\) \(2\)
deg\((q(x))\) \(=\) \(5\)
We know that, deg\((r(x))\) is less than \(2\).
deg(\(p(x)\) \(=\) deg(\(g(x)\))\(+\)deg(\(q(x)\))
\(7\) \(=\) \(5\) \(+\) \(2\)
Thus, \(x^2 + 5\) can be the quotient of the polynomial.