ࡱ> AC@5@ I>bjbj22 "nXXI6t D=2Z&X=Z=Z=Z=Z=Z=Z=$ ?R]Ab~=~==&&&X=&X=&&@'RX:<N 0aM2%~L;<=0=j;A&A<<A< &~=~=& Lesson Plans for Polynomial Division Real Zeros of Polynomial and Complex Numbers Objectives: Upon completion of this lesson, the students will be able to: Polynomial Division Divide polynomials using long division Divide polynomials using synthetic division (where appropriate) Use the Remainder Theorem to evaluate a polynomial Use the Factor Theorem to factor a polynomial Complex Numbers Perform operations with complex number and write the results in standard form Solve a quadratic equation involving complex zeros Polynomials Find all possible rational zeros of a polynomial function using the Rational Root Test Find the real zeros of a polynomial function algebraically Approximate the real zeros of a polynomial function using the Intermediate Value Theorem Approximate the real zeros of a polynomial using a graphing utility Prior to these sections the students should know how to evaluate a polynomial at a given point, factor basic polynomials, and find zeros of quadratics and factorable third degree polynomials. Polynomial Division: (50 minutes) Review: Remind the students that in the early school grades they were taught how to write numbers in expanded form ( EMBED Equation.3 ) and how to do long division. Point out that a number written in expanded form looks very much like a polynomial with the x replaced with 10. Polynomial long division works that same way that long division of numbers works. Also remind students of the vocabulary that is used with division (dividend, divisor, quotient and remainder). These have the same meaning they did when they used them in numerical division. Material: We start with long division: Work through the first example explaining how each step works. The first example is a simple, straightforward example with no remainder. Example 1: Divide by long division. Dividend Divisor  EMBED Equation.3   EMBED Equation.3  Set up the second example with the students and then let then complete the problem. The second example requires using a placeholder for a missing term in the divisor. Example 2: Divide by long division. Dividend Divisor  EMBED Equation.3   EMBED Equation.3  When going over the example, talk with the students about how to deal with the remainder. Give the students the third example to do on their own. Example 3: Divide by long division. Dividend Divisor  EMBED Equation.3   EMBED Equation.3  Since long division takes a good deal of time. You can introduce synthetic division as a time saver for certain types of polynomial division. It works well to introduce it as a short hand when the divisor is of the form  EMBED Equation.3 . Work through the fourth example explaining how each step works. The fourth example is a simple, straightforward example. Example 4: Divide by long division. Dividend Divisor  EMBED Equation.3   EMBED Equation.3  Make sure that the students have to write out the polynomial that is the quotient (not just the coefficients). Ask the students how to set up the fifth example (make sure they see that placeholders are necessary in synthetic division as well) and then let then complete the problem. The fifth example requires using a placeholder for a missing term in the dividend. Example 5: Divide by long division. Dividend Divisor  EMBED Equation.3   EMBED Equation.3  Now is a good time to have the students explore The Remainder Theorem and Factor Theorem. A good way to do this is to have the students evaluate the dividend in example 4 ( EMBED Equation.3 ) at  EMBED Equation.3 . Ask the students what kind of number 4 is. (expect answers like root or zero) Point out that the reminder for the division problem and the value when  EMBED Equation.3  were the same. You can then introduce the Factor Theorem Factor Theorem A polynomial  EMBED Equation.3  has a factor  EMBED Equation.3  if and only if  EMBED Equation.3 . You can now explore if the value of the polynomial at a point, c, is the same as the remainder when the polynomial is divided by x(c is also true if the divisor is not a factor. Ask the students to evaluate the dividend in example 5 ( EMBED Equation.3 ) at  EMBED Equation.3 . Again point out that the reminder for the division problem and the value when  EMBED Equation.3  were the same. You can now state the Remainder Theorem The Remainder Theorem If a polynomial  EMBED Equation.3  is divided by  EMBED Equation.3 , the remainder is  EMBED Equation.3 . Polynomials: (45- 60 minutes) Review: Ask the students to find the real zeros of the polynomial function  EMBED Equation.3 . They should be able to factor this quadratic. Next ask the students to find the real zeros of the polynomial function  EMBED Equation.3  by factoring. You can also review factoring a third degree polynomial by finding the real zeros of  EMBED Equation.3 . In all cases insist that the answers be left in fractional form. Material: Now ask the students if they know how to find the real zeros of the polynomial function  EMBED Equation.3 . We need to try something different since we cannot factor this polynomial. Now draw their