6. keith determines the zeros of the function f(x) to be -6 and 5. what could be keith’s function?
a. $f(x) = x + 5)(x + 6)$
b. $f(x) = (x + 5)(x - 6)$
c. $f(x) = (x - 5)(x + 6)$
d. $f(x) = (x - 5)(x - 6)$

7. the zeros of the function $f(x) = 2x^3 + 12x - 10x^2$ are
a. ${2, 3}$
b. ${-1, 6}$
c. ${0, 2, 3}$
d. ${0, -1, 6}$

8. which polynomial function has the zeros at -3, 0 and 4?
a. $f(x) = (x + 3)(x^2 + 4)$
b. $f(x) = (x^2 - 3)(x - 4)$
c. $f(x) = x(x + 3)(x - 4)$
d. $f(x) = x(x - 3)(x + 4)$

9. explain how to determine the zeros of $f(x) = (x + 3)(x - 1)(x - 8)$. state the zeros of the function.

10. the graph of the function $f(x) = ax^2 + bx + c$ is given. could the factors of $f(x)$ be $(x + 2)$ and $(x - 3)$? based on the graph, explain why or why not.
graph of a parabola opening upwards, vertex on the y-axis, crossing the x-axis at two points symmetric about the y-axis

Question

6. keith determines the zeros of the function f(x) to be -6 and 5. what could be keith’s function?
a. $f(x) = x + 5)(x + 6)$
b. $f(x) = (x + 5)(x - 6)$
c. $f(x) = (x - 5)(x + 6)$
d. $f(x) = (x - 5)(x - 6)$

7. the zeros of the function $f(x) = 2x^3 + 12x - 10x^2$ are
a. ${2, 3}$
b. ${-1, 6}$
c. ${0, 2, 3}$
d. ${0, -1, 6}$

8. which polynomial function has the zeros at -3, 0 and 4?
a. $f(x) = (x + 3)(x^2 + 4)$
b. $f(x) = (x^2 - 3)(x - 4)$
c. $f(x) = x(x + 3)(x - 4)$
d. $f(x) = x(x - 3)(x + 4)$

9. explain how to determine the zeros of $f(x) = (x + 3)(x - 1)(x - 8)$. state the zeros of the function.

10. the graph of the function $f(x) = ax^2 + bx + c$ is given. could the factors of $f(x)$ be $(x + 2)$ and $(x - 3)$? based on the graph, explain why or why not.
graph of a parabola opening upwards, vertex on the y-axis, crossing the x-axis at two points symmetric about the y-axis

Accepted Answer

### Question 6
# Explanation:
## Step1: Recall zero - factor property
If a function $y = f(x)$ has a zero at $x = a$, then $(x - a)$ is a factor of $f(x)$.
## Step2: Find factors for given zeros
The zeros of the function are $x=-6$ and $x = 5$.
For $x=-6$, the factor is $(x-(-6))=(x + 6)$.
For $x = 5$, the factor is $(x - 5)$.
So the function $f(x)=(x - 5)(x + 6)$.
# Answer:
c. $f(x)=(x - 5)(x + 6)$

### Question 7
# Explanation:
## Step1: Rewrite the function
Given $f(x)=2x^{3}+12x - 10x^{2}$, we first rewrite it in standard form (descending powers of $x$): $f(x)=2x^{3}-10x^{2}+12x$.
## Step2: Factor out the GCF
Factor out the greatest common factor $2x$ from each term: $f(x)=2x(x^{2}-5x + 6)$.
## Step3: Factor the quadratic
Factor the quadratic $x^{2}-5x + 6$. We need two numbers that multiply to $6$ and add up to $- 5$. The numbers are $-2$ and $-3$. So $x^{2}-5x + 6=(x - 2)(x - 3)$.
## Step4: Find the zeros
Set $f(x)=0$, then $2x(x - 2)(x - 3)=0$.
Using the zero - factor property:
$2x=0$ gives $x = 0$;
$x - 2=0$ gives $x = 2$;
$x - 3=0$ gives $x = 3$.
So the zeros are $0$, $2$ and $3$.
# Answer:
c. $\{0,2,3\}$

