Day Topic(s) /Classwork Homework

1Introduction to Polynomials, Review adding, subtracting, multiplying polynomials, and naming

2Long Division, Synthetic Division

3Solving Polynomials by Factoring (Sum & Diff of Cubes, Quartics that are factorable)

4Quiz 1Remainder and Factor Theorems

5Finding Roots in Graphing Calc, Zeros/Roots of Polynomials

6 Extra Practice

7Quiz 2 and Flex

8 Sketching Polynomials, Find “a”

9Writing equations of Polynomials from the given roots

10Group Quiz

11Review

12 Unit 3 TEST 

Math 3 Unit 3 Polynomials Name _______________________________

Day 1 Notes Introduction to Polynomials1

Classifying by degree Classify by number of terms

Degree Name Example 1 Monomial0 Constant 5 2 Binomial1 Linear 3x + 2 3 Trinomial2 Quadratic x² + 7x – 2; 5x2 4+ polynomial3 Cubic 7x3 + 4x2 – 3x + 8; 2x²

– x3

4 Quartic 5x4 + 3x3 – 5x² + 7x – 65+ nth degree 6x7+3x 7th degree

binomialPolynomial:

Degree of a polynomial:

Standard form of a polynomial: p(x) = -3x3 + 2x2 + 6x – 2

Write each polynomial in standard form. Name by degree and number of terms.1) 4x – (x + 2) 2) (3 + 12x2 ) + (5x – 3x2 – x4) 3) x2 – (7x – 2x3 + 9)

4) x3 + 2x2 – x2 + x 5) (x2 – 5)(x + 3) 6) x (x – 8)2

7) – 2(x + 3) (x + 4) 8) (2x + 1)(4x2 + x – 5) 9. (x – 2)(x – 1)(x + 3)

10) (x – 2)3 11) (x2 – 6)(x2 + 4)

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If f(x) =x3 – 4x2 + x – 4, find each of the following and then plot them on the coordinate plane below.

12) f(0)= 13) f( 4) =

14) f(-1) = 15) f(1) =

16) f(3) = 17) f(2) =

Find but don’t graph.18) f(5) = 19) f(-4) =

Day 1 HWWrite each polynomial in standard form. Classify the polynomial by degree and number of terms.1. 7x + 3x + 5 2. –x3 + x6 + x 3) y = (4 – 8x) – (x2 – 5x + 2)

4. (2 x2−5 x+7 )−(3 x3+x2+2) 5. (x + 3) (x + 4) (x + 5)

5. x (x + 2)2 6. x (x - 1)(x + 1) 7. (x – 12)(2x2 + 10x – 4)

8. (x – 3)3 9. (2x2 – 3)(4x2 – 5) 10. (x+2)( x−2)(2x+5)

Day 2 Notes on Divide using long division:

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1) (9x³ – 18x² – x + 2) ÷ (3x + 1) 2) 6 x3+2 x2−11 x+12

3 x+4 3) (X4 – 3X3 + 5X – 6) ÷ (X + 2)

Synthetic Division: A shortcut way to divide that uses just the coefficients. WARNING: Use ONLY if dividing by (x ± #)Steps to divide P(x) by (x ± a):

1) Example: If P(x) = 3x5 – x4 – 5x + 10 ÷(x+2)

-2 3 -1 0 0 -5 10

Remainder

3) (x³ – 8x² + 17x – 10) ÷ (x – 5) 4) x4+2 x3+x−3

x−1

5) x3+5 x2−x−9

x+2 6) (x5 – 1) ÷ (x – 1)

Day 2 Homework: Divide. Use synthetic division if possible, otherwise use long division. Do on separate sheet of paper!1) (2x³ + 9x² +14x + 7) ¿ (2x+1) 2) (x³ − 2x² −5x + 6) ¿ (x – 1) 3) (-2x³ + 5x² − x + 2) ¿ (x+2) 4) (6x³ – 8x – 2) ÷ (x – 2) 5) (3x4 – 5x3 + 2x² + 3x – 2) ¿ (3x – 2) 6) (3x³ + 17x² + 21x – 9) ¿ (x+3)

x³–x² + 1 x² – x – 66x² + 12x + 16 +

30x+2 -2x² + 9x – 19 +

40x+2 x² + 4x + 5 +

22 x+1

3x² + 8x – 3

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Change sign of a, put in top box.1)Write coefficients of P(x) ***Need a coef. for every degree! Use 0 for missing degree.2) Bring down first number3) Multiply by box, add to next column, repeat4) Your answer starts one degree less than original.

