Read All Real Zeros from a Cubic Factorization
A cubic already written as a product of linear factors exposes its real zeros through the zero-product property.
About 25 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: P(x)=(x−3)(x−2)(x+2). List all real zeros. Candidate answers: (A) 0, 2, 3 (B) 3 only (C) −2, 2, 3 (D) −3, −2, 2.
- Independent check inputs: P(x)=(x−4)(x−2)(x+2). List all real zeros. Candidate answers: (A) 0, 2, 4 (B) 4 only (C) −2, 2, 4 (D) −4, −2, 2.
Learn the idea
Polynomial functions connect symbolic factors with graph features. Division, remainder evaluation, real zeros and multiplicity reveal structure without requiring every coefficient to be expanded, but domains and repeated roots still need attention. A cubic already written as a product of linear factors exposes its real zeros through the zero-product property. Each distinct factor gives a candidate that can be verified without expanding the cubic.
Here is the method on the supplied teaching case. Each factor can be zero: x=3, x=2, or x=−2. These are distinct because 3>2. Multiplying the factors would not change these zeros.
Match each factor’s sign correctly. A list of zeros should not invent a root at zero unless x is actually a factor; repeated factors change multiplicity rather than create additional distinct zero values.
- 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: P(x)=(x−3)(x−2)(x+2). List all real zeros. Candidate answers: (A) 0, 2, 3 (B) 3 only (C) −2, 2, 3 (D) −3, −2, 2. Solution: Each factor can be zero: x=3, x=2, or x=−2. These are distinct because 3>2. Multiplying the factors would not change these zeros. The final answer is −2, 2, 3. Match each factor’s sign correctly. A list of zeros should not invent a root at zero unless x is actually a factor; repeated factors change multiplicity rather than create additional distinct zero values.
Check your reasoning
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Helpful tips
- A cubic already written as a product of linear factors exposes its real zeros through the zero-product property.
- Match each factor’s sign correctly.
See the answer and explanation
−2, 2, 4
Each factor can be zero: x=4, x=2, or x=−2. These are distinct because 4>2. Multiplying the factors would not change these zeros.
Take it further
Changed-input task: P(x)=(x−10)(x−2)(x+2). List all real zeros. Candidate answers: (A) 0, 2, 10 (B) 10 only (C) −10, −2, 2 (D) −2, 2, 10. Show the mathematical reason, not only the selected answer. Expected result: −2, 2, 10. Each factor can be zero: x=10, x=2, or x=−2. These are distinct because 10>2. Multiplying the factors would not change these zeros.
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More practice: try new cases
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