Roots & Multiplicity

Roots & Multiplicity

Raíces y multiplicidad

Multiple Choice

What goes in the box with the question mark?

• 40 x 5

• 30 x 6

• 6 x 40

Multiple Choice

What number goes in the box with the question mark?

• 36

• 3,600

• 360

Multiple Choice

What expressions should replace A?

• A = 2 a

• A = 2x

• A = 2x²

Multiple Choice

What expressions should replace B?

• B = -1

• B = -5

• B = 5

Multiple Choice

What expression goes in the missing box?

• 7

• -7

• 7x

• -7x

What do we know?

¿Qué sabemos?

 $y=\left(x+3\right)\left(x-4\right)$  

(even degree)  (grado par)
positive leading coefficient
coeficiente principal positivo
roots at -3 and 4  raíces en -3 y 4

Look at the factors

Mira los factores

The factors & roots are different

The graph shows the function goes through the x-axis

Los factores y las raíces son diferentes. El gráfico muestra que la función pasa por el eje x

Multiple Choice

In the equation

(x + 2)(x - 3) = 0, the binomials are called...?

• Factors

• Products

• Multiples

• Zeros

Multiple Choice

Solve for the values of x:

(x + 2)(x - 3) = 0

• { 2, 3 }

• { -2, -3 }

• { 2, 3 }

• { -2, 3 }

Multiple Choice

Solve and find your x-intercepts:

0=x(x - 6)

• x=0 and x=-6

• x=1 and x=-6

• x=0 and x=6

• x=6

Multiple Choice

Solve and find your zeros:

z(z - 15)=0

• z=0 and z=15

• z=0 and z=-15

• z=-15

• z=15

Multiple Choice

Solve for x: (2x - 6)(x + 1)

• -6, 1

• 3, -1

• -3, 1

• -3, -1

¿Qué sabemos?

What do we know?

 $y=\left(x+3\right)^2$  

(even degree) (grado par)
positive leading coefficient

coeficiente principal positivo
root at -3  raíz en -3

Look at the factors.

Mira los factores.

This function has repeated factors.

The graph shows the function touches the x-axis.

Esta función tiene factores repetidos.

El gráfico muestra que la función toca el eje x.

Multiple Choice

True or false:
The solution, root, x-intercept, and zero of a problem are all the same thing

• True
• False, because the solution and root are the same but the x-intercept and zero are different
• False, because the x-intercept and root are the same but the zero and solution are different
• False, they are all different

Multiple Choice

What mathematical process did they use?

• Quadratic Formula

• Completing the Square

• Factoring

• Differentiation

Multiple Choice

 $y=-\left(x-0.5\right)^2$ 
 How many solutions does this graph have?

• one

• two

• no solutions

• cannot be determined

Multiple Choice

How many solutions does this graph have?

• one

• two

• no solutions

• cannot be determined

¿Qué sabemos?

What do we know?

 $y=\left(x+3\right)^3$  
(odd degree) (grado impar)
positive leading coefficient
coeficiente principal positivo
root at -3 raíz en -3

Look at the factors.

Mira los factores.

This function has repeated factors.

The graph shows the function goes through the x-axis.

Esta función tiene factores repetidos.

El gráfico muestra que la función pasa por el eje x.

Multiple Choice

Choose the correct function that represents this graph.

• f(x) = x²

• f(x) = x³

• f(x) = 1/x²

• f(x) = ∛x

Multiple Choice

Determine the roots of the polynomial.

f(x) = -x(x+1)(x-3)

• -1, 3

• 0, -1, 3

• 1, -3

• 0, -3, 1

Multiple Choice

What is the degree of the polynomial?

9x⁵ - x⁴ + 2x

• 9

• 5

• 4

• 2

Multiple Choice

What is the leading coefficient of the polynomial?

7x⁵ - 9x² + 6x

• 7

• 5

• 9

• 2

Multiple Choice

Determine the sign of the leading coefficient of the polynomial shown.

• Positive

• Negative

• Not enough information to tell

Multiple Choice

Determine the sign of the leading coefficient of the polynomial shown.

• Positive

• Negative

• Not enough information to tell

Multiple Choice

Determine if the degree of the polynomial shown is even or odd.

• Even

• Odd

• Not enough information to tell

Multiple Choice

Classify the function:

• quadratic
• cubic 
• quartic
• linear

¿Qué sabemos?