Day 2 Extra Practice Worksheet.1. Dividing by a monomial: (−30 x3 y+12 x2 y2−18 x2 y )÷ (−6 x2 y )

2. Divide using Long Division. a. (6 x2−x−7)÷(3 x+1) b. (4 x2−2x+6)÷(2 x−1)

c. (4 x3−8 x2+3 x−8)÷(2 x−1) d. (2 x5−3 x2−18x−8)÷(x−4)

3. Divide using Synthetic division. a. (2 x2+3 x−4 )÷ ( x−2 ) b. (x4−3 x3+5 x−6 )÷ ( x+2 )

c. (2 x3+4 x−6)÷(x+3) d. (x4−2x3+6 x2−8 x+10)÷(x+2)

e. (x4−10x2+2 x+3)÷(x−3) f. (x3−13 x−12)÷(x−4)

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Day 3 Notes on Solving Polynomials by Factoring

multiple zero:multiplicity:Find the zeros and state the multiplicity of multiple zeros. Then state the degree of the polynomial.1. y = (x + 8) 2 2. y = x4 (x – 2)3 (x + 7) 3. y = x3 – 4x2 + 4x

4. y = (x – 1) (x + 2) 5. y = (x + 4)3 6. y = x2 + 8x + 16

Recall: difference of two squares: a2 – b2 = (a + b) (a-b)

S O A PSum of cubes: a3 + b3 =

Difference of cubes: a3 – b3 =

Find all complex roots (solutions).7) 27x3 + 1 = 0 8) 4x3 – 32 = 0

9) 8x3 + 27 = 0 10) 2x4 = 128x

Factoring Quartic Trinomials (Degree of 4 with 3 terms)GCF: x4 + 5x³ – 6x² = 0 No GCF(in quadratic form) : x4 – 2x² – 8 = 0

Solve each equation.11) x4 + x2 – 12 = 0 12) x4 + 11x2 + 18 = 0 13) x4 – 4x2 = 45 14) x4 – 3x2 = 28

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Common perfect cubes:13= 23 =33 =43 =53 =63 = 103 =

Fundamental Theorem of Algebra: A polynomial of degree n has exactly n roots.(i.e. the number of roots = degree of the polynomial)

Day 3 HW Write each polynomial in standard form B) Classify by degree & by number of terms1) 6x4 – 1 2) 6 – 2x3 – 4 + x3 3) 2 + 3x2 – 2 4) 7

Write the polynomial function in standard form with the given zeros5) 5, -4, 1 6) 0, 3, 2 7) use graph

List the zeros, include multiplicity.8) y =(x + 1) (x – 2) (x + 4) 9) y = (x + 3)2 (x – 5)3 10) y = -5x (x - 3) (x - 1)

11) y = x4 + 4x3 – 12x2 12) y = x3 – 64x 13) y = x2 - 6x + 8

Solve each equation. 14) 125x3 – 64 = 0 15) 4x3 + 108 = 0 16) 27x3 + 1 = 0

17) x4 + 2x2 – 24 = 0 18) x4 – 12x2 + 11 = 0 19) x4 + 3x2 – 4 = 0

Write each complex number in the form a + bi 20)16−√−16 21) √−40−3

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1) quartic binomial 2) -x³ + 2 cubic binomial 3) 3x² quadratic monomial 4) constant 5) y = x³ − 2x² −19x+20 6) y = x³ − 5x² + 6x 7) y = x³ + 9x² + 15x −25 8) x = -1, 2, -4; deg 3 rhr + 9) x = -3 (mult 2), 5(mult 3); deg 5 rhr + 10) x = 0, 3, 1 deg: 3; rhr −

11) x=0(mult 2) , -6, 2 deg:4 rhr + 12) x=0, -8, 8 deg: 3 rhr + 13) x = 2, 4 deg: 2; rhr + 14) x =

45;−2

5±2 i√3

5 15) x = -3, 32±3 i √3

2

16) x=−13; 1

6± i √3

6 17) ±i √6 ; ±2 18) ±1 ; ±√11 19) ±2 i ; ±1 20) 16 – 4i 21) −3+2 i √5

Day 3 Extra Practice: Solve each polynomial by factoring.

