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Solving Higher Order Polynomials

Solving Higher Order Polynomials

Assessment

Presentation

•

Mathematics

•

9th - 11th Grade

•

Medium

Created by

Theresa Lentz

Used 6+ times

FREE Resource

4 Slides • 40 Questions

1

Solving Higher Order Polynomials

We will be able to...

~Use synthetic division to solve polynomials

~Factor to solve polynomials

~Determine functions and factors from graphs

Slide image

2

Open Ended

Question image

Given the graph, state all the x values where the function intersects the x axis. (You can click on the graph to make it bigger.)

3

Multiple Choice

Question image

The graph has a zero at x=-7. The factor that goes with this zero is...?

1

(x+7)

2

(x-7)

3

(x+0)

4

Multiple Choice

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The graph has another zero at x=2. The factor that goes with this zero is...?

1

(x+2)

2

(x-2)

3

(x+0)

5

Multiple Select

Question image

There are two more zeros, choose the factors that go with the zeros that are left over.

1

(x+1)

2

(x-1)

3

(x+9)

4

(x-9)

6

Building a Function

~We can put all these factors together to write a function that could represent the graph.

~We also need to consider the a value!

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7

Multiple Choice

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Considering all the factors and the end behavior, what is a function that could match the graph?

1

f(x)=8(x+1)(x+7)(x−2)(x−9)f\left(x\right)=8\left(x+1\right)\left(x+7\right)\left(x-2\right)\left(x-9\right)

2

f(x)=(x−1)(x−7)(x+2)(x+9)f\left(x\right)=\left(x-1\right)\left(x-7\right)\left(x+2\right)\left(x+9\right)

3

f(x)=−2(x+1)(x+7)(x−2)(x−9)f\left(x\right)=-2\left(x+1\right)\left(x+7\right)\left(x-2\right)\left(x-9\right)

8

Open Ended

Question image

Analyze the new graph.  Considering all the the zeros, the factors that go with them and the end behavior, write a function that could possibly represent the graph.

9

Open Ended

Question image

Analyze the new graph.  Considering all the the zeros, the factors that go with them and the end behavior, write a function that could possibly represent the graph.

10

Open Ended

Question image

Analyze the new graph.  Considering all the the zeros, the factors that go with them and the end behavior, write a function that could possibly represent the graph.

11

Solve:

 0=2x3+3x2−18x−270=2x^3+3x^2-18x-27  

  • Graph, determine as many zeros as possible from graph

  • Synthetically divide until you run out of zeros

  • Solve with factoring, quadratic formula, or inverse operations

12

Multiple Select

Solve the problem with me, then select ALL the solutions to let me know you're there!

 0=2x3+3x2−18x−270=2x^3+3x^2-18x-27  

1

x=2

2

x=-3

3

x=3

4

x=-2/3

5

x=-3/2

13

Multiple Select

Graph the function to find whole number zeros, do not include decimals.

 0=2x3−x2−27x+360=2x^3-x^2-27x+36  

1

x=3

2

x=-2

3

x=1

4

x=-4

14

Multiple Choice

You should have seen that the graph crosses at x=-4 and 3.  We can take one of those zeros and divide.  Doesn't matter which one... Let's divide by 3 first.

 0=2x3−x2−27x+360=2x^3-x^2-27x+36  

1

 2x2+5x+122x^2+5x+12  

2

 2x2+5x−122x^2+5x-12  

3

 2x3+5x2+12x2x^3+5x^2+12x  

4

 3x2+8x−33x^2+8x-3  

15

Fill in the Blanks

Type answer...

16

Fill in the Blanks

Type answer...

17

Open Ended

Question image

Write a function in factored form that COULD represent the graph based on it's zeros and end behavior.

18

Multiple Choice

Solve: First graph. What is the only zero we can see?

 x3−x2+49x−49=0x^3-x^2+49x-49=0  

1

x=2

2

x=-2

3

x=1

4

x=-1

19

Multiple Choice

Solve: Since x=1 is the only zero, we need to divide by it.  What is the result?

 x3−x2+49x−49=0x^3-x^2+49x-49=0  

1

 x3+49xx^3+49x  

2

 x3+49x2x^3+49x^2  

3

 x2+49xx^2+49x  

4

 x2+49x^2+49  

20

Multiple Choice

Solve: We are left a quadratic we can solve with inverse operations. What is the first step to solve for x?

 x2+49=0x^2+49=0  

1

take the square root of both sides

2

add 49

3

subtract 49

4

divide by 49

21

Multiple Choice

Solve: Now we can take the square root of both sides.  What would x equal?

 x2=−49x^2=-49  

1

7

2

 ±7i\pm7i  

3

 7\sqrt{7}  

4

 ±7\pm7  

22

Multiple Choice

Solve: First graph. What is the only zero we can see?

 x3−7x2+37x+45=0x^3-7x^2+37x+45=0  

1

x=2

2

x=-2

3

x=1

4

x=-1

23

Multiple Choice

Solve: Synthetically divide by the zero x=-1. What is the result?

