BetterGrades Precalculus · Unit 5 · Lesson

Remainder and factor theorems

Use P(c) as the remainder on division by x-c and test whether x-c is a factor.

Opening

Start with the situation

The remainder on division by x-c is P(c), and x-c is a factor exactly when $P(c)=0$ .

Polynomial structure links symbolic factors to visible graph behavior. Reading that structure efficiently makes it possible to sketch, solve, and model without treating every problem as a blind numerical search.

Before you begin

Prerequisite check

Factor polynomial expressions.
Read zeros and graph behavior.
Distinguish exact and approximate forms.

Core explanation

Explanation

Evaluate the candidate or use synthetic division, then divide confirmed factors out and continue.

Repeated successful division by the same factor reveals multiplicity.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through remainder theorem substitution, factor-test loop, or another equivalent representation.

Conceptual reading

What the idea is really doing

Polynomial formulas contain structural information before a graph is drawn. Degree and leading coefficient control the ends, factors reveal zeros, multiplicity predicts crossing or touching, and selected values settle the remaining shape.

This lesson narrows that lens to one goal: use P(c) as the remainder on division by x-c and test whether x-c is a factor. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Evaluate the candidate or use synthetic division.
Then divide confirmed factors out.
Continue.

Verification: Compare the proposed graph with the factorization and leading term. Every real zero, sign interval, end direction, and y-intercept should agree with the same formula.

Foundation walkthrough

Plan before calculating

Problem

Is $x+2$ a factor of $x^3-3x+2$ ?

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Evaluate the candidate or use synthetic division, then divide confirmed factors out and continue.
Conclusion: Yes, because $P(-2)=0$ .
Why the check works: Zero remainder confirms the factor.

Worked examples

See the idea in three forms

foundation example

Is $x+2$ a factor of $x^3-3x+2$ ?

SolutionYes, because $P(-2)=0$ .

Zero remainder confirms the factor.

representation example

If $P(4)=7,$ remainder by $x-4$ .

Solution $7$

This example expresses remainder and factor theorems in a second form.

transfer example

If $P(-2)=0,$ factor.

Solution $x+2$

Repeated successful division by the same factor reveals multiplicity.

Remainder theorem substitution. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Zero remainder confirms the factor. — **Anchor figure · Remainder theorem substitution**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Zero remainder confirms the factor.

Factor-test loop. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for remainder and factor theorems. — **Mechanism figure · Factor-test loop**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for remainder and factor theorems.

Multiplicity by repeated division. Compare the valid path with the tempting shortcut. The figure shows why calling every rational candidate a factor before checking the remainder leads to a false conclusion. — **Comparison and error figure · Multiplicity by repeated division**
Compare the valid path with the tempting shortcut. The figure shows why calling every rational candidate a factor before checking the remainder leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is calling every rational candidate a factor before checking the remainder.

Check yourself

Remainder of $x^2+3x+1$ by $x-2$ .

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Practice

Ten concrete questions

Practice 101

Remainder of $x^2+3x+1$ by $x-2$ .

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Practice 202

If $P(4)=7,$ remainder by $x-4$ .

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Practice 303

If $P(-2)=0,$ factor.

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Practice 404

Meaning repeated division.

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Practice 505

Explain why this conclusion is valid: Yes, because $P(-2)=0$ . Use the foundation problem as evidence: Is $x+2$ a factor of $x^3-3x+2$ ?

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Practice 606

Solve the representation example, then name the feature of remainder and factor theorems that it illustrates: If $P(4)=7,$ remainder by $x-4$ .

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Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is calling every rational candidate a factor before checking the remainder.

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Practice 808

Connect two representations for this example: Is $x+2$ a factor of $x^3-3x+2$ ? Describe what a graph, table, mapping, or algebraic form would have to show.

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Practice 909

Create a nearby example by changing one number or condition in this prompt: If $P(-2)=0,$ factor. Predict the effect, solve your new example, and compare it with the original.

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Practice 1010

Write a short verification checklist for remainder and factor theorems, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, Rational-root candidates and root search, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Lippman and Rasmussen, Precalculus Volume 1
Stitz and Zeager, Precalculus
Redden, Advanced Algebra

No long source passage is reproduced.

Is x+2x+2x+2 a factor of x3−3x+2x^3-3x+2x3−3x+2?

Is x+2x+2x+2 a factor of x3−3x+2x^3-3x+2x3−3x+2?

If P(4)=7,P(4)=7,P(4)=7, remainder by x−4x-4x−4.

If P(−2)=0,P(-2)=0,P(−2)=0, factor.

Original BetterGrades manuscript, rights-separated references.

Is $x+2$ a factor of $x^3-3x+2$ ?

Is $x+2$ a factor of $x^3-3x+2$ ?

If $P(4)=7,$ remainder by $x-4$ .

If $P(-2)=0,$ factor.