1. 8x3 + 125 = 0 2. X4 – 6x2 + 8 = 0 3. X3 = 27

4. x4 + 18 = -11x2 5. 16x4 = 81 6. 2x4 – 5x3 = 3x2

7. 125x3 + 216 = 0 8. X4 – 12x2 – 64 = 0 9. X3 + 64 = 0

10. x4 – 64 = 0 11. 2x3 = 250 12. X4 – 4x2 = 12

13. 4x3 – 16x2 + 12x = 0 14. 81x3 – 192 = 0 15. 5x3 = 5x2 + 12x

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Day 4 NotesRemainder Theorem: The value of the polynomial P(x ) at x=a is the same as the remainder you get when you divide that polynomial P(x ) by x−a.Factor Theorem: If P (a )=0 ,then x−a is a factor of P (x ) .

a. If P ( x )=3 x5−x4−5 x+10, find P(−2) using synthetic division.

b. If f ( x )=2x4−8 x2+5 x−7, find f (3 ). c. If p ( x )=3 x47+2 x32+4 x , find p (−1 ) .

d. Determine whether x=2 is a zero of p ( x )=3 x7−x4+2 x3−5x2−4.

e. Determine whether x=−4 is a solution of x6+5 x5+5 x4+5 x3+2x2−10 x−8=0.

f. Is x−2 a factor of f ( x )=x4−4 x3+5 x2+4 x−12?

g. Use synthetic division and the given factor to find all the zeros of the function. y = x3 + 3x² – 5x – 4; (x + 4)

Find the values for k so that each remainder is 5.h. (2x2 –8x + k) ¿ (x – 7) i. (x3 + 4x2 + kx + 8) ¿ (x + 2)

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Day 4 HW:1. If P ( x )=x3−7 x2+15 x−9, find P(3) using synthetic division. 2. If f ( x )=3 x3+10 x2−x−12, find f (−3 ) .

3. If p ( x )=x4+3 x2+x+4 , find p (−3 ) . 4. Determine whether x=−2 is a zero of p ( x )=x3+4 x2+4 x .

5. Determine whether x=−3 is a solution of 2 x4++x3+5 x2−45=0.

6. Factor f ( x )=x3+2 x2−5x−9 given that f (−1 )=0. 7. Factor f ¿ given that x+2 is a factor.

Use synthetic division and the given factor to find all the zeros of the function.

8) y = x3 + 3x² – 13x – 15; (x + 5) 9) y = x3 – 13x + 12; (x + 4) 10) y = 2x3 + 14x² + 13x + 6; (x+ 6)

Find the values for k so that each remainder is 5.11. (x4 + kx3 – 7x2 + 8x + 25) ¿ (x – 2) 12. (x2 + 2x + 6) ¿ (x + k)

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Day 4 More PracticeMORE REMAINDER & FACTOR THEOREM Name:__________________________

Use Synthetic Division to find the quotient. Is the binomial a factor of the polynomial?1. (4x3 – 9x2 – 10x – 2) ¿ (x – 3) 2. (2x3 + 5x2 – 9x + 20) ¿ (x + 4)

3. (x4 – 6x3 – 2x – 10) ¿ (x + 1) 4. (3x4 – 9x3 – 32x2 + 54) ¿ (x – 5)

5. (x3 – 64) ¿ (x – 4)

Given a polynomial and one of its factors, find the remaining factors of the polynomial. 6. x3 + 6x2 – x – 30; x + 5 7. x3 – 11x2 + 36x – 36; x – 6

8. 2x3 + 3x2 – 65x + 84; x – 4 9. 2x3 + 15x2 – 14x – 48; x – 2

Given a polynomial and one of its factors, find all the zeros of the polynomial.

10. y = 16x5 + 32x4 –x – 2; x + 2 11. y = x3 – 3x + 2; x – 1

12. y = 6x3 – 25x2 + 2x + 8; 3x – 2 13. y = 16x5 – 32x4 – 81x + 162; x – 2

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Day 5 Notes on Finding roots using a graphing calculator To solve a polynomial, first ask yourself: How many answers should I have?Use a graphing calc. to find the rational roots of the function, then divide using these zeros to get to a quadratic or linear expression. To solve a quadratic that cannot be factored use the quadratic formula.

1) y = 4x3 + 16x2 – 22x – 10

Use synthetic to get quadratic factor and solve!

Find all the zeros of each function.