 x3−7x2+37x+45=0x^3-7x^2+37x+45=0  

1

 x2−8x+45x^2-8x+45  

2

 x2−6x+43x^2-6x+43  

3

 x2−8x+29x^2-8x+29  

4

 x2+8x+29x^2+8x+29  

24

Multiple Choice

Solve: We can't solve this with inverse operations, we have to factor or use...dun dun DUN... QUADRATIC FORMULA!?!?!?

 x2−8x+45=0x^2-8x+45=0  

1

 −8±(−8)2+4(1)(45)2(1)\frac{-8\pm\sqrt{\left(-8\right)^2+4\left(1\right)\left(45\right)}}{2\left(1\right)}  

2

 8±(−8)2−4(1)(45)2(1)\frac{8\pm\sqrt{\left(-8\right)^2-4\left(1\right)\left(45\right)}}{2\left(1\right)}  

3

 8±(−8)+4(1)(45)2(1)\frac{8\pm\sqrt{\left(-8\right)+4\left(1\right)\left(45\right)}}{2\left(1\right)}  

25

Multiple Choice

Solve: When we do the quadratic formula, what do you get as the discriminate?  (Number under the square root.)

 8±(−8)2−4(1)(45)2(1)\frac{8\pm\sqrt{\left(-8\right)^2-4\left(1\right)\left(45\right)}}{2\left(1\right)}  

1

-172

2

-224

3

-116

4

-360

26

Multiple Choice

Solve: Now we have to break down the number under the square root! Remember!?! What perfect square can we divide -116 by?

 8±−1162(1)\frac{8\pm\sqrt{-116}}{2\left(1\right)}  

1

4

2

9

3

16

4

25

27

Multiple Choice

Solve: We can use 4 and 29 to break down the radical.  What would our answer look like after taking the square roots?

 8±−14292\frac{8\pm\sqrt{-1}\sqrt{4}\sqrt{29}}{2}  

1

 8±4i292\frac{8\pm4i\sqrt{29}}{2}  

2

 8±2i292\frac{8\pm2i\sqrt{29}}{2}  

3

 8±2292\frac{8\pm2\sqrt{29}}{2}  

28

Multiple Choice

Solve: Last step!  Reduce the numerator by the denominator if you can. 

 8±2i292\frac{8\pm2i\sqrt{29}}{2}  

1

 4±i294\pm i\sqrt{29}  

2

 4±i292\frac{4\pm i\sqrt{29}}{2}  

3

 8±i298\pm i\sqrt{29}  

4

 4±2i294\pm2i\sqrt{29}  

29

That was ALOT of steps... but it's not so bad.

  • Graph to find whole number zeros

  • Continually synthetically divide until you run our of zeros

  • Solve by factoring, quadratic formula, or inverse operations

30

Multiple Choice

Solve: First graph. What is the only zero we can see?

 x3−2x+4=0x^3-2x+4=0  

1

x=2

2

x=-2

3

x=1

4

x=-1

31

Multiple Choice

Solve: x=-2 is the only zero on the graph so divide by it.  What do you get?

 x3−2x+4=0x^3-2x+4=0  

1

 x2−4x^2-4  

2

 x2−4x+12x^2-4x+12  

3

 x2−2x+2x^2-2x+2  

4

 x3+2x2−6x+16x^3+2x^2-6x+16  

32

Multiple Choice

Solve: Now we decide how to solve.  Factor if you can, otherwise we need to do quadratic formula.

 x2−2x+2=0x^2-2x+2=0  

1

 2±(−2)2−4(1)(2)2(1)\frac{2\pm\sqrt{\left(-2\right)^2-4\left(1\right)\left(2\right)}}{2\left(1\right)}  

2

 −2(−2)2+4(1)(2)2(1)\frac{-2\sqrt{\left(-2\right)^2+4\left(1\right)\left(2\right)}}{2\left(1\right)}  

3

 −2±(−4)2−4(1)(−2)2(1)\frac{-2\pm\sqrt{\left(-4\right)^2-4\left(1\right)\left(-2\right)}}{2\left(1\right)}  

33

Multiple Choice

Solve: Determine the discriminant

 2±(−2)2−4(1)(2)2(1)\frac{2\pm\sqrt{\left(-2\right)^2-4\left(1\right)\left(2\right)}}{2\left(1\right)}  

1

 2±−42(1)\frac{2\pm\sqrt{-4}}{2\left(1\right)}  

2

 2±42(1)\frac{2\pm\sqrt{4}}{2\left(1\right)}  

3

 2±−322(1)\frac{2\pm\sqrt{-32}}{2\left(1\right)}  

4

 2±322(1)\frac{2\pm\sqrt{32}}{2\left(1\right)}  

34

Multiple Choice

Solve: Break down the radical.

