Questions: Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around, at each zero. f(x)=3(x+4)(x+7)^3 A. -4 , multiplicity 1 , crosses x-axis; -7 , multiplicity 3 , crosses x-axis B. 4, multiplicity 1 , crosses x-axis; 7 , multiplicity 3 , crosses x-axis C. -4 , multiplicity 1 , crosses x-axis; -7 , multiplicity 3 , touches x-axis and turns around D. 4, multiplicity 1 , touches x-axis; 7 , multiplicity 3 , touches x-axis and turns around

Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around, at each zero.

f(x)=3(x+4)(x+7)^3

A. -4 , multiplicity 1 , crosses x-axis; -7 , multiplicity 3 , crosses x-axis

B. 4, multiplicity 1 , crosses x-axis; 7 , multiplicity 3 , crosses x-axis

C. -4 , multiplicity 1 , crosses x-axis; -7 , multiplicity 3 , touches x-axis and turns around D. 4, multiplicity 1 , touches x-axis; 7 , multiplicity 3 , touches x-axis and turns around

Transcript text: Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the $x$-axis or touches the $x$-axis and turns around, at each zero. \[ f(x)=3(x+4)(x+7)^{3} \] A. -4 , multiplicity 1 , crosses $x$-axis; -7 , multiplicity 3 , crosses $x$-axis B. 4, multiplicity 1 , crosses $x$-axis; 7 , multiplicity 3 , crosses $x$-axis C. -4 , multiplicity 1 , crosses $x$-axis; -7 , multiplicity 3 , touches $x$-axis and turns around D. 4, multiplicity 1 , touches $x$-axis; 7 , multiplicity 3 , touches $x$-axis and turns around

Solution

Solution Steps

To find the zeros of the polynomial function $ f(x) = 3(x+4)(x+7)^3 $, we need to set the function equal to zero and solve for $ x $. The zeros occur where each factor of the polynomial is zero. The multiplicity of each zero is determined by the exponent of the corresponding factor. If the multiplicity is odd, the graph crosses the x-axis at that zero; if even, it touches the x-axis and turns around.

Step 1: Identify the Zeros of the Polynomial

The given polynomial function is:

\[ f(x) = 3(x+4)(x+7)^3 \]

To find the zeros, we set $ f(x) = 0 $:

\[ 3(x+4)(x+7)^3 = 0 \]

This equation is satisfied when either $ x+4 = 0 $ or $ (x+7)^3 = 0 $.

Step 2: Solve for Each Zero

Zero from $ x+4 = 0 $:

\[ x+4 = 0 \implies x = -4 \]

The zero is $ x = -4 $.
Zero from $ (x+7)^3 = 0 $:

\[ (x+7)^3 = 0 \implies x+7 = 0 \implies x = -7 \]

The zero is $ x = -7 $.

Step 3: Determine the Multiplicity and Behavior at Each Zero

Multiplicity of $ x = -4 $:

The factor $ (x+4) $ appears once, so the multiplicity is 1. A zero with an odd multiplicity (1 in this case) means the graph crosses the $ x $-axis at this point.
Multiplicity of $ x = -7 $:

The factor $ (x+7) $ appears three times, so the multiplicity is 3. A zero with an odd multiplicity (3 in this case) also means the graph crosses the $ x $-axis at this point.