2) f(x) = x3 + 3x2 – 6x – 8 3) y = x3 – 5x2 + 17x – 13

4) f(x) = x3 + x2 – 7x + 2 5) y = 2x4 – 3x3 – 2x2 + x – 2

6) g(x) = x3 – 5x2 + 5x – 4 7) m(x) = x3 + 4x2 + x – 6

8) y = x4 – 5x³ – 13x² + 62x – 120 9) f(x) = x3 – 6x2 + 13x – 20

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Graph the polynomial

Find a zero 2nd CALC

Day 5 Extra Practice: Find all the zeros of each function. Please show all your work.

1. y=x3+x+10 2. y=x3+4 x2−x−10

3. y=x3−3 x2−15 x+125 4. y=x3−27

5. y=x3−2 x2+4 x−3 6. y=x4−3 x3+4 x2−6 x+4

7. y=x4−3 x3−6 x2+14 x−12 8. y=x4−6 x3+6 x2+24 x−40

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Day 6 Notes on writing the polynomial given the graph.Steps to find the equation given a graph: A. Find zeros B. Set factors = to y with leading coefficient as “a”C. Plug in pt (shown on graph) and solve for “a” D. Write equation in factored form (with a value for “a”)

Ex 1. Ex. 2.

1. ______________________________ 2. ______________________________

Ex. 3 Ex. 4

3. __________________________ 4. ____________________________

5. Write the equation of the polynomial that has the given zeros and passes through the given point. (factored form) {4,-2,3} ; (0,6)

END BEHAVIOR:

Even Degree: Positive leading coefficient: ______________ Negative leading coefficient: _____________

Odd Degree: Positive leading coefficient: ______________ Negative leading coefficient: _____________14

Sketch the following polynomials on the axis provided without a graphing calculator. When stating the zeros, indicate any multiplicities other than 1.

a. f ( x )=(x+1)(x−2) b. g ( x )=−( x+3)(x+2)(x−1) c. h ( x )=−x (x−2)(x+4)( x+1)

d. p ( x )=( x−2 )2(x+3) e. j ( x )=−3 (x+1)3x2 f. f ( x )− (x+3 )5(x−1)

Day 6 HW. I. A complete graph of a polynomial is shown.

a. Is the degree even or odd?

b. Is the leading coefficient positive or negative?c. What are the real zeros? d. What is the smallest possible degree?

1. 2.

15

Degree: ____

Leading term: pos or neg

End Behavior:

Zeros: ________________

y-int: ______________

a. _______

b. ________

c. ____________

Degree: ____

Leading term: pos or neg

End Behavior:

Zeros: ________________

y-int: ______________

Degree: ____

Leading term: pos or neg

End Behavior:

Zeros: ________________

y-int: ______________

Degree: ____

Leading term: pos or neg

End Behavior:

Zeros: ________________

y-int: ______________

Degree: ____

Leading term: pos or neg

End Behavior:

Zeros: ________________

y-int: ______________

Degree: ____

Leading term: pos or neg

End Behavior:

Zeros: ________________

y-int: ______________

a. _______

b. ________

c. ____________

II. Sketch each function. a. y=−2(x2−9)(x+4) b. y=( x2−4 ) (x+3 )c . y=−1(x2−9)(x2−4)

d. y=14(x+2)(x−1)2 e. y=

15(x−3)2(x+1)2 f. y= (x+1 )3(x−4)

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a. _______

b. ________

c. ____________

a. _______

b. ________

c. ____________

Degree: ____ Leading term: pos or neg

End Behavior:Zeros: ________________

y-int: ______________

Degree: ____ Leading term: pos or neg

End Behavior:Zeros: ________________

y-int: ______________

Degree: ____ Leading term: pos or neg

End Behavior:Zeros: ________________

y-int: ______________

Degree: ____ Leading term: pos or neg

End Behavior:

Zeros: ________________

y-int: ______________

Degree: ____ Leading term: pos or neg

End Behavior:

Zeros: ________________

y-int: ______________

Degree: ____ Leading term: pos or neg

End Behavior:

Zeros: ________________

y-int: ______________

Day 6 Extra Practice with sketching polynomials.1) y = (x + 3)(x + 4)(x – 2) 2) y = (x + 5)(x + 2) (x – 1) (x – 4) 3) y = (x + 3) 3 (x – 1)2 4) y = -2x (x + 3)(x -1)2 5) y = 5x (x – 3)(x + 2)

6) y = -7x(x – 5)(x + 3)³ 7) y = x5 – 2x4 – 8x3 8) y = x3 + 7x2 + 10x 9) y = x4 + 8x3 + 16x2

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Day 7 Notes on Writing an Equation with integral coefficients given the roots.Standard form of a quadratic equation: y=a x2+bx+c

Irrational root theorem: If a+√b is a root of P(x), then its conjugate, a−√b , is also a root of P(x).

