11.5 Notes - arrowheadschools.org · Notes 115 Finding Zeros by Factoring 1. Factor x2 ─ 9 = 0 2. Factor 7x2 ─ 28 = 0 3. Factor 5p3 + 12 p2 + 7p = 0. AdvAlg11.5TheFactorTheorem.notebook

AdvAlg11.5TheFactorTheorem.notebook

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11.5 Notes.tns


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The product of two number equals zero if and only if at least one of the numbers is zero.

If the product of two number is zero then at least one of the number must be zero

If at least one of two number is zero then the product of the numbers must be zero.


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The final solution is x = 0, 5, or 6


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The xintercepts of the graph of a function are the reals zeros of the function.


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If xr is a factor of a polynomial then r is a zero of the polynomial.

If r is a zero of a polynomial then xr is a factor of the polynomial.

7.


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8.9.


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The zeros are x=0, x=3, and x=2

The xint. are x=0, x=3, and x=2


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Where a is a number or a polynomial in x.


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1. Write a polynomial function that would have zeroes of ( ,0) ( 5/3, 0) (7/2,0)

2. Write a polynomial function represented by the following: The graph crosses through the xaxis at ( 9,0) and then touches the xaxis at (5, 0) and goes back down.


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Zero Product Theorem: For all a and b, ab = 0 if and only if a = 0 or b = 0

Notes 115

Finding Zeros by Factoring

1. Factor x2 ─ 9 = 0 2. Factor 7x2 ─ 28 = 0

3. Factor 5p3 + 12 p2 + 7p = 0


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Finding Zeros by Factoring

4. Factor 10c2 ─ 11c + 3 = 0

5. Factor x2 + 4x = 21

6. Factor m2 ─ 12m = ─36


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Factoring By Finding Zeros

1. Graph P(x) = x3 ─ 9x2 + 18x

Find the zeros _____________

Find the factors _____________

2. Graph 2x3 ─ 19x2 + 35x

Find the zeros _____________

Find the factors _____________

x r is a factor of a polynomial P(x) if and only if P(r) = 0


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3. Graph the function P(x) = x5 ─ 13x3 + 36x Give the real zeros of the function


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Finding Equations from Zeros1. Find the general form of the equation with zeros at 1, 0, 5

2. Give equations for 3 different polynomial functions with zeros at 8, , and

3. Give equations for 3 different polynomial functions with zeros at 0, , 4, and 3

P(x) =

P(x) =P(x) = P(x) =

P(x) =P(x) =P(x) =


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11.111.10 WS A.pdf


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Attachments

11.111.10 WS A.pdf

11.5 Notes.tns

Name

L E S S O NM A S T E R

Properties Objective E

1. True or false. 9x2 1 4x 2 1 1 3x -1 is a polynomial.

In 2–4, a polynomial is given. a. Give its degree.b. Name its leading coefficient.

2. m3 2 3m7 3. 4n 2 15 2 3n2 4. 8p 2 2 1

a. a. a.

b. b. b.

Uses Objective H

5. Diane Chang invested her savings in an account paying r%interest compounded annually. Suppose she invests $90 at the beginning ofeach year for six years. No additional money is added or withdrawn.

a. Write a polynomial expression for the total amountin Diane’s account at the end of the sixth year.

b. Evaluate how much Diane will have if theaccount earns 3.9% interest each year.

Representations Objective J

In 6 and 7, graph the function given.

6. p(x) 5 x4 2 .2x3 2 8x 2 2 1.5x 1 5 7. k(x) 5 5x5 2 3x 2 3

Questions on SPUR ObjectivesSee pages 741-745 for objectives.11-1

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Skills Objective A

In 1–4, expand and write in standard form.

In 5 and 6, multiply and simplify.

Properties Objective E

In 7–9, an expression is given. a. Classify it as amonomial, a binomial, or a trinomial. b. Give its degree.

7. 13t 2 1 4t 3 8. 384m6n2 2 m6n2 9. r 5t 5u2 2 u 2 1

a. a. a.

b. b. b.

Uses Objective I

10. The largest figure at the right is a rectangle.

a. What are its dimensions?

b. What is its area?

11. From a sheet of notebook paper 26.7 cmby 20.3 cm, squares of side x are removedfrom each corner, forming an open box.

a. Sketch a diagram of this situation.

b. Write a formula for the volume V(x) of the box.

c. Write a formula for its surface area S(x).


