name____________________ geometry midterm review date_________period____
1. answer the questions about the following polynomial.
$-3 - x + \frac{1}{3}x^2 + x^4$
the expression represents a $\underline{\quad\quad}$ polynomial
with $\underline{\quad\quad}$ terms. the constant term is $\underline{\quad\quad}$, the
leading term is $\underline{\quad\quad}$, and the leading coefficient is
$\underline{\quad\quad}$
this word bank also applies to question 2.
word bank: (a) cubic, (b) quartic, (c) linear, (d) quadratic, (e) quintic

2. answer the questions about the following polynomial.
$-4x^5 - x^4 - 7 - \frac{1}{4}x$
the expression represents a $\underline{\quad\quad}$ polynomial
with $\underline{\quad\quad}$ terms. the constant term is $\underline{\quad\quad}$, the
leading term is $\underline{\quad\quad}$, and the leading coefficient is
$\underline{\quad\quad}$

rewrite the following polynomial in standard form.
$10 + \frac{x^5}{5} + x^3$

rewrite the following polynomial in standard form.
$2 - 8x - \frac{x^2}{5}$

5. use the box method to distribute and simplify
$(2x - 6)(-6x + 3)$.

6. use the box method to distribute and simplify
$(x - 3)(-5x + 1)$.

7. use the box method to distribute and simplify
$(-6x^2 + 4x)(3x + 5)$.

Question

name____________________ geometry midterm review date_________period____
1. answer the questions about the following polynomial.
$-3 - x + \frac{1}{3}x^2 + x^4$
the expression represents a $\underline{\quad\quad}$ polynomial
with $\underline{\quad\quad}$ terms. the constant term is $\underline{\quad\quad}$, the
leading term is $\underline{\quad\quad}$, and the leading coefficient is
$\underline{\quad\quad}$
this word bank also applies to question 2.
word bank: (a) cubic, (b) quartic, (c) linear, (d) quadratic, (e) quintic

2. answer the questions about the following polynomial.
$-4x^5 - x^4 - 7 - \frac{1}{4}x$
the expression represents a $\underline{\quad\quad}$ polynomial
with $\underline{\quad\quad}$ terms. the constant term is $\underline{\quad\quad}$, the
leading term is $\underline{\quad\quad}$, and the leading coefficient is
$\underline{\quad\quad}$

rewrite the following polynomial in standard form.
$10 + \frac{x^5}{5} + x^3$

rewrite the following polynomial in standard form.
$2 - 8x - \frac{x^2}{5}$

5. use the box method to distribute and simplify
$(2x - 6)(-6x + 3)$.

6. use the box method to distribute and simplify
$(x - 3)(-5x + 1)$.

7. use the box method to distribute and simplify
$(-6x^2 + 4x)(3x + 5)$.

Accepted Answer

### Question 1: Analyze the polynomial $-3 - x + \frac{1}{3}x^2 + x^4$
# Explanation:
## Step 1: Determine the degree of the polynomial
The degree of a polynomial is the highest power of $x$ with a non - zero coefficient. In the polynomial $-3 - x + \frac{1}{3}x^2 + x^4$, the highest power of $x$ is $4$ (from the term $x^4$). So the polynomial is a quartic (since degree 4) polynomial.
## Step 2: Count the number of terms
A term is a single number, variable, or the product of a number and variable(s). The terms in the polynomial are $-3$, $-x$, $\frac{1}{3}x^2$, and $x^4$. So there are 4 terms.
## Step 3: Identify the constant term
The constant term is the term without a variable. In the polynomial, the constant term is $-3$.
## Step 4: Identify the leading term
The leading term is the term with the highest degree. Here, the term with the highest degree (degree 4) is $x^4$.
## Step 5: Identify the leading coefficient
The leading coefficient is the coefficient of the leading term. For the term $x^4$, the coefficient is $1$.

# Answer:
quartic, 4, $-3$, $x^4$, $1$

### Question 2: Analyze the polynomial $-4x^5 - x^4 - 7-\frac{1}{4}x$
# Explanation:
## Step 1: Determine the degree of the polynomial
The highest power of $x$ in the polynomial $-4x^5 - x^4 - 7-\frac{1}{4}x$ is $5$ (from the term $-4x^5$). So the polynomial is a quintic (degree 5) polynomial.
## Step 2: Count the number of terms
The terms are $-4x^5$, $-x^4$, $-7$, and $-\frac{1}{4}x$. So there are 4 terms.
## Step 3: Identify the constant term
The constant term is $-7$ (the term without a variable).
## Step 4: Identify the leading term
The leading term is the term with the highest degree, which is $-4x^5$.
## Step 5: Identify the leading coefficient
The coefficient of the leading term $-4x^5$ is $-4$.