Complex Conjugate root theorem: If a+bi is a root of P(x), then its conjugate, a−bi , is also a root of P(x).

I. Write the equation with integral coefficients given the roots.

a. -3, 4 b. 0, -9 c. 13,2

d. 7±√2 e. 1+3 i f. √5

f. 5, −4 i g. -1, 0, 1 h. 0(m2),3+√7

Day 7 HW. Write an equation that has integral coefficients and the given the roots.

1. –3(m2) _____________________________ 2. ±7 i _______________________

3. 4 , 3

4 _____________________________ 4. 2±5 i ________________________

5. √3 , 4 _____________________________ 6. 2−√2 ________________________

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Day 7 Extra Practice: Find a polynomial equation of least possible degree, with integer coefficients, that has the given numbers as roots.1. −3 ,3−√6 2. √5 ,2 i

3. −4 ,2+3i 4. 3 ,1+ i

5. √5 ,−√7 6. 0 ,−2i ,3+√2

7. 1 ,3 i 8. 3+i ,−3

9. −1 ,2−i 10. 4−√6 ,√3

11. 2 ,1−√10 12. −2 ,−5+√3

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More ReviewSTANDARD FORM FACTORED FORM X-INTERCEPTS

y = x2 + 5x + 6 y = (x + 2) (x + 3) x = -2, -3

Find the zeros of each function.1. y = (x+1) (x-1) (x-3) 2. y = (x+2) (x-3) 3. y = x (x – 3) (x + 1)Find the zeros and state the multiplicity of multiple zeros. Then state the degree of the polynomial.2. y = (x + 1)(x – 1) (x – 2) 3. y = x3 – 36x 4. y = x4 + 3x3 – 4x2

5. y = (x – 4)2 6. y = x3(x - 2)4 7. y = x3 + 7x2 + 10x8. y = x (x + 17) 9. y = x4 – 7x3 – 18x21 10. y = x4 + 6x3 + 8x2

Write a polynomial in factored form with the given zeros. Then expand into standard form.11. –4, -2 12. –3, -2, 1 13. x = 0, 2, -6 14. x = -1, 3, 4Name the least degree and the number of real solutions of the polynomial graphed. 15. 16. 17.

Write a polynomial in factored form on top, then standard form for the curve graphed on bottom. 18. 19. 20.

18. 19. 20.20

-1

-1-1

1

11

SIGNS ZEROSSOLUTIONSROOTS

Study GuideWrite each polynomial in standard form. Classify by degree and number of terms.1. 3x2 + 6x – 3x3 +4 2. 2x – 5 + 3x – 2 3. (x + 1)(x – 1)(x – 3) 4. 3 – 4(5x – 4) + 2x2 -5x

Graph.

5. y = x(x – 2)(x – 1) 6. y = (x+1)2(x-3)2 7. y = –x3(x + 2) 8. y = -2(x-3)(x+4)2

Write the equation of the polynomial that has the given zeros and passes through the given point.9. 2,-3,1; (-1,2) 10. 0,1,-2 (mult. 2); (-1,3)

Write a polynomial in standard form for the given zeros or graph. 11) 0, -4, 1 12)

Solve each equation (without using synthetic division). 13) 8x3 – 27 = 0 14) x3 + 64 = 0 15) x4 – 12x2 – 64 = 0 16) x4 + 7x2 – 18 = 0

21

x

y

Divide using long division. Divide using synthetic division.17) 6x3 + 2x2 – 11x + 12 ¿ (3x + 4) 18) x3 + 3x + 4 ¿ (x + 2)

Use synthetic division and the given factor to find all the zeros. 19) x3 – 2x2 – 19x + 20 = 0, (x – 1) 20) x3 – 37x + 84 = 0; (x – 4)

21) Use the remainder theorem to find f(-2) given f(x) = x3 – 5x – 3.

22) Show that x + 5 is a factor of x3 +2x2 – 13x + 10 and find the remaining linear factors.

A polynomial equation with integral coefficients has the given roots. Find any additional roots.

23) 1 + 4i 24) 3 -

Find a third-degree polynomial equation that has the given roots.25) –7, 3i 26) 3, -4i

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