Divide polynomials and interpret the remainder

Try

Ordinary division records both a quotient and a remainder: 23=5×4+3. Polynomial division has the same structure. To divide P(x) by D(x), seek a quotient Q(x) and remainder R(x) such that P(x)=D(x)Q(x)+R(x), with the remainder’s degree smaller than the divisor’s degree.

Start with x²+3x+5 divided by x+1. Multiplying (x+1)(x+2) gives x²+3x+2, so the remainder is 3. The exact identity is x²+3x+5=(x+1)(x+2)+3.

Predict

Predict the remainder degree when the divisor is linear. It must be constant, or zero. Predict what happens when x is the zero of the divisor: the product D(x)Q(x) disappears, leaving only the remainder. This observation will connect division to evaluation.

Experiment

Divide x³−2x²+0x+4 by x−1. Keep the zero coefficient as a place holder so that columns remain aligned.

Stage	Quotient term chosen	Expression after subtraction
Cancel x³	x²	−x²+0x+4
Cancel −x²	−x	−x+4
Cancel −x	−1	3

The quotient is x²−x−1 and remainder 3. Verify by expanding (x−1)(x²−x−1)+3. It gives x³−2x²+4 exactly. A second check evaluates the original at 1: 1−2+4=3.

Now change only the constant term from 4 to 1. Predict the new remainder without repeating every division step. Evaluation at 1 gives zero, so x−1 is now a factor.

Understand

In long division, divide the leading term of the current remainder by the leading term of the divisor. Multiply the entire divisor by that quotient term, subtract every resulting term and repeat. Stop when the remainder degree is lower than the divisor degree. Subtraction signs are a common failure point: write parentheses around the whole polynomial being subtracted.

For a divisor x−a, polynomial division gives P(x)=(x−a)Q(x)+r with constant r. Substituting x=a proves r=P(a). This is the remainder theorem. If P(a)=0, the identity becomes P(x)=(x−a)Q(x), so x−a is a factor. Conversely, such a factor makes P(a)=0. This equivalence is the factor theorem.

A nonmonic linear divisor works similarly. For 2x−3, evaluate at x=3/2 to find its constant remainder. Do not evaluate at 3 merely because that number appears in the divisor. The remainder is determined by where the entire divisor vanishes.

The division identity is a polynomial equality valid at every input. Rewriting a rational expression as P(x)/D(x)=Q(x)+R(x)/D(x) additionally requires D(x) nonzero. If the remainder is zero, the quotient may simplify to a polynomial, but its original excluded inputs do not return. This distinction will matter when classifying holes in graphs.

Apply

The practice mixes evaluation shortcuts with full division and a theorem explanation. In the parameter question, impose P(a)=0 to make a proposed factor valid. In the written long division, show the leading-term cancellations and verify the entire identity. A matching remainder alone does not prove every quotient coefficient is correct.

If a polynomial has a missing power, enter a zero coefficient in your working table. This is bookkeeping, not a new term that changes the function.

Continue

Known factors identify roots. The next lesson reads multiplicities and end behavior from a factored polynomial while distinguishing what those facts do and do not determine about its graph.