# Answer:
quintic, 4, $-7$, $-4x^5$, $-4$

### Question 3: Rewrite $10+\frac{x^5}{5}+x^3$ in standard form
# Explanation:
## Step 1: Recall the standard form of a polynomial
The standard form of a polynomial is written in descending order of the exponents of the variable.
## Step 2: Rearrange the terms
For the polynomial $10+\frac{x^5}{5}+x^3$, we arrange the terms from the highest degree to the lowest degree. The term with degree 5 is $\frac{x^5}{5}$, the term with degree 3 is $x^3$, and the constant term (degree 0) is $10$. So the standard form is $\frac{1}{5}x^5 + x^3+10$.

# Answer:
$\frac{1}{5}x^5 + x^3 + 10$

### Question 4: Rewrite $2 - 8x-\frac{x^2}{5}$ in standard form
# Explanation:
## Step 1: Recall the standard form of a polynomial
Standard form is in descending order of exponents.
## Step 2: Rearrange the terms
For the polynomial $2 - 8x-\frac{x^2}{5}$, the term with degree 2 is $-\frac{x^2}{5}$, the term with degree 1 is $-8x$, and the constant term (degree 0) is $2$. So the standard form is $-\frac{1}{5}x^2-8x + 2$.

# Answer:
$-\frac{1}{5}x^2-8x + 2$

### Question 5: Use the box method to simplify $(2x - 6)(-6x + 3)$
# Explanation:
## Step 1: Set up the box
Draw a 2x2 box. Write the terms of the first binomial ($2x$ and $-6$) along the top of the box, and the terms of the second binomial ($-6x$ and $3$) along the left - hand side of the box.
|       | $2x$ | $-6$ |
|-------|-------|-------|
| $-6x$ | $2x\times(-6x)=-12x^2$ | $-6\times(-6x) = 36x$ |
| $3$ | $2x\times3 = 6x$ | $-6\times3=-18$ |
## Step 2: Combine like terms
Now, we add up all the terms inside the box: $-12x^2+36x + 6x-18$. Combine the like terms $36x$ and $6x$ (since they have the same degree of $x$). $36x+6x = 42x$. So the simplified form is $-12x^2 + 42x-18$.

# Answer:
$-12x^2+42x - 18$

### Question 6: Use the box method to simplify $(x - 3)(-5x + 1)$
# Explanation:
## Step 1: Set up the box
Draw a 2x2 box. Write $x$ and $-3$ along the top, and $-5x$ and $1$ along the left.
|       | $x$ | $-3$ |
|-------|-------|-------|
| $-5x$ | $x\times(-5x)=-5x^2$ | $-3\times(-5x)=15x$ |
| $1$ | $x\times1=x$ | $-3\times1 = - 3$ |
## Step 2: Combine like terms
Add the terms inside the box: $-5x^2+15x+x - 3$. Combine $15x$ and $x$: $15x + x=16x$. So the simplified form is $-5x^2+16x - 3$.

# Answer:
$-5x^2+16x - 3$

### Question 7: Use the box method to simplify $(-6x^2 + 4x)(3x + 5)$
# Explanation:
## Step 1: Set up the box
Draw a 2x2 box. Write $-6x^2$ and $4x$ along the top, and $3x$ and $5$ along the left.
|       | $-6x^2$ | $4x$ |
|-------|-------|-------|
| $3x$ | $-6x^2\times3x=-18x^3$ | $4x\times3x = 12x^2$ |
| $5$ | $-6x^2\times5=-30x^2$ | $4x\times5 = 20x$ |
## Step 2: Combine like terms
Add the terms inside the box: $-18x^3+12x^2-30x^2 + 20x$. Combine $12x^2$ and $-30x^2$: $12x^2-30x^2=-18x^2$. So the simplified form is $-18x^3-18x^2 + 20x$.

# Answer:
$-18x^3-18x^2 + 20x$

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QUESTION IMAGE

Question

Explanation:

Question 1: Analyze the polynomial \(-3 - x + \frac{1}{3}x^2 + x^4\)

Step 1: Determine the degree of the polynomial

Step 2: Count the number of terms

Step 3: Identify the constant term

Step 4: Identify the leading term

Step 5: Identify the leading coefficient

Step 1: Determine the degree of the polynomial

Step 2: Count the number of terms

Step 3: Identify the constant term

Step 4: Identify the leading term

Step 5: Identify the leading coefficient

Step 1: Recall the standard form of a polynomial

Step 2: Rearrange the terms

Answer:

Question 2: Analyze the polynomial \(-4x^5 - x^4 - 7-\frac{1}{4}x\)