Content
Introduction to the notion and technique of dividing polynomials, and the factor and remainder theorems that apply to polynomial division.
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If one polynomial P(x) is the product of two polynomials D(x) and Q(x), then we can say that D(x) is a divisor of P(x).
In general polynomials are not perfect divisors of each other, but instead leave a remainder:
Here, P(x) is the dividend, D(x) is the divisor, Q(x) is the quotient and R(x) is the remainder.
The quotient is the "largest" possible polynomial that can multiply D(x) without exceeding the degree of P(x).
The quotient and remainder of a polynomial division can be found using long division, an iterative process where we cancel out the highest powers of the dividend one by one.
The steps in the above division are
Compare the leading terms 11x2 and x, and write 11x in the space for the quotient.
Subtract 11x⋅(x−5)
Compare the new leading term 29x to x, and write +29 in the space for the divisor.
Subtract 29⋅(x−5), leaving a constant of 110.
Stop the division as the leading power is not smaller than x1.
When the divisor is linear, the remainder is constant.
Polynomial long division stops when the remainder cannot be further divided by the divisor, that is the degree of the remainder is less than that of the divisor.
And since the degree of a product of polynomials is the sum of their degrees, the degree of the quotient and divisor must add up to the degree of the original polynomial. So if
then
The degree of R is less than the degree of D. 🚫
The degree of Q plus the degree of D equals the degree of P 🚫.
The polynomial factor theorem states that
The remainder theorem states that