Lesson 2 of 4
Use long or synthetic division and verify dividend=divisor times quotient+remainder.
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Big ideas
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Arrange dividend and divisor in descending powers. Include a zero coefficient for a missing power so each subtraction lines up correctly.
In polynomial division, divide leading terms, multiply the divisor by that quotient term, then subtract the whole product. Continue until the remainder degree is less than the divisor degree.
The standard coefficient shortcut applies to a divisor . Use c, including its sign, and bring down, multiply and add across the coefficients.
The identity is . For division by , the Remainder Theorem gives remainder ; if that is zero, is a factor.
Step-through
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Worked example: a quotient and remainder
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Check .
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Check , and .
Do this: Explain why the tempting shortcut fails.
Wrong idea: dividing by means using synthetic value 3. Rewrite the divisor as , so the value is −3. Check the sign by plugging it into the divisor: .
Tip
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What does the remainder theorem let you calculate without completing long division? It gives the remainder as for division by .
Worked example
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Divide by .
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Flashcards
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Rebuild the starting polynomial.
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What identity checks polynomial division?
Dividend equals divisor times quotient plus remainder.
Why include a zero coefficient for a missing power?
It keeps coefficient positions aligned during division.
For divisor , what synthetic value is used?
4.
For divisor , what synthetic value is used?
−6.
How small must the final remainder degree be?
Less than the divisor degree, unless the remainder is zero.
What does the Remainder Theorem say for ?
The remainder equals .
What does remainder zero tell you for ?
is a factor of .
Divide by .
Quotient and remainder 0.
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Self-test
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Self-test
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6 questions. One at a time.
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