Factor a Difference of Two Squares
Opposite middle terms cancel in (u−v)(u+v), leaving u²−v².
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Factor x²−4. Candidate answers: (A) (x−2)(x+2) (B) (x−2)² (C) (x−4)(x+1) (D) x(x−4).
- Independent check inputs: Factor x²−9. Candidate answers: (A) (x−3)(x+3) (B) (x−3)² (C) (x−9)(x+1) (D) x(x−9).
Learn the idea
Polynomials combine terms with nonnegative integer variable powers. Their degree describes the highest power, while addition, multiplication and factoring preserve equivalent expressions that can be checked by expansion or substitution. Opposite middle terms cancel in (u−v)(u+v), leaving u²−v². Recognizing both terms as squares turns that cancellation pattern into a reliable factorization.
Here is the method on the supplied teaching case. Use u²−v²=(u−v)(u+v) with u=x and v=2. The opposite middle terms cancel when the factors are multiplied.
A difference of squares differs from the square of a difference. Expanding (x−a)² produces a middle term; expanding the conjugate pair cancels it, so the factorizations are not interchangeable.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Factor x²−4. Candidate answers: (A) (x−2)(x+2) (B) (x−2)² (C) (x−4)(x+1) (D) x(x−4). Solution: Use u²−v²=(u−v)(u+v) with u=x and v=2. The opposite middle terms cancel when the factors are multiplied. The final answer is (x−2)(x+2). A difference of squares differs from the square of a difference. Expanding (x−a)² produces a middle term; expanding the conjugate pair cancels it, so the factorizations are not interchangeable.
Check your reasoning
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Helpful tips
- Opposite middle terms cancel in (u−v)(u+v), leaving u²−v².
- A difference of squares differs from the square of a difference.
See the answer and explanation
(x−3)(x+3)
Use u²−v²=(u−v)(u+v) with u=x and v=3. The opposite middle terms cancel when the factors are multiplied.
Take it further
Changed-input task: Factor x²−81. Candidate answers: (A) (x−81)(x+1) (B) (x−9)(x+9) (C) (x−9)² (D) x(x−81). Show the mathematical reason, not only the selected answer. Expected result: (x−9)(x+9). Use u²−v²=(u−v)(u+v) with u=x and v=9. The opposite middle terms cancel when the factors are multiplied.
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