DPST1012 Chap.6 Polynomial Division, Factor Theorem and Sketching
Polynomial Division, Factor Theorem and Sketching
Dividing polynomials
Higher-degree polynomials appear when a parabola is not flexible enough; the course extends its Sydney temperature model from April to September into a cubic covering January to September. Dividing a polynomial follows the same routine as long division of numbers: divide the leading terms, multiply, subtract and bring down.
The division algorithm records the result as p(x) = q(x)·d(x) + r(x), where the remainder has lower degree than the divisor, and a remainder of zero means the divisor is a factor.
Remainder and factor theorems
The remainder theorem replaces a whole division with a single substitution: dividing f(x) by x − k leaves remainder f(k).
The factor theorem follows: x − k is a factor exactly when f(k) = 0. Together they give a four-step toolbox for solving cubics: guess a root by testing small integers, confirm the factor, divide to get a quadratic, and finish with the quadratic methods.
The same theorems find unknown coefficients and rebuild a polynomial from its roots and one more point, the idea behind the worksheet's cryptography example.
Sketching from the factors
Two features give a sketch without plotting dozens of points. The leading term sets the end behaviour: even degree means both ends point the same way, odd degree opposite ways, and the sign of the leading coefficient decides which.
The multiplicity of each root sets the local shape: a single root cuts through the axis, a double root touches and turns back, and a triple root flattens and crosses. A finished sketch then solves polynomial inequalities by reading where the graph lies above or below the axis.
What this chapter covers
- 01
Long division of polynomials
- 02
The division algorithm and remainder degree
- 03
The remainder theorem
- 04
The factor theorem and unknown coefficients
- 05
Solving cubics with the four-step toolbox
- 06
End behaviour from the leading term
- 07
Root multiplicity: cut, touch, flatten
- 08
Polynomial inequalities from a sketch
Worked example · free
Use the factor theorem to solve and sketch a cubic
- 1p(3) = 27 − 18 − 15 + 6 = 0, so (x − 3) is a factor by the factor theorem.
- 1Dividing p(x) by (x − 3) gives x² + x − 2 with remainder 0.
- 1x² + x − 2 = (x + 2)(x − 1), so p(x) = (x − 3)(x + 2)(x − 1).
- 2A positive cubic with single roots −2, 1 and 3 is negative before −2 and between 1 and 3, so p(x) ≤ 0 for x ≤ −2 or 1 ≤ x ≤ 3.
Key terms
- Division Algorithm
- The statement that any polynomial divided by a non-zero polynomial equals divisor times quotient plus a remainder of lower degree.
- Remainder Theorem
- The result that dividing f(x) by x − k leaves a remainder equal to f(k).
- Factor Theorem
- The result that f(k) = 0 exactly when x − k divides f(x) with no remainder.
- Multiplicity
- The number of times a factor is repeated in a polynomial, which decides how the graph behaves at that root.
- End Behaviour
- The direction a polynomial graph heads as x becomes very large or very negative, set by its leading term.
Polynomial Division, Factor Theorem and Sketching FAQ
When should I use long division instead of the remainder theorem?
Use the remainder theorem when only the remainder or a yes-or-no factor test is needed. Use long division when you need the quotient, for example to reduce a cubic to a quadratic after finding one root.
How many roots can a cubic have?
A cubic always has at least one real root and at most three. It can have one, two or three distinct real roots, where two distinct roots means one of them is repeated and the graph touches the axis there.
How do I find the smallest-degree polynomial from a graph?
Read every x-intercept and its behaviour: a crossing gives a single factor, a touch a squared factor and a flattened crossing a cubed one. Then use the y-intercept to find the constant multiplier in front.
What is the maximum degree of a remainder?
One less than the degree of the divisor. Dividing by a linear factor leaves a constant, and dividing by a quadratic leaves at most a linear expression, which you can state before doing any working.
Why write zero placeholders before dividing?
Long division lines up terms by power, so a missing power shifts every later column. Writing the gap as a zero term, such as 0x² in x³ − 8, keeps like powers under each other and stops the subtraction going wrong.
Exam move
Pick three cubics from their roots, expand them, then hand them back to yourself a day later to factorise with the toolbox and sketch from the factors. Check each division by multiplying back before you trust the quotient.
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