A

1. (a 2 3)(2a3 2 3a2)

3. (7 2 c)2 (2 2 c)

5. (10e 1 2f )(6f 2 3g 1 1)

2. (b 1 7)(b 2 1)(b 1 4)

4. (-d 1 1)(5d 2 2 2d 2 3)

6. (2h 1 j 2 k)(h 2 j 2 k)

p

r

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Vocabulary

1. Is a2 2 39 prime

a. over the set of polynomials with rational coefficients?

b. over the set of polynomials with real coefficients?

c. Explain your answers to Parts a and b.

2. The Discriminant Theorem for Factoring Quadraticsapplies to quadratics with coefficients.

Skills Objective B

In 3–6, fill in the blanks.

3. 19m2n 2 114mn2 5 19mn ( 2 )

4. 24p3t 1 60p3 5 (2t 1 5)

5. 5wz 1 25w2z 2 35w3z 5 5wz ( 1 2 )

6. (3 2 2h)3 1 (3 2 2h)4 5 (3 2 2h)3( 1 )

In 7–12, factor.

7. a2 2 12a 1 36 8. 9c2 1 6c 1 1 9. 30e3 2 60e2 1 30e

10. g2 2 64h6 11. k 4 2 25k2 12. 5r2 1 9r 2 18

13. a. Write t 4 2 16 as the product oftwo binomials.

b. Write t 4 2 16 as the product ofthree binomials.

14. True or false. 3x2 2 y 2 5 (x=3 1 y)(x=3 2 y)

15. True or false. 9a2 1 b 2 5 (3a 1 bi)(3a 2 bi)

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Representations Objective K

In 1 and 2, estimate the real zeros of thedescribed function to the nearest tenth.

1. f(x) 5 2x4 2 10x2 1 3 2. y 5 2x5 2 5x3 2 2x2 1 3x 1 1

In 3 and 4, use the graph of the functionto determine its integer zeros.

3. y 5 x3 2 9x 2 1 23x 2 15 4. h(x) 5 4 2 3x2 2 x3

5. a. Complete the table of values for the function Qwith equation Q(x) 5 .09x3 2 2x 1 16.

x -10 -8 -6 -4 -2 0 2 4 6

Q (x )

b. Use the table to tell how many zeros the function has.

c. For each zero, indicate the two consecutiveeven integers between which the zero must lie.

d. Describe how you could use a graph to justify your response to Part b.

e. Use technology to find eachzero to the nearest hundredth.


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Skills Objective C

In 1–3, give the exact zeros of the function described.

1. A(x) 5 x(x 2 3)(x 1 4)(2x 2 1)

2. B(x) 5 x2 2 100

3. C(x) 5 x3 2 x

Skills Objective D

4. Give equations for 3 different polynomial functionswith zeros at -8, , and .

Properties Objective F

In 5 and 6, consider the functions with equations.

M( x)5 x(x2 1)(x2 2) and N(x)5 x2(x2 1)(x2 2).

5. True or false. M and N have the same graphs.

6. True or false. M and N have the same zeros.

7. True or false. 3 is not a solution to (g 2 3)(g 1 4) 5 7.

8. Consider the polynomial R(x) 5 2x 32 19x 2 1 35x .

a. List its factors.

b. List its zeros.

c. Which theorem allows you to proceedfrom Part a to Part b without graphing?

Representations Objectives J and K

9. At the right, graph the function Pwith P(x) 5 x 5 2 13x3 1 36x. Givethe real zeros of the function.

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-5 ≤ x ≤ 5 x-scale 5 1-50 ≤ y ≤ 50 y-scale 5 10

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Skills Objective B and C

In 1–4, a polynomial is given. a. Factor over the set of complex numbers. b. Identify all real zeros.c. Check by graphing.

1. A(r) 5 6r2 1 5r 2 4 2. B(t) 5 6t3 1 33t2 2 18t

a. a.

b. b.

c. c.

3. A(x) 5 x2 1 5x 2 7 4. T(y) 5 y 2 3y2 2 4

a. a.

b. b.

c. c.


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Properties Objective G

In 1 and 2, use the Rational Zero Theoremto factor the polynomial.

1. L(x) 5 30x3 2 31x2 1 10x 2 1 2. M(x) 5 x4 1 2x3 1 x2

In 3–7, a polynomial is given. a. Use the RationalZero theorem to list all possible rational zeros.b. Find all rational zeros.

3. P(x) 5 7x 5 2 3x 4 2 2 4. Q(m) 5 64m3 2 1

a. a.

b. b.

5. R(n) 5 3n2 2 15n 2 18 6. S(x) 5 2x 4 2 7x 3 1 5x 2 2 7x 1 3

a. a.

b. b.