 2±−42(1)\frac{2\pm\sqrt{-4}}{2\left(1\right)}  

1

 2±22(1)\frac{2\pm2}{2\left(1\right)}  

2

 2±2i2(1)\frac{2\pm2i}{2\left(1\right)}  

3

 2±4i2(1)\frac{2\pm4i}{2\left(1\right)}  

4

 2±42(1)\frac{2\pm4}{2\left(1\right)}  

35

Multiple Choice

Solve: Reduce if you can

 2±2i2(1)\frac{2\pm2i}{2\left(1\right)}  

1

 ±i\pm i  

2

 2±2i2\pm2i  

3

 1±i1\pm i  

4

 2±i2\pm i  

36

Fill in the Blanks

Type answer...

37

Multiple Choice

Solve: x=-3 is the only zero on the graph so divide by it.  What do you get?

 2x3+9x2+14x+15=02x^3+9x^2+14x+15=0  

1

 2x2+3x+52x^2+3x+5  

2

 2x2+15x+59+162x+32x^2+15x+59+\frac{162}{x+3}  

3

 2x3+3x2+5x2x^3+3x^2+5x  

4

  x2+3x−5x^2+3x-5  

38

Multiple Choice

Solve: Now we decide how to solve.  Factor if you can, otherwise we need to do quadratic formula.

 2x2+3x+5=02x^2+3x+5=0  

1

 −3±(3)2−4(2)(5)2(2)\frac{-3\pm\sqrt{\left(3\right)^2-4\left(2\right)\left(5\right)}}{2\left(2\right)}  

2

 3±(3)2−4(2)(5)2(1)\frac{3\pm\sqrt{\left(3\right)^2-4\left(2\right)\left(5\right)}}{2\left(1\right)}  

3

 −3(3)2+4(2)(5)2(2)\frac{-3\sqrt{\left(3\right)^2+4\left(2\right)\left(5\right)}}{2\left(2\right)}  

39

Multiple Choice

Solve: Determine the discriminant

 −3±(3)2−4(2)(5)2(2)\frac{-3\pm\sqrt{\left(3\right)^2-4\left(2\right)\left(5\right)}}{2\left(2\right)}  

1

 −3±−312(2)\frac{-3\pm\sqrt{-31}}{2\left(2\right)}  

2

 −3±−492(2)\frac{-3\pm\sqrt{-49}}{2\left(2\right)}  

3

 −3±−372(2)\frac{-3\pm\sqrt{-37}}{2\left(2\right)}  

4

 −3±492(2)\frac{-3\pm\sqrt{49}}{2\left(2\right)}  

40

Multiple Choice

Solve: Break down the radical.

 −3±−312(2)\frac{-3\pm\sqrt{-31}}{2\left(2\right)}  

1

 −3±i314\frac{-3\pm i\sqrt{31}}{4}  

2

 −3±31i4\frac{-3\pm31i}{4}  

3

 −3±314\frac{-3\pm31}{4}  

4

 −3±314\frac{-3\pm\sqrt{31}}{4}  

41

Multiple Select

Solve: First graph. What TWO zeros do you see on the graph?

 x4−3x3+12x−16=0x^4-3x^3+12x-16=0  

1

x=-2

2

x=2

3

x=4

4

x=-4

5

x=0

42

Multiple Choice

Solve: Since there are two zeros, you will divide twice in any order.  Just be sure to continue dividing, don't divide the original problem twice

 x4−3x3+12x−16=0x^4-3x^3+12x-16=0  

1

 x2−3x+4x^2-3x+4  

2

 x3−3x2+4xx^3-3x^2+4x  

3

 3x2+2x−53x^2+2x-5  

4

  x2−x+4x^2-x+4  

43

Multiple Choice

Solve: Now we decide how to solve.  Factor if you can, otherwise we need to do quadratic formula.

 x2−3x+4=0x^2-3x+4=0  

1

 3±(−3)2−4(1)(4)2(1)\frac{3\pm\sqrt{\left(-3\right)^2-4\left(1\right)\left(4\right)}}{2\left(1\right)}  

2

 3(−3)2+4(1)(4)2(1)\frac{3\sqrt{\left(-3\right)^2+4\left(1\right)\left(4\right)}}{2\left(1\right)}  

3

 −3±(−3)2+4(1)(4)4(1)\frac{-3\pm\sqrt{\left(-3\right)^2+4\left(1\right)\left(4\right)}}{4\left(1\right)}  

44

Multiple Choice

Solve: Now simplify as much as possible

 3±(−3)2−4(1)(4)2(1)\frac{3\pm\sqrt{\left(-3\right)^2-4\left(1\right)\left(4\right)}}{2\left(1\right)}  

1

 3±i72\frac{3\pm i\sqrt{7}}{2}  

2

 3±72\frac{3\pm\sqrt{7}}{2}  

3

 3±7i2\frac{3\pm7i}{2}  

Solving Higher Order Polynomials

We will be able to...

~Use synthetic division to solve polynomials

~Factor to solve polynomials

~Determine functions and factors from graphs

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