What do we know?

degree is 4 (even degree)

grado es 4 (grado par)

positive leading coefficient

coeficiente principal positivo

¿Qué sabemos?

What do we know?

degree is 5 (odd degree)

grado es 5 (grado impar)

positive leading coefficient

coeficiente principal positivo

Notice 

• (x+4) is a single factor, degree 1 (odd) so function goes through the x-axis

• (x-3) is a single factor, degree 1 (odd) so function goes through the x-axis

•  $\left(x+1\right)^2$  is a repeated factor, degree 2 (even) so the graph touches the x-axis

Aviso

• (x + 4) es un factor único, grado 1 (impar) por lo que la función pasa por el eje x
  

• (x-3) es un factor único, grado 1 (impar) por lo que la función pasa por el eje x
  

•  $\left(x+1\right)^2$  es un factor repetido, grado 2 (par) por lo que el gráfico toca el eje x

Factores con grados pares TOQUE el eje x

•   $y=\left(x+3\right)^2\left(x-1\right)\left(x+2\right)$  

• Los factores individuales con un grado impar de 1 pasan por el eje x.
  

Factors with even degrees TOUCH the x-axis

•   $y=\left(x+3\right)^2\left(x-1\right)\left(x+2\right)$  

• The single factors with an odd degree of 1, go through the x-axis

Factors with an odd degree pass through the x-axis.

•    $y=\left(x+3\right)^1\left(x-1\right)^3\left(x+2\right)^5$  

Los factores con un grado impar pasan por el eje x.

•    $y=\left(x+3\right)^1\left(x-1\right)^3\left(x+2\right)^5$  

Multiple Choice

 $y=\left(x+1\right)^2\left(x-2\right)\left(x-4\right)$ 

Which root has a multiplicity of 2?

• -1

• 2

• 4

• -4

Multiple Choice

What is the degree of the polynomial?

 $m\left(x\right)=x\left(x+5\right)\left(x-6\right)$  

• 3

• 2

• 9

• 6

Multiple Choice

What is the degree of the polynomial?

 $g\left(x\right)=\left(x-4\right)^2\left(x+7\right)^3$  

• 6

• 5

• 1

• 9

Multiple Choice

 $y=\left(x+4\right)\left(x+1\right)\left(x-1\right)^2\left(x-2\right)$  

Which root has even multiplicity?

• 1

• -4

• -1

• 2

Multiple Choice

What is the multiplicity of the zero x=-5 for the function $f\left(x\right)=x^2\left(x-1\right)^4\left(x+5\right)$ ?

• 1

• 2

• 3

• 4

Multiple Choice

What could be the function for this graph in factored form?

• f(x) = (x-4)²(x+2)

• f(x) = (x+4)²(x-2)

• f(x) = (x-4)(x+2)²

• f(x) = (x+4)(x-2)²

Multiple Choice

What is the multiplicity of the zero x=1 for the function  $f\left(x\right)=x^2\left(x-1\right)^4\left(x+5\right)$  ?

• 2

• 1

• 4

• 3

Multiple Choice

Which one of these are not a zero of  $f\left(x\right)=x^2\left(x-1\right)^2\left(x+5\right)$  ?

• 0

• 1

• -5

• -1

Multiple Choice

Odd Multiplicities have which of the following effects on the x-axis?

• Cross the x-axis

• Turn on the x-axis

Multiple Choice

If the graph of a function crosses the x-axis, what does that mean about the multiplicity of that zero?

• Even

• Odd

• None

Multiple Choice

If the graph of a function crosses the x-axis, what does that mean about the multiplicity of that zero?

• Even

• Odd

• None

Multiple Choice

Which factors BEST represent the graph?

• f(x) = (x − 1)²(x + 2)(x + 4)³

• f(x) = (x + 1)(x − 2)(x − 4)

• f(x) = (x + 1)²(x − 2)(x − 4)³

• f(x) = (x + 1)(x − 2)(x − 4)³

Multiple Choice

Which function might have the given graph?

• f(x) = (x - 6)(x + 4)²

• f(x) = (x - 6)²(x + 4)

• f(x) = (x + 6)²(x - 4)

• f(x) = x(x - 6)(x + 4)

Poll

I understand how to find roots.

Poll

I understand multiplicity.