Lesson 1 of 3
Connect terms, degree, leading coefficient, factors, and zeros.
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Worked example
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For , the degree is 3, leading coefficient is 4 and constant term is 9. Check each label against its meaning.
Big ideas
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A PolynomialA sum of terms with nonnegative whole-number powers of the variable.Exactly: A finite sum of terms , with nonnegative integer exponents and real coefficients .Mix-up alert: A variable in a denominator or a negative power of x does not make a polynomial.See this in Key words is a sum of terms where n is a nonnegative integer. Variable denominators, negative variable exponents, and square roots of x are not polynomial terms.
Combine like terms and write descending powers. The degree is the largest exponent with nonzero coefficient; its coefficient is the leading coefficient. A nonzero constant has degree 0.
In a product such as , add factor degrees to get degree 3. Multiply the leading coefficients to get leading coefficient 2.
A zero r satisfies . A factor gives zero r with MultiplicityHow many times the same zero appears as a factor.Exactly: A zero has multiplicity when divides the polynomial but does not.Mix-up alert: Even multiplicity touches the axis; odd multiplicity crosses it.See this in Key words m. Count distinct zeros separately from their multiplicities. Expand or substitute to check.
Worked example
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Construct a polynomial with leading coefficient 3, simple zero 2 and double zero . Use . Its degree is 3 and . Substitution at 2 and verifies both zeros.
Step-through
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Describe .
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Worked example
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For , the cubic terms cancel. The result is , so degree is 2 and leading coefficient is 2. Reading the first written term without combining would be wrong.
Worked example
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Build a monic cubic with zeros -2, 1, and 5, each once. Use . The leading term is . Check , and each required input makes one factor zero.
Mix-up alert
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Wrong idea: gives a zero of 6. Repair: solve , giving -6. A factor is an expression; a zero is an input.
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Check each exponent.
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Is a polynomial?
Yes, all variable exponents are nonnegative integers.
Is a polynomial?
No, has a negative exponent.
What is the degree of ?
6.
What is the leading coefficient of ?
-4.
What zero comes from x+7?
.
What is the degree of ?
5.
How many distinct zeros does have?
One: .
What is called on the graph?
The output coordinate of the y-intercept (0,).
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Self-test
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6 questions. One at a time.
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Self-test
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6 questions. One at a time.
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Connect the ideas
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