Key takeaway: The factor theorem states that x − c divides a polynomial P(x) with no remainder exactly when P(c) = 0, so every zero of the polynomial gives a linear factor.
Grade 11 – Algebra 2A-APR.B.2Expressions and Equations
Definition of Factor Theorem
The factor theorem answers one question: does a given binomial \(x-c\) divide a polynomial evenly? It does precisely when plugging \(c\) into the polynomial returns zero.
This is the remainder theorem seen from a special angle. The leftover after dividing by \(x-c\) is \(P(c)\), and a leftover of nothing means an exact division. Because it runs both ways, you can start from a zero and build a factor, or start from a factor and read off a zero.
Formula and notation
For a polynomial \(P(x)\) and a number \(c\), the factor theorem reads
\[P(c)=0 \iff (x-c) ext{ is a factor of } P(x).\]
Then \(P(x)=(x-c)\,Q(x)\) with \(Q(x)\) one degree lower. Each such \(c\) is also an x-intercept on the graph of \(y=P(x)\).
Key facts
- Every real zero c produces one linear factor x − c.
- Guess small whole numbers for c, then divide to lower the degree.
- A factor like 2x − 5 matches the zero c = 5/2.
- A repeated factor makes the graph touch or flatten at that zero.
Factor Theorem example
Show that \(x-4\) divides \(P(x)=x^3-5x^2+2x+8\) evenly, then write \(P(x)\) as a product of three binomials.
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Test: \(P(4)=64-80+8+8=0\), so \(x-4\) is a factor.
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Synthetic division by 4 on the coefficients \(1,-5,2,8\) gives the row \(1,-1,-2\) and a final 0, so the quotient is \(x^2-x-2\).
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Split the quotient: \(x^2-x-2=(x-2)(x+1)\).
The full product is \((x+1)(x-2)(x-4)\).
Common mistake
Watch out
Omar tests whether \(x+3\) divides \(x^2+x-6\). He computes \(P(3)=9+3-6=6\), which is not zero, so he rejects it.
Why it is wrong
The binomial \(x+3\) is \(x-(-3)\), so the test number is \(-3\). Then \(P(-3)=9-3-6=0\) and \(x^2+x-6=(x+3)(x-2)\). Use the opposite of the constant term.
Related terms and lessons
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