solve the problem.
8) a rectangle with width 2x + 1 inches has an area of 2x^4 + 9x^3 - 18x^2 - 71x - 30 square inches. write a polynomial that represents its length.
a) x^3 - 11x^2 + 4x - 30 inches
b) x^3 + 4x^2 - 11x - 30 inches
c) x^3 - 10x^2 + 2x - 30 inches
d) x^3 + 2x^2 - 10x - 30 inches
write the equation of a polynomial function with the given characteristics. use a leading coefficient of 1 or -1 and make the degree of the function as small as possible.
9) crosses the x - axis at -2, 0, and 4; lies below the x - axis between -2 and 0; lies above the x - axis between 0 and 4.
a) f(x)=x^3 - 2x^2 - 8x
b) f(x)=-x^3 + 2x^2 + 8x
c) f(x)=x^3 + 2x^2 - 8x
d) f(x)=-x^3 - 2x^2 + 8x
determine the maximum possible number of turning points for the graph of the function.
10) f(x)=(5x + 3)^2(x^2 + 5)(x + 7)
a) 4
b) 2
c) 25
d) 5
compute the discriminant. then determine the number and type of solutions for the given equation.
11) 6x^2=-2x - 7
a) 0; one real solution
b) 172; two unequal real solutions
c) -164; two complex imaginary solutions
solve the equation by factoring.
13) -6x - 2=(3x + 1)^2
solve the formula for the specified variable.
14) p = 2l + 2w for w
a) w = p - 2l
b) w = p - l
c) w = (p - 2l)/2
d) w = (p - 2l)/2
solve the problem. (show work )
15) the sum of the angles of a triangle is 180°. find the three angles of the triangle if one angle is twice the smallest angle and the third angle is 28° greater than the smallest angle.
graph the polynomial function.
12) f(x)=x^4 + 16x^3 + 64x^2 (nc)

Question

Accepted Answer

### 8.
# Explanation:
## Step1: Recall area formula
The area of a rectangle is $A = lw$, where $A$ is area, $l$ is length and $w$ is width. Given $w=2x + 1$ and $A=2x^{4}+9x^{3}-18x^{2}-71x - 30$. Then $l=\frac{A}{w}=\frac{2x^{4}+9x^{3}-18x^{2}-71x - 30}{2x + 1}$.
## Step2: Use polynomial long - division
Dividing $2x^{4}+9x^{3}-18x^{2}-71x - 30$ by $2x + 1$ gives $x^{3}+4x^{2}-11x - 30$.
# Answer:
$x^{3}+4x^{2}-11x - 30$ inches

### 9.
# Explanation:
## Step1: Write the polynomial form
Since the polynomial crosses the $x$-axis at $x=-2,0,4$, the polynomial can be written in factored form as $f(x)=a(x + 2)(x-0)(x - 4)$.
## Step2: Determine the leading coefficient
We want the leading coefficient $a = 1$ or $a=-1$. Expanding $f(x)=a(x + 2)(x)(x - 4)=a(x^{3}-2x^{2}-8x)$. Since the polynomial lies below the $x$-axis between $0$ and $4$ and above the $x$-axis between $-2$ and $0$, $a=-1$. So $f(x)=-x^{3}+2x^{2}+8x$.
# Answer:
B) $f(x)=-x^{3}+2x^{2}+8x$

### 10.
# Explanation:
## Step1: Recall the rule for turning points
The maximum number of turning points of a polynomial function $y = f(x)$ of degree $n$ is $n - 1$. First, expand $f(x)=(5x + 3)^{2}(x^{2}+5)(x + 7)$. The degree of $(5x + 3)^{2}=2$, the degree of $x^{2}+5 = 2$ and the degree of $x + 7=1$. The degree of the product is $2+2 + 1=5$.
## Step2: Calculate the number of turning points
The maximum number of turning points is $n-1=4$.
# Answer:
A) 4

### 11.
# Explanation:
## Step1: Recall the discriminant formula
For a quadratic equation $ax^{2}+bx + c = 0$, the discriminant is $\Delta=b^{2}-4ac$. Rewrite $6x^{2}=-2x - 7$ as $6x^{2}+2x+7 = 0$, where $a = 6$, $b = 2$ and $c = 7$.
## Step2: Calculate the discriminant
$\Delta=(2)^{2}-4	imes6	imes7=4 - 168=-164$. Since $\Delta<0$, there are two complex imaginary solutions.
# Answer:
D) $-164$; two complex imaginary solutions

### 13.
# Explanation:
## Step1: Expand the right - hand side
Expand $(3x + 1)^{2}=9x^{2}+6x + 1$. The equation $-6x-2=(3x + 1)^{2}$ becomes $-6x-2=9x^{2}+6x + 1$.
## Step2: Rearrange to standard quadratic form
Rearrange to $9x^{2}+12x+3 = 0$, then divide by $3$ to get $3x^{2}+4x + 1 = 0$.
## Step3: Factor the quadratic equation
Factor $3x^{2}+4x + 1=(3x + 1)(x + 1)=0$.
## Step4: Solve for $x$
Setting each factor equal to zero gives $3x+1 = 0$ or $x + 1=0$. So $x=-\frac{1}{3}$ or $x=-1$.
# Answer:
$x=-\frac{1}{3},-1$

### 14.
# Explanation:
## Step1: Isolate the terms with $W$
Start with $P = 2L+2W$. Subtract $2L$ from both sides: $P-2L=2W$.
## Step2: Solve for $W$
Divide both sides by $2$: $W=\frac{P - 2L}{2}$.
# Answer:
D) $W=\frac{P - 2L}{2}$

### 15.
# Explanation:
## Step1: Let the smallest angle be $x$
Let the smallest angle of the triangle be $x$. Then one angle is $2x$ and the third angle is $x + 28$.
## Step2: Use the angle - sum property of a triangle
Since the sum of the angles of a triangle is $180^{\circ}$, we have the equation $x+2x+(x + 28)=180$.
## Step3: Combine like terms
$4x+28 = 180$.
## Step4: Solve for $x$
Subtract $28$ from both sides: $4x=180 - 28=152$. Divide by $4$: $x = 38$.
The angles are $x = 38^{\circ}$, $2x=76^{\circ}$ and $x + 28=66^{\circ}$.
# Answer:
$38^{\circ},76^{\circ},66^{\circ}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

8.

Step1: Recall area formula

Step2: Use polynomial long - division

Step1: Write the polynomial form

Step2: Determine the leading coefficient

Step1: Recall the rule for turning points

Step2: Calculate the number of turning points

Answer:

9.