1. simplify the expression below and write it in standard form.
7jk² - 5(m + 4) - 20n
2. the volume of a shipping box can be found using the formula, v = lwh. the expression 5(x² - 6)(x² - 4) represents the volume of a shipping box.
what is the meaning of the coefficient 5?
a. volume of the shipping box
b. dimension of the shipping box
c. area of the base
3. the volume of a cylindrical water tower can be found using the formula, v = πr²h. the expression π(h + 3)²h, represents the volume where h is the height of the water tower in feet. which statement below is true?
a. the radius of the water tower is 3 feet greater than the height.
b. the height of the water tower is 3 feet greater than the radius.
c. the height of the water tower is 3 feet.
4. what does it mean that polynomials are closed under multiplication?
a. when adding any two polynomials, the result will always be a polynomial.
b. when multiplying any two polynomials, the result will always be a polynomial.
c. any two polynomials can be multiplied.
5. simplify 3x³ - 2x² + 7x plus 3x² - 5x + 3.
6. simplify (2x² + 3x + 4)(3x³ + x² - x).
7. simplify (2a³ + 4b²)(4a³ - 2b²).

Question

Accepted Answer

### 1.
# Explanation:
## Step1: Expand the expression
$$7jk^{2}-5(m + 4)-20n=7jk^{2}-5m-20 - 20n$$
Since there are no like - terms to combine further, the standard form (where terms are ordered in a way that variables are in alphabetical order if possible) is $7jk^{2}-5m-20n - 20$

# Answer:
$7jk^{2}-5m-20n - 20$

### 2.
# Brief Explanations:
In the volume formula $V = lwh$, if the volume is given by $V=5(x^{2}-6)(x^{2}-4)$, and we compare it to $V = lwh$. The coefficient $5$ represents one of the dimensions (length, width, or height) of the shipping box.

# Answer:
B. Dimension of the Shipping Box

### 3.
# Brief Explanations:
Given the volume formula $V=\pi r^{2}h$ and the expression $V = \pi(h + 3)^{2}h$. By comparing $r$ and $h$ in the formula, we have $r=h + 3$, which means the radius of the water tower is 3 feet greater than the height.

# Answer:
A. The radius of the water tower is 3 feet greater than the height.

### 4.
# Brief Explanations:
The closure property for multiplication of polynomials means that when we multiply any two polynomials, the result is also a polynomial. Option A is about the closure property of addition, and option C is just a statement about the operation being possible, not about the closure property.

# Answer:
B. When multiplying any two polynomials, the result will always be a polynomial.

### 5.
# Explanation:
## Step1: Combine like - terms
$$
\begin{align*}
&(3x^{3}-2x^{2}+7x)+(3x^{2}-5x + 3)\
=&3x^{3}+(-2x^{2}+3x^{2})+(7x-5x)+3
\end{align*}
$$
## Step2: Simplify each group of like - terms
For the $x^{2}$ terms: $-2x^{2}+3x^{2}=x^{2}$
For the $x$ terms: $7x - 5x=2x$
So the simplified expression is $3x^{3}+x^{2}+2x + 3$

# Answer:
$3x^{3}+x^{2}+2x + 3$

### 6.
# Explanation:
## Step1: Use the distributive property (FOIL for multiple terms)
$$
\begin{align*}
&(2x^{2}+3x + 4)(3x^{3}+x^{2}-x)\
=&2x^{2}(3x^{3}+x^{2}-x)+3x(3x^{3}+x^{2}-x)+4(3x^{3}+x^{2}-x)
\end{align*}
$$
## Step2: Distribute each term
$$
\begin{align*}
&2x^{2}\times3x^{3}+2x^{2}\times x^{2}-2x^{2}\times x+3x\times3x^{3}+3x\times x^{2}-3x\times x+4\times3x^{3}+4\times x^{2}-4\times x\
=&6x^{5}+2x^{4}-2x^{3}+9x^{4}+3x^{3}-3x^{2}+12x^{3}+4x^{2}-4x
\end{align*}
$$
## Step3: Combine like - terms
For $x^{5}$ terms: $6x^{5}$
For $x^{4}$ terms: $2x^{4}+9x^{4}=11x^{4}$
For $x^{3}$ terms: $-2x^{3}+3x^{3}+12x^{3}=13x^{3}$
For $x^{2}$ terms: $-3x^{2}+4x^{2}=x^{2}$
For $x$ terms: $-4x$
The simplified expression is $6x^{5}+11x^{4}+13x^{3}+x^{2}-4x$

# Answer:
$6x^{5}+11x^{4}+13x^{3}+x^{2}-4x$

### 7.
# Explanation:
## Step1: Use the FOIL method (First, Outer, Inner, Last)
$$
\begin{align*}
&(2a^{3}+4b^{2})(4a^{3}-2b^{2})\
=&2a^{3}\times4a^{3}+2a^{3}\times(-2b^{2})+4b^{2}\times4a^{3}+4b^{2}\times(-2b^{2})
\end{align*}
$$
## Step2: Calculate each product
$$
\begin{align*}
&8a^{6}-4a^{3}b^{2}+16a^{3}b^{2}-8b^{4}
\end{align*}
$$
## Step3: Combine like - terms
For the $a^{3}b^{2}$ terms: $-4a^{3}b^{2}+16a^{3}b^{2}=12a^{3}b^{2}$
The simplified expression is $8a^{6}+12a^{3}b^{2}-8b^{4}$

# Answer:
$8a^{6}+12a^{3}b^{2}-8b^{4}$

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QUESTION IMAGE

Question

Explanation:

1.

Step1: Expand the expression

Answer:

2.