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MathChapter 13: Polynomial and Radical Functions

Notes

Remainder Theorem and Factor Theorem Notes

Use function evaluation to find remainders, test factors, solve parameters, and connect zeros, roots, intercepts, and divisibility.

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The Remainder Theorem replaces polynomial division by one focused evaluation. The Factor Theorem is its zero-remainder case, connecting algebra, equations, factors, and graphs.

Remainder Theorem

Remainder Theorem identity

The division identity labels the polynomial dividend, linear divisor, polynomial quotient, and constant remainder.

\[\underbrace{f(x)}_{\text{dividend}}=\underbrace{(x-c)}_{\text{divisor}}\underbrace{q(x)}_{\text{quotient}}+\underbrace{f(c)}_{\text{remainder}}\]
Dividend
\(f(x)\)

The polynomial being divided.

Divisor
\(x-c\)

Its zero is c; preserve the sign.

Quotient
\(q(x)\)

The polynomial result of division.

Remainder
\(f(c)\)

Evaluate the dividend at the divisor's zero.

Worked example

Use the correct sign

Find the remainder when \(f(x)=2x^3-5x+7\) is divided by \(x+2\).

  1. Identify c

    \(x+2=x-(-2)\), so \(c=-2\).

  2. Evaluate

    \(f(-2)=2(-2)^3-5(-2)+7=-16+10+7=1\).

The remainder is \(1\).

Factor Theorem and equivalent statements

One zero, five equivalent descriptionsA five-stage flow connects a zero function value to root language, an intercept, a factor, and zero remainder.
  1. \(f(c)=0\)

    Start with a zero output.

  2. \(c\text{ is a zero/root}\)

    It solves the equation f(x)=0.

  3. \((c,0)\text{ is an x-intercept}\)

    The graph meets the horizontal axis.

  4. \(x-c\text{ is a factor}\)

    Factor Theorem connects the zero to divisibility.

  5. \(\operatorname{remainder}=0\)

    Division by x-c is exact.

Equivalent descriptions when f(c) equals zero
ViewpointEquivalent statement
Equation\(c\) solves \(f(x)=0\)
Function\(c\) is a zero or root
Graph\((c,0)\) is an \(x\)-intercept
Factoring\(x-c\) is a factor
DivisionDivision by \(x-c\) has remainder \(0\)

Known factors determine parameters

Worked example

Solve a parameter from a factor

If \(x-2\) is a factor of \(p(x)=x^3+kx-14\), find \(k\).

  1. Apply Factor Theorem

    Because \(x-2\) is a factor, \(p(2)=0\).

  2. Substitute

    \(2^3+2k-14=0\), so \(2k-6=0\).

  3. Solve

    \(k=3\).

\(k=3\).

Factor evidence from a table

Original function values for a factor decision
xf(x)Consequence
\(-3\)\(5\)\(x+3\) is not certified as a factor
\(-1\)\(0\)\(x+1\) is a factor
\(2\)\(-7\)\(x-2\) is not certified as a factor
\(4\)\(0\)\(x-4\) is a factor

Common mistakes and traps

  • Using \(f(4)\) for division by \(x+4\).
  • Computing \(f(x-c)\) instead of \(f(c)\).
  • Calling \(x-c\) a factor when \(f(c)\ne0\).
  • Confusing the root \(c\) with the intercept \((c,0)\).
  • Substituting correctly but making sign errors in odd powers.
Mini check

Factor or remainder

If \(p(-3)=0\), which binomial must be a factor of \(p(x)\)?

  1. \(x-3\)
  2. \(x+3\)
  3. \(3x-1\)
  4. \(x\)
Show answer and explanation

Answer: \(x+3\)

Here \(c=-3\), so the matching factor is \(x-c=x-(-3)=x+3\).

Key takeaways

Key takeaways

What to remember

  • Division by \(x-c\) leaves remainder \(f(c)\).
  • \(x-c\) is a factor exactly when \(f(c)=0\).
  • A zero, root, solution, intercept, factor, and zero remainder describe the same relationship.
  • For \(x+c\), evaluate at \(-c\).
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.