10G (2.4) PART 2 Introduction to Polynomials

Introduction to Polynomials CONT.

​(2.4 PART2)

In the past, we could move things around and get x by itself on one side. If we try that here, we get:

I can't see anything to do next.

This is because there are actually two values that x could be.

!Probably the most mathy thing of the course!

Multiple Choice

How many values of x COULD the following polynomial have?

$$x^2+x\ -2\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$2x^1\ +1\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$9x^1\ +7\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$x^1\ -4\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$4x^1\ +4\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$x^2+4x^1\ +4\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$x^2+2x^1\ -8\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$2x^2+4x^1\ -4\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$x^{3\ }+2x^2+4x^1\ -4\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$x^{3\ }+x^2+4x^1\ -4\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$3x^{3\ }+2x^2+4x^1\ +10\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$2x^2+4x^1\ +10\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$4x^1\ +10\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$x^4+2x^3+4x^{2\ }+8x^1\ +10\ =0$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

Multiple Choice

How many values of x COULD the following polynomial have?

$$7+10\ =17$$

• 1,

  because the highest exponent is

  1

• 2,

  because the highest exponent is

  2

• 3,

  because the highest exponent is

  3

• 4,

  because the highest exponent is

  4

• 0,

  because the highest exponent is

  0

We can't do what we did in the past to find x. We need to find BOTH values of x




The simplest way to do that is to plug in values for x until we find two that work. Not fun

​So coming back to this:

Multiple Choice

For the polynomial

$$x^2+x-2=0$$

Does x = 0 work?

• Yes, because

  $$0^2+0\ -2\ =0$$

• No, because

  $$0^2+0\ -2\ \ne0$$

Multiple Choice

For the polynomial

$$x^2+x-2=0$$

Does x = 1 work?

• Yes, because

  $$1^2+1\ -2\ =0$$

• No, because

  $$1^2+1\ -2\ \ne0$$

Multiple Choice

For the polynomial

$$x^2+x-2=0$$

Does x = -1 work?

• Yes, because

  $$\left(-1\right)^2-1\ -2\ =0$$

• No, because

  $$\left(-1\right)^2-1\ -2\ \ne0$$

Multiple Choice

For the polynomial

$$x^2+x-2=0$$

Does x = 2 work?

• Yes, because

  $$\left(2\right)^2+2\ -2\ =0$$

• No, because

  $$\left(2\right)^2+2\ -2\ \ne0$$

Multiple Choice

For the polynomial

$$x^2+x-2=0$$

Does x = -2 work?

• Yes, because

  $$\left(-2\right)^2-2\ -2\ =0$$

• No, because

  $$\left(-2\right)^2-2\ -2\ \ne0$$

So it took us four guesses to get the two answers.
Not bad but that was a very easy example. It could easily take 40 guesses on a regents problem and they also want to see you using a smarter method.
Before we can learn better methods, let's try to understand binomials better

Poll

First, can you guess the values of x for

$$x^2+10x\ +\ 9\ =0$$

• x = -9

  and

  x = -1

• x = -9

  and

  x = 1

• x = 9

  and

  x = -1

• x = 2

  and

  x = 3

• x = 9

  and

  x = 10

Multiple Choice

Hopefully you remember foiling from last year.

( x + 9 )( x + 1 ) = 0

• $$x^2+10x+9\ =0$$

• $$x^2+2x\ +2$$

• $$x^2+x+2$$

Multiple Choice

Hopefully you remember foiling from last year.

( x - 2 )( x - 1 ) = ?

• $$x^2-3x+2$$

• $$x^2-6x\ +2$$

• $$x^{2\ }-x+2$$

• $$x^{2\ }+3x-2$$

• $$x^2+3x+2$$

Multiple Choice

( x - 2 )( x - 1 ) = $$x^2-3x+2$$

Can you guess the values of x that solve the equation:

$$x^2-3x+2=0$$

[HINT: Look for numbers that makes one of those parentheses = 0 ]

• x = 2

  and

  x = 1

• x = 10

  and

  x = 12

• x = 50

  and

  x = 34

• x = 7

  and

  x = 11

• x = 15

  and

  x = 19

Multiple Choice

Hopefully you remember foiling from last year.

( x - 4)( x - 2 ) = ?