### Question 8
# Explanation:
## Step1: Recall zero - factor property
If a function $y = f(x)$ has a zero at $x=a$, then $(x - a)$ is a factor of $f(x)$.
## Step2: Find factors for given zeros
The zeros are $x=-3$, $x = 0$ and $x = 4$.
For $x=-3$, the factor is $(x-(-3))=(x + 3)$.
For $x = 0$, the factor is $x$ (since when $x = 0$, $x=0$ implies the factor is $x$).
For $x = 4$, the factor is $(x - 4)$.
So the function $f(x)=x(x + 3)(x - 4)$.
# Answer:
c. $f(x)=x(x + 3)(x - 4)$

### Question 9
# Explanation:
## Step1: Recall zero - factor property
To find the zeros of a factored function $f(x)=(x + 3)(x - 1)(x - 8)$, we use the zero - factor property. The zero - factor property states that if $ab = 0$, then either $a = 0$ or $b = 0$ (or both).
## Step2: Set each factor equal to zero
Set $x+3 = 0$, then $x=-3$.
Set $x - 1=0$, then $x = 1$.
Set $x - 8=0$, then $x = 8$.
# Brief Explanations:
To find the zeros of $f(x)=(x + 3)(x - 1)(x - 8)$, we use the zero - factor property. Set each factor $(x + 3)$, $(x - 1)$ and $(x - 8)$ equal to zero and solve for $x$.
# Answer:
To determine the zeros, set each factor equal to zero: $x + 3=0\Rightarrow x=-3$, $x - 1=0\Rightarrow x = 1$, $x - 8=0\Rightarrow x = 8$. The zeros of the function are $-3$, $1$ and $8$.

### Question 10
# Explanation:
## Step1: Recall the relationship between factors and zeros
If $(x + 2)$ and $(x - 3)$ are factors of $f(x)=ax^{2}+bx + c$, then the zeros of the function are $x=-2$ (from $x + 2=0$) and $x = 3$ (from $x - 3=0$).
## Step2: Analyze the graph
The graph of a quadratic function $y = ax^{2}+bx + c$ intersects the $x$ - axis at its zeros. From the given graph, we observe the $x$ - intercepts (zeros) of the parabola. If the factors were $(x + 2)$ and $(x - 3)$, the parabola should cross the $x$ - axis at $x=-2$ and $x = 3$. But from the graph (assuming the parabola is symmetric about the $y$ - axis or has $x$ - intercepts that are symmetric with respect to the $y$ - axis, or at least not at $-2$ and $3$), we can see that the $x$ - intercepts do not match $-2$ and $3$. For example, if the parabola is symmetric about the $y$ - axis, the $x$ - intercepts should be of the form $(h,0)$ and $(-h,0)$. So the factors $(x + 2)$ and $(x - 3)$ do not match the $x$ - intercepts shown in the graph.
# Brief Explanations:
If $(x + 2)$ and $(x - 3)$ are factors, the zeros are $x=-2$ and $x = 3$. The graph of $f(x)$ (a parabola) has $x$ - intercepts that do not match $-2$ and $3$ (e.g., if symmetric about $y$ - axis, intercepts are symmetric about $y$ - axis, not $-2$ and $3$). So the factors cannot be $(x + 2)$ and $(x - 3)$.
# Answer:
No, the factors cannot be $(x + 2)$ and $(x - 3)$. Because if $(x + 2)$ and $(x - 3)$ were factors, the function would have zeros at $x=-2$ and $x = 3$, but the graph of the parabola does not have $x$ - intercepts at these values (e.g., the parabola's $x$ - intercepts appear to be symmetric with respect to the $y$ - axis, not at $-2$ and $3$).

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QUESTION IMAGE

Question

Explanation:

Question 6

Step1: Recall zero - factor property

Step2: Find factors for given zeros

Step1: Rewrite the function

Step2: Factor out the GCF

Step3: Factor the quadratic

Step4: Find the zeros

Step1: Recall zero - factor property

Step2: Find factors for given zeros

Answer:

Question 7