Zeros, Multiplicity, and Graph Shape

Develop zeros as factors, even multiplicity, odd multiplicity.

Lesson 2 of 43 parts Practice Flashcards

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Zeros as Factors

Objective: Explain zeros as factors, apply the rule, and justify relevant restrictions.

A real zero r corresponds to factor (x−r) for a polynomial. Substitute r or divide by x−r to verify the claim. Evaluate the polynomial at that input. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

p(c)=0 ⇔ (x−c) is a factor of p(x)

Worked examples

Try itFor p(x)=(x−7)(x+3), find p(7).Show answer

Step 1: The factor x−7 is zero at x=7. Step 2: The product is 0·(10). Step 3: Thus p(7)=0.

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Zeros as Factors

Objective: Explain zeros as factors, apply the rule, and justify relevant restrictions.

Formulas

p(c)=0 ⇔ (x−c) is a factor of p(x)

Worked examples

Try itFor p(x)=(x−7)(x+3), find p(7).Show answer

Step 1: The factor x−7 is zero at x=7. Step 2: The product is 0·(10). Step 3: Thus p(7)=0.

Even Multiplicity

Objective: Explain even multiplicity, apply the rule, and justify relevant restrictions.

An even-multiplicity root typically touches the x-axis and turns, because the factor does not change sign across that root. No, it generally touches and turns locally. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$p(x)=(x−c)^m g(x), g(c)≠0; even m: touch$

Worked examples

Try itFor p(x)=(x−3)²(x+7), find the multiplicity of x=3.Show answer

Step 1: The factor x−3 is squared. Step 2: The root x=3 appears twice. Step 3: Its multiplicity is 2.

Odd Multiplicity

Objective: Explain odd multiplicity, apply the rule, and justify relevant restrictions.

An odd-multiplicity real root crosses the axis locally; a greater multiplicity may make the crossing flatter. Multiplicity counts repeated factors. The exponent on the associated linear factor. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$p(x)=(x−c)^m g(x), g(c)≠0; odd m: cross$

Worked examples

Try itFor p(x)=(x−7)^7(x+5), what is the multiplicity of the root x=7?Show answer

Step 1: The factor x−7 has exponent 7. Step 2: The other factor is nonzero at x=7. Step 3: The multiplicity is 7, which is odd, so the graph crosses there.

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