attention to the real zeros of the polynomial functions that were used in the review. Ask them what they notice about the numerators of each of the zeros in relation to the original polynomials. (Guide them toward the numerators are factors of the constant term of the polynomial). Also ask them what they notice about the denominators of the zeros in relation to the original polynomials. Make sure to point out that integer zeros have a denominator of 1. (Guide them toward the denominators are factors of the leading coefficient of the original polynomial.) You can now make the generalization to the Rational Zero Test. The Rational Zero Test If the polynomial  EMBED Equation.3  has integer coefficients, then every rational zero of f has the form  EMBED Equation.3  where p and q have no common factors other than 1 and p is a factor of the constant term  EMBED Equation.3  and q is a factor of the leading coefficient  EMBED Equation.3 . Now go back to the problem of finding the real zeros of the polynomial function  EMBED Equation.3 . Ask the students how we might start to answer this problem using the new information that we have. (Hopefully they will suggest writing all the possible rational zeros.) Next ask the students how we might check to see if the possible rational zeros are really rational zeros. (Most students will tell you to just plug in the number to check, but you want to encourage them to find out more that just if it is a zero, but to also factor out the linear factor associated with the zero.) Complete this example with the students by using polynomial division to factor out each of the three factors associated with the three rational roots. Now ask the students to complete example 1 by finding all the possible rational roots and checking to see if they are really roots. Example 1: List all the possible rational roots of the polynomial function  EMBED Equation.3 . Determine which of the possible rational zeros is actually a zero of the function. Ask the students to complete example 2. Example 2: List all the possible rational roots of the polynomial function  EMBED Equation.3 . Determine which of the possible rational zeros is actually a zero of the function. This particular example has two rational roots  EMBED Equation.3  and  EMBED Equation.3 . Now you can expand example 2 to find all the real zeros. You can point out to the students that after dividing out the rational zeros, you are left with the quadratic polynomial expression  EMBED Equation.3 . This is a polynomial that the students can find the zeros of. Ask the students to find all the zeros of the polynomial in example 2. Ask the students what they might do with example 3 (They will likely groan). Example 3: Find all the real zeros of the polynomial function  EMBED Equation.3 . This example leads well into bypassing the rational zero test and using the calculator to determine which possible rational zeros to test. Note that this is not using the calculators ability to determine roots, but just looking at the graph to see what to test and the divide out zeros that work. 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Polynomial Division: (50 minutes) Review: Remind the students that in the early school grades they were taught how to write numbers in expanded form ( EMBED Equation.3 ) and how to do long division. Point out that a number written in expanded form looks very much like a polynomial with the x replaced with 10. Polynomial long division works that same way that long division of numbers works. Also remind students of the vocabulary that is used with division (dividend, divisor, quotient and remainder). These have the same meaning they did when they used them in numerical division. Material: We start with long division: Work through the first example explaining how each step works. The first example is a simple, straightforward example with no remainder. Example 1: Divide by long division. Dividend Divisor  EMBED Equation.3   EMBED Equation.3  Set up the second example with the students and then let then complete the problem. The second example requires using a placeholder for a missing term in the divisor. Example 2: Divide by long division. Dividend Divisor  EMBED Equation.3   EMBED Equation.3  When going over the example, talk with the students about how to deal with the remainder. Give the students the third example to do on their own. Example 3: Divide by long division. Dividend Divisor  EMBED Equation.3   EMBED Equation.3  Since long division takes a good deal of time. You can introduce synthetic division as a time saver for certain types of polynomial division. It works well to introduce it as a short hand when the divisor is of the form  EMBED Equation.3 . Work through the fourth example explaining how each step works. The fourth example is a simple, straightforward example. Example 4: Divide by long division. Dividend Divisor  EMBED Equation.3   EMBED Equation.3  Make sure that the students have to write out the polynomial that is the quotient (not just the coefficients). Ask the students how to set up the fifth example (make sure they see that placeholders are necessary in