polynomial p(x) binomial factors polynomial p(x) binomial factors solutions of p(x)=0 zeros of p(x)

2 2 0x x

1 2 0x x

1 2x or x

1 2 0x x

1 2x or x

2 2 ( )x x P x Polynomial P(x)

2 2 0x x

2 2 0x x

1 2 0x x

1 2x or x

Polynomial P(x)

When 0P x

2 2 ( )x x P x

1 2 0x x

1 2x or x

Polynomial P(x)

When 0P x

Binomial Factors

2 2 ( )x x P x 2 2 0x x

1 2 0x x

1 2x or x

Polynomial P(x)

When 0P x

Binomial Factors

Solutions of P(x)=0

Zeros of P(x)

2 2 ( )x x P x 2 2 0x x

1 2 0x x

1 2x or x

Polynomial P(x)

When 0P x

Binomial Factors

Solutions of P(x)=0

Zeros of P(x)

2 2 ( )x x P x 2 2 0x x

If in polynomial P(x), P(c) = 0

is a solution to the equation ( ) 0c Px x

is a zero of ( )x c P x

is a factor of ( )x c P x

1 2x or x

Zero Zero

P(x)=0 P(x)=0

1 2P x x x

Zero Zero

P(x)=0 P(x)=0

Binomial Factors

1 2x or x

2 4 4 0x x

2 2 0x x

2 2x or x

2( ) 4 4P x x x Polynomial P(x)

When 0P x

Binomial Factors

Solutions of P(x)=0

Zeros of P(x)

2 2x or x Solutions of P(x)=0

Zeros of P(x)

2 2x or x Solutions of P(x)=0

Zeros of P(x)

2, . .d r

multiplicity 22,

2 2x or x Solutions of P(x)=0

Zeros of P(x)

mult. 22,

Zeros

P(x)=0

mult. 22,

The graph is tangent (touches) the x-axis but doesn’t cross the x-axis at that point

Zeros

P(x)=0

mult. 22,

The graph is tangent (touches) the x-axis but doesn’t cross the x-axis at that point

( ) 2 2P x x x Binomial Factors

2( ) 2P x x x 2( ) 4 4P x x x

( ) 1 2P x x x ( ) 2 2P x x x

( ) 1 2P x x x 2( ) 2P x x

( ) 1 2P x x x 2( ) 2P x x

( ) 1 2P x x x 2( ) 2P x x

2( ) 1 2P x x x

2

( ) 1 2P x x x

-1

P(x)=0

mult. 11,

2

( ) 1 2P x x x

P(x)=0

mult. 22,-1

P(x)=0

mult. 11,

2( ) 1 2P x x x

2( ) 1 2P x x x

( ) 1 2 2P x x x x

2( ) 1 4 4P x x x x

2( ) 1 2P x x x

( ) 1 2 2P x x x x

2( ) 1 4 4P x x x x

3 2 2( ) 4 4 4 4P x x x x x x

3 2( ) 3 4P x x x

3 2( ) 3 4P x x x

P(x)=0

mult. 22,-1

P(x)=0

mult. 11,

P(x)=0

mult. 22,-1

P(x)=0

2( ) 1 2P x x x

mult. 11,

2

( ) 2 1P x x x

mult. 22,

2( ) 2 1P x x x

2

( ) 2 1P x x x

mult. 22,

2( ) 2 1P x x x

2

( ) 2 1P x x x

mult. 22,

2( ) 2 1P x x x

mult. 11,

2

( ) 2 1P x x x

mult. 22, mult. 11,

2

( ) 2 1P x x x

mult. 22, 1mult. 11,

3 2( ) 3 4P x x x

mult. 22, 1mult. 11,

2

( ) 2 1P x x x

mult. 22, 1mult. 11,

2 2

( ) 2 1P x x x

mult. 22,

2 2

( ) 2 1P x x x

mult. 22,

2 2

( ) 2 1P x x x

mult. 22, mult. 21,

2 2

( ) 2 1P x x x

mult. 22, mult. 21,

2 2

( ) 2 1P x x x

mult. 22, mult. 21,

4 3 2( ) 2 3 4 4P x x x x x

mult. 22, mult. 21,

2 2

( ) 2 1P x x x

mult. 22, mult. 21,

– 3, multiplicity 2

2, multiplicity 1

– 3, multiplicity 2

2, multiplicity 1

2( ) 3 2P x x x

– 3, multiplicity 2

2, multiplicity 1

2( ) 3 2P x x x

2( ) 3 2P x x x

2( 6) 29xP x x x

2( ) 3 2P x x x

2( 6) 29xP x x x

2( ) 3 2P x x x

2( 6) 29xP x x x

x2 6x 9

x

–2

3x 26x 9x22x 12x 18

x2 6x 9

x

–2

3x 26x 9x22x 12x 18

x2 6x 9

x

–2

3x 26x 9x22x 12x 18

3 2( ) 4 3 18P x x x x

3

( ) 2P x x

Mother Cubic Function

Shifted 2 to the left

3( )f x x 3

( ) 2P x x

3

( ) 2P x x

mult. 32,

The graph intersects the x-axis and

flattens out

as it passes through

that point.

3( ) 2P x x ( ) 2 2P x x x

3( ) 2P x x ( 2 2)P x x x

1

3( ) 2P x x ( 2 2)P x x x

1

If the degree of the binomial is odd and greater than 1, the graph will intersect the x-axis and flatten out

as it passes through that point.

If the degree of the binomial is odd and greater than 1, the graph will intersect the x-axis and flatten out

as it passes through that point.

–1 1 2 3

–1 1 2 3

3 2( ) 1 2 3P x x x x

mult. 31,

–1 1 2 3

3 2( ) 1 2 3P x x x x

mult. 22,

3 2

( ) 1 2 3P x x x x

–1 1 2 3

mult. 13,

1

4

( ) 2P x x

4

( ) 2P x x

4

( ) 2P x x

4

( ) 2P x x

The graph is tangent (touches) the x-axis but doesn’t cross the x-axis at that point

4

( ) 2P x x

mult. 42,

4

( ) 2P x x

mult. 42,

The graph is tangent (touches) the x-axis but doesn’t cross the x-axis at that point

4( ) 2P x x 2

( ) 2P x x

2( ) 3P x x

4( ) 3P x x

Example 1

4 3 26 9f x x x x

– 3

– 3 2nd TABLE

2 2 223 0 3f x x x x x

2 2 223 0 3f x x x x x

2nd TABLE

2 2 223 0 3f x x x x x

2 2 223 0 3f x x x x x

Example 2

4 23 2f x x x x

ZOOM 1: ZBox

22 0 1f x x x x

22 0 1f x x x x

22 0 1f x x x x

22 0 1f x x x x

22 0 1f x x x x

22 0 1f x x x x

2

2 1f x x x x

4 3 23) 5 9 7 2f x x x x x

5 4 3 24) 5 5 25 4 20f x x x x x x

4 3 23) 5 9 7 2f x x x x x

5 4 3 24) 5 5 25 4 20f x x x x x x

32 1f x x x

4 3 23) 5 9 7 2f x x x x x

5 4 3 24) 5 5 25 4 20f x x x x x x

32 1f x x x

5 2 1 1 2f x x x x x x