Math101 · Independent Ontario learning library
Math101learn.math101.caFactor Theorem
The factor theorem connects a polynomial zero P(k)=0 with the linear factor x−k.
Evaluating one number can prove that an entire linear expression divides a polynomial.
The theorem
For a polynomial $P(x)$,
The statement works in both directions. A zero produces a factor, and a factor produces a zero. It links equations, graphs, and algebraic division.
Why it works
The division algorithm says
where $R$ is a constant remainder. Substitute $x=k$:
Therefore if $P(k)=0$, the remainder is zero and division by $x-k$ is exact.
Testing a proposed factor
To determine whether $x-3$ is a factor, evaluate $P(3)$. To test $x+4=x-(-4)$, evaluate $P(-4)$. The sign change is built into the form $x-k$.
A nonzero result proves that the proposed expression is not a factor; it also gives the remainder.
Worked example: factor a cubic
The zeros are $1$, $4$, and $-1$.
Synthetic division connection
Once $P(k)=0$ identifies $x-k$, synthetic division efficiently finds the remaining factor. Write every coefficient in descending degree order, including zeros for missing powers.
The last synthetic value is the remainder. A zero at the end confirms exact division and the other values form the quotient coefficients.
Finding possible rational zeros
For a polynomial with integer coefficients, the rational root theorem lists candidates
The factor theorem tests those candidates. The rational root theorem proposes possibilities; it does not guarantee that every candidate is a zero.
Repeated factors
If $x-k$ divides $P(x)$ more than once, then $k$ is a repeated zero. After one division, test the quotient at $k$ again. A factor $(x-k)^m$ gives multiplicity $m$.
Even multiplicity usually makes the graph touch the axis; odd multiplicity makes it cross.
Building a polynomial from zeros
If a polynomial has zeros $-2$, $1$, and $5$, then
for some nonzero leading multiplier $a$. A supplied point determines $a$. Zeros alone determine factors but not the vertical scale.
Solving polynomial equations
Set the polynomial equal to zero, find one factor, divide, and continue until the remaining factors can be solved. The zero-product property then turns the factorization into individual equations.
Always check whether the problem asks for real zeros, rational zeros, or all complex zeros.
Graphical interpretation
The condition $P(k)=0$ means $(k,0)$ is an $x$-intercept. Factorization records the same fact algebraically. A graph can suggest candidate zeros, but exact evaluation establishes them.
Approximate graphing values should not be mistaken for proof when an exact factor is required.
Common mistakes
Testing the wrong sign. For factor $x+3$, use $k=-3$.
Assuming $P(k)=0$ gives factor $x+k$. The matching factor is $x-k$.
Omitting zero coefficients in synthetic division. This shifts every place value.
Stopping after finding one factor. Factor the quotient as far as the question requires.
Treating a rational-root candidate as confirmed. Evaluate it first.
Quick self-check
- Is the proposed factor written as $x-k$?
- Did I evaluate $P(k)$ accurately?
- Does a zero remainder confirm exact division?
- Have missing powers been represented with zero coefficients?
- Does multiplying the factors reproduce the original polynomial?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
For P(x) = x³ − 4x² − x + 4, which statement follows from P(1) = 0?
- Here k = 1.
- P(1) = 0 means division by x − 1 has zero remainder.
- Therefore x − 1 is a factor.
End of lesson
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