7. T(x) 5 x5 1 2x3 2 x2 2 2

a.

b.

Representations Objective J

8. Consider U(x) 5 5x 6 1 6x4 1 x2 1 12.

a. Use the Rational Zero Theorem to listall possible rational zeros.

b. Graph this polynomial on thegrid at the right.

c. Use Parts a and b to find all rational zeros of this polynomial.


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Vocabulary

1. Multiple choice. Consider the equation

(x 1 1)6(x 2 2)(x 2 1) 5 0. Which is true?

(a) 1 is a double root. (b) -1 is a double root.

(c) 1 is a root with multiplicity 6. (d) -1 is a root with multiplicity 6.

Properties Objective F

In 2 and 3, use the equation =6 x3 2 5ix1 .4x2 2 2 5 0.

2. True or false. This equation has at least one complex solution.

3. This equation has exactly roots.

4. Consider the equation (x 2 3)6(x 2 2 3)(x 2 1 9) 5 0.

a. This equation has exactly roots ifmultiplicities of multiple roots are counted.

b. is a rational root with multiplicity .

is an irrational root with multiplicity .

is an irrational root with multiplicity .

is a complex root with multiplicity .

is a complex root with multiplicity .

Culture Objective L

In 5–10, match each mathematician with his contributiontoward solving all polynomial equations.

5. Omar Khayyam, 1100s

6. Ludovico Ferrari, 1500s

7. Niccolo Tartaglia, 1500s

8. Karl Friedrich Gauss, 1700s

9. Évariste Galois, 1800s

10. Niels Abel, 1800s

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a. discovered how to solve quarticequations using complex numbers

b. proved the Fundamental Theoremof Algebra

c. discovered how to solve all cubicequations

d. first showed how to solve manycubic equations

e. proved the general quartic equationcannot be solved using a formula

f. described method for determiningwhich polynomials of degree 5 ormore can be solved with a formula

Name


Skills Objective D

In 1 and 2, a function is described. a. Determinewhether the function is a polynomial function ofdegree ≤ 5. b. If so, give the degree of the function.

1. the function(n, an) where a1 5 2 and an 5 an21 1 3

a. b.

2.

a. b.

3. Suppose the sequence 3, 4, 11, 24, 43, 68, 99, . . .has a formula with degree ≤ 4. Use the method offinite differences to predict the next term.

4. Can the method of finite differences be used withthis set of data? Explain why or why not.

Uses Objective H

In 5 and 6, use the results of an experiment involvingrolling a marble down an inclined plane.

5. What degree polynomial would you use to best modelthese data? Explain your answer.

6. Use your polynomial to predict how far the marblewould be from the given point after 10 seconds.

Time Passed (seconds) 0 1 2 3 4 5 6

Distance of Marble from Given Point (cm) 10 12.4 14.8 17.2 19.6 22 24.4

x 1 3 6 10 15 21 28 36 45

y 24 49 74 99 124 149 174 199 224

x 10 20 30 40 50 60 70 80 90

y -200 -209 -280 -443 -584 -325 1096 4945 13,112


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Skills Objective D

1. Consider the values in the table below.

a. This data can be modeled by a polynomial equationof the form y 5 ax2 1 bx 1 c. List three equationswhich can be used to solve for a, b, and c.

b. Solve the system to find a formulawhich models the data above.

In 2–4, write a formula of degree n ≤ 5 whichmodels the data.

2.

3.

4.

Uses Objective H

5. Suppose that Sylvia stacks soccer balls in a triangularpyramid display. That is, one ball is in the top row, threeare in the second row, six are in the third row, and so on.

a. Complete the table.

b. How many soccer balls are needed for n rows?

Number of rows (n ) 1 2 3 4 5 6

Total Number of Balls (T ) 1 4 10

x 1 2 3 4 5 6 7 8

y -2 3 4 -5 -30 -77 -152 -261

x 1 2 3 4 5 6 7 8

y -3 -16 -27 0 125 432 1029 2048

x 1 2 3 4 5 6 7 8

y -22.5 -24 -15.5 12 67.5 160 298.5 492

x 1 2 3 4 5 6 7 8

y 2 20 50 92 146 212 290 380

Questions on SPUR ObjectivesSee pages 741-745 for objectives.

A11-10

SMART Notebook

SMART Notebook

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11.5 Notes - arrowheadschools.org · Notes 115 Finding Zeros by Factoring 1. Factor x2 ─ 9 = 0 2. Factor 7x2 ─ 28 = 0 3. Factor 5p3 + 12 p2 + 7p = 0. AdvAlg11.5TheFactorTheorem.notebook

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