• $$x^2-6x+8$$

• $$x^2-6x\ -8$$

• $$x^{2\ }-4x+8$$

• $$x^{2\ }-2x+8$$

• $$x^2+2x+8$$

Multiple Choice

( x - 4 )( x - 2 ) = $$x^2-6x\ +8$$

Can you guess the values of x that solve the equation:

$$x^2-6x+8=0$$

[HINT: Look for numbers that makes one of those parentheses = 0 ]

• x = 4

  and

  x = 2

• x = -4

  and

  x = -2

• x = 4

  and

  x = -2

• x = -4

  and

  x = 2

• x = 0

  and

  x = 0

Multiple Choice

Hopefully you remember foiling from last year.

( x - 3)( x + 2 ) = ?

• $$x^2-x-6$$

• $$x^2+x-6$$

• $$x^{2\ }-5x\ -6$$

• $$x^{2\ }-5x+6$$

• $$x^2-x+6$$

Multiple Choice

( x - 3 )( x + 2 ) = $$x^2-x-6$$

Can you guess the values of x that solve the equation:

$$x^2-x-6=0$$

[HINT: Look for numbers that makes one of those parentheses = 0 ]

• x = 3

  and

  x = -2

• x = -3

  and

  x = -2

• x = -3

  and

  x = 2

• x = 3

  and

  x = 2

• x = 0

  and

  x = 0

Multiple Choice

Hopefully you remember foiling from last year.

( x - 5)( x - 4 ) = ?

• $$x^2-9x\ +20$$

• $$x^2-x+20$$

• $$x^{2\ }-x-20$$

• $$x^2-9x-20$$

• $$x^2+x+20$$

Multiple Choice

( x - 5 )( x - 4) = $$x^2-9x+20$$

Can you guess the values of x that solve the equation:

$$x^2-9x+20$$

[HINT: Look for numbers that makes one of those parentheses = 0 ]

• x = 5

  and

  x = 4

• x = -5

  and

  x = -4

• x = -5

  and

  x = 4

• x = 5

  and

  x = -4

• x = 0

  and

  x = 0

Multiple Choice

Hopefully you remember foiling from last year.

( x + 6)( x +1) = ?

• $$x^2+7x+6$$

• $$x^2+7x-6$$

• $$x^{2\ }-7x-6$$

• $$x^2-10x-6$$

• $$x^2+5x\ +6$$

Multiple Choice

( x + 6 )( x + 1) = $$x^2+7x\ +6$$

Can you guess the values of x that solve the equation:

$$x^2+7x\ +6$$

[HINT: Look for numbers that makes one of those parentheses = 0 ]

• x = -6

  and

  x = -1

• x = 6

  and

  x = -1

• x = -6

  and

  x = 1

• x = 6

  and

  x = 1

• x = 0

  and

  x = 0

Multiple Choice

Hopefully you remember foiling from last year.

( x - 10)( x + 3) = ?

• $$x^2-7x-30$$

• $$x^2+7x-30$$

• $$x^{2\ }-13x-30$$

• $$x^2-10x+30$$

• $$x^2-7x+30$$

Multiple Choice

( x - 10)( x + 3) = $$x^2+7x\ +6$$

Can you guess the values of x that solve the equation:

$$x^2+7x\ +6$$

[HINT: Look for numbers that makes one of those parentheses = 0 ]

• x = 10

  and

  x = -3

• x = -10

  and

  x = -3

• x = 10

  and

  x = 3

• x = -10

  and

  x = 3

• x = 0

  and

  x = 0

Multiple Choice

Hopefully you remember foiling from last year.

( x - 9)( x - 2) = ?

• $$x^2-11x+18$$

• $$x^2-7x+18$$

• $$x^{2\ }-11x-18$$

• $$x^2-7x-18$$

• $$x^2+7x-18$$

Multiple Choice

( x - 9)( x - 2 ) = $$x^2-11x+18$$

Can you guess the values of x that solve the equation:

$$x^2-11x+18$$

[HINT: Look for numbers that makes one of those parentheses = 0 ]

• x = 9

  and

  x = 2

• x = -9

  and

  x = -2

• x = 9

  and

  x = -2

• x = -9

  and

  x = 2

• x = 0

  and

  x = 0

For these problems, I gave you the stuff in the parentheses.

The real work is finding the stuff in the parentheses
This is called finding the factors
or
Finding the zeroes

Next time, we're going to work on doing just that