Factor & remainder theorems

This chapter introduces two connected ideas for polynomial division: Remainder Theorem and Factor Theorem. They help you test factors quickly, find roots, and break a cubic (or higher) polynomial into simpler factors.

Quick recap

A polynomial in \(x\) is an expression like \(P(x)=2x^3-3x^2+5x-7\). If you divide \(P(x)\) by \((x-a)\), you get:

\[ P(x) = (x-a)Q(x) + R, \]

where \(Q(x)\) is the quotient and \(R\) is a constant remainder.

Remainder Theorem

Example 1. Find the remainder when \(P(x)=x^3-4x+1\) is divided by \((x-2)\).

By the theorem, remainder \(=P(2)=2^3-4(2)+1=8-8+1=1\).

Example 2. Find the remainder when \(2x^3+x^2-5\) is divided by \((x+1)\).

\((x+1)=(x-(-1))\), so use \(a=-1\): \(P(-1)=2(-1)^3+(-1)^2-5=-2+1-5=-6\).

Factor Theorem

This is a direct consequence of the Remainder Theorem: factor means remainder \(0\).

Example 3. Is \((x-3)\) a factor of \(P(x)=x^3-4x^2-3x+18\)?

\(P(3)=27-36-9+18=0\), so yes, \((x-3)\) is a factor.

Finding factors and roots (high-school workflow)

1. List possible rational roots using factors of constant term (and leading coefficient).
2. Test candidates with \(P(a)\).
3. When \(P(a)=0\), divide by \((x-a)\) to reduce degree.
4. Factor the smaller polynomial and solve for all roots.

Example 4. Solve \(x^3-6x^2+11x-6=0\).

Try integer candidates \(\pm1,\pm2,\pm3,\pm6\). We get \(P(1)=0\), so \((x-1)\) is a factor. Dividing gives \(x^2-5x+6\), which factorises to \((x-2)(x-3)\). Hence:

\[ x^3-6x^2+11x-6=(x-1)(x-2)(x-3), \]

so roots are \(x=1,2,3\).

Synthetic division (quick table method)

Synthetic division is a compact way to divide by \((x-a)\). It is fast for checking many candidates in exam settings.

Example 5. Divide \(x^3-6x^2+11x-6\) by \((x-1)\):

\[ \begin{array}{r|rrrr} 1 & 1 & -6 & 11 & -6\\ & & 1 & -5 & 6\\ \hline & 1 & -5 & 6 & 0 \end{array} \]

Quotient is \(x^2-5x+6\), remainder \(0\).

Exam tips

• Always rewrite \((x+b)\) as \((x-(-b))\) before substituting.
• If remainder is not zero, that candidate is not a factor.
• After finding one factor, reduce degree before continuing.
• Check final roots by substitution when time permits.

History

Polynomial equations became central in algebra from the Renaissance onward, when mathematicians studied systematic methods for solving cubic and quartic equations. As symbolic algebra matured, methods for division and factor testing became standard tools.

What we now call the factor and remainder theorems are part of the broader polynomial division framework developed in 18th–19th century algebra texts. They remain important in modern high-school courses because they connect substitution, division, and root-finding in one clear structure.

Derivation

From the polynomial division algorithm

For any polynomial \(P(x)\), dividing by \((x-a)\) gives: \[ P(x)=(x-a)Q(x)+R. \] Now substitute \(x=a\): \[ P(a)=(a-a)Q(a)+R=R. \] So the remainder equals \(P(a)\), proving the Remainder Theorem.

If \((x-a)\) is a factor, remainder \(R=0\), so \(P(a)=0\). Conversely, if \(P(a)=0\), then remainder is zero, so \((x-a)\) divides \(P(x)\). This proves the Factor Theorem.

Checkpoints

Pause and reason before continuing. Discuss with a classmate or write your reasoning in a notebook.

Applications

Graph intercepts and curve sketching

Solving \(P(x)=0\) gives x-intercepts of polynomial graphs. Factor Theorem helps find those intercepts quickly, then sketch end behavior and turning points more accurately.

Constraint values in models

In applied models, a condition like “output is zero at \(x=a\)” means \(P(a)=0\), so \((x-a)\) must be a factor. This appears in physics calibration, optimization constraints, and coding of polynomial approximations.

Gateway to numerical methods

Understanding factor tests and polynomial reduction supports later methods such as Newton’s method and polynomial interpolation, where roots and remainders are central ideas.

References

• Standard high-school algebra curricula (IB, A-Level, DSE extension) on polynomial theorems.
• Classical algebra texts covering polynomial division and root-factor relationships.