synthetic division as well) and then let then complete the problem. The fifth example requires using a placeholder for a missing term in the dividend. Example 5: Divide by long division. Dividend Divisor  EMBED Equation.3   EMBED Equation.3  Now is a good time to have the students explore The Remainder Theorem and Factor Theorem. A good way to do this is to have the students evaluate the dividend in example 4 ( EMBED Equation.3 ) at  EMBED Equation.3 . Ask the students what kind of number 4 is. (expect answers like root or zero) Point out that the reminder for the division problem and the value when  EMBED Equation.3  were the same. You can then introduce the Factor Theorem Factor Theorem A polynomial  EMBED Equation.3  has a factor  EMBED Equation.3  if and only if  EMBED Equation.3 . You can now explore if the value of the polynomial at a point, c, is the same as the remainder when the polynomial is divided by x(c is also true if the divisor is not a factor. Ask the students to evaluate the dividend in example 5 ( EMBED Equation.3 ) at  EMBED Equation.3 . Again point out that the reminder for the division problem and the value when  EMBED Equation.3  were the same. You can now state the Remainder Theorem The Remainder Theorem If a polynomial  EMBED Equation.3  is divided by  EMBED Equation.3 , the remainder is  EMBED Equation.3 . Polynomials: (45- 60 minutes) Review: Ask the students to find the real zeros of the polynomial function  EMBED Equation.3 . They should be able to factor this quadratic. Next ask the students to find the real zeros of the polynomial function  EMBED Equation.3  by factoring. You can also review factoring a third degree polynomial by finding the real zeros of  EMBED Equation.3 . In all cases insist that the answers be left in fractional form. Material: Now ask the students if they know how to find the real zeros of the polynomial function  EMBED Equation.3 . We need to try something different since we cannot factor this polynomial. Now draw their attention to the real zeros of the polynomial functions that were used in the review. Ask them what they notice about the numerators of each of the zeros in relation to the original polynomials. (Guide them toward the numerators are factors of the constant term of the polynomial). Also ask them what they notice about the denominators of the zeros in relation to the original polynomials. Make sure to point out that integer zeros have a denominator of 1. (Guide them toward the denominators are factors of the leading coefficient of the original polynomial.) You can now make the generalization to the Rational Zero Test. The Rational Zero Test If the polynomial  EMBED Equation.3  has integer coefficients, then every rational zero of f has the form  EMBED Equation.3  where p and q have no common factors other than 1 and p is a factor of the constant term  EMBED Equation.3  and q is a factor of the leading coefficient  EMBED Equation.3 . Now go back to the problem of finding the real zeros of the polynomial function  EMBED Equation.3 . Ask the students how we might start to answer this problem using the new information that we have. (Hopefully they will suggest writing all the possible rational zeros.) Next ask the students how we might check to see if the possible rational zeros are really rational zeros. (Most students will tell you to just plug in the number to check, but you want to encourage them to find out more that just if it is a zero, but to also factor out the linear factor associated with the zero.) Complete this example with the students by using polynomial division to factor out each of the three factors associated with the three rational roots. Now ask the students to complete example 1 by finding all the possible rational roots and checking to see if they are really roots. Example 1: List all the possible rational roots of the polynomial function  EMBED Equation.3 . Determine which of the possible rational zeros is actually a zero of the function. Ask the students to complete example 2. Example 2: List all the possible rational roots of the polynomial function  EMBED Equation.3 . Determine which of the possible rational zeros is actually a zero of the function. This particular example has two rational roots  EMBED Equation.3  and  EMBED Equation.3 . Now you can expand example 2 to find all the real zeros. You can point out to the students that after dividing out the rational zeros, you are left with the quadratic polynomial expression  EMBED Equation.3 . This is a polynomial that the students can find the zeros of. Ask the students to find all the zeros of the polynomial in example 2. Ask the students what they might do with example 3 (They will likely groan). Example 3: Find all the real zeros of the polynomial function  EMBED Equation.3 . This example leads well into bypassing the rational zero test and using the calculator to determine which possible rational zeros to test. Note that this is not using the calculators ability to determine roots, but just looking at the graph to see what to test and the divide out zeros that work. 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