please answer all questions on this sheet and remember to show all your work!
1. (a) which of the following is a polynomial? (circle one.)
$\frac{7}{2}x^2 + x^{-1} - 1$, $5x^{\frac{1}{2}}y$, $x^2 - 5$, $\frac{1.1y}{x}$
(b) for the polynomial above, state whether it is a monomial, a binomial, or a trinomial.
(c) for the polynomial identified in (a), state the degree and the constant term.
2. add and/or subtract the following polynomials:
(a) $(9 - 24x^2 + 4x) - 7y^2 - 9x + 2y + 16x^2$
(b) $(-3 + 12d^2) - (e^2 + d^2) - (4d - 8e^2 + 7)$
3. expand and simplify:
(a) $-3a^2(2a^3 - 7a + 9)$
(b) $(2x + 4)(x - 2)$
(c) $(3x - 2)^2$

Question

Accepted Answer

### 1. (a)
# Explanation:
## Step1: Recall Polynomial Definition
A polynomial is an expression of the form $a_nx^n + a_{n - 1}x^{n - 1}+\dots+a_1x + a_0$, where exponents are non - negative integers and coefficients are real numbers.
- For $\frac{7}{2}x^{2}+x^{-1}-1$, the term $x^{-1}=\frac{1}{x}$ has a negative exponent, so it is not a polynomial.
- For $5x^{\frac{1}{2}}y$, the exponent $\frac{1}{2}$ is a fraction (not a non - negative integer), so it is not a polynomial.
- For $x^{2}-5$, it can be written as $1x^{2}+0x - 5$, where exponents are non - negative integers and coefficients are real numbers. So it is a polynomial.
- For $\frac{1.1y}{x}=1.1yx^{-1}$, the term $x^{-1}$ has a negative exponent, so it is not a polynomial.

## Step2: Identify the Polynomial
From the above analysis, the polynomial is $x^{2}-5$.

# Answer: $x^{2}-5$

### 1. (b)
# Explanation:
## Step1: Recall Monomial, Binomial, Trinomial Definitions
- A monomial has 1 term.
- A binomial has 2 terms.
- A trinomial has 3 terms.
The polynomial $x^{2}-5$ can be written as $x^{2}+(- 5)$, which has two terms.

# Answer: Binomial

### 1. (c)
# Explanation:
## Step1: Find the Degree of the Polynomial
The degree of a polynomial is the highest power of the variable. In the polynomial $x^{2}-5$, the highest power of $x$ is 2, so the degree is 2.
## Step2: Find the Constant Term
The constant term is the term without a variable. In the polynomial $x^{2}-5$, the constant term is $-5$.

# Answer: Degree is 2, Constant term is $-5$

### 2. (a)
# Explanation:
## Step1: Remove Parentheses
$(9 - 24x^{2}+4x)-7y^{2}-9x + 2y+16x^{2}=9 - 24x^{2}+4x-7y^{2}-9x + 2y+16x^{2}$
## Step2: Combine Like Terms
- For $x^{2}$ terms: $-24x^{2}+16x^{2}=(-24 + 16)x^{2}=-8x^{2}$
- For $x$ terms: $4x-9x=(4 - 9)x=-5x$
- The constant term is 9, and we also have $-7y^{2}$ and $2y$
So the simplified form is $-8x^{2}-5x-7y^{2}+2y + 9$

# Answer: $-8x^{2}-5x-7y^{2}+2y + 9$

### 2. (b)
# Explanation:
## Step1: Remove Parentheses
$(-3 + 12d^{2})-(e^{2}+d^{2})-(4d - 8e^{2}+7)=-3 + 12d^{2}-e^{2}-d^{2}-4d + 8e^{2}-7$
## Step2: Combine Like Terms
- For $d^{2}$ terms: $12d^{2}-d^{2}=(12 - 1)d^{2}=11d^{2}$
- For $e^{2}$ terms: $-e^{2}+8e^{2}=(-1 + 8)e^{2}=7e^{2}$
- For constant terms: $-3-7=-10$
- The term with $d$ is $-4d$
So the simplified form is $11d^{2}+7e^{2}-4d - 10$

# Answer: $11d^{2}+7e^{2}-4d - 10$

### 3. (a)
# Explanation:
## Step1: Use the Distributive Property ($a(b + c + d)=ab+ac + ad$)
$-3a^{2}(2a^{3}-7a + 9)=-3a^{2}\times2a^{3}-3a^{2}\times(-7a)-3a^{2}\times9$
## Step2: Simplify Each Term
- For $-3a^{2}\times2a^{3}$: Using the rule $x^{m}\times x^{n}=x^{m + n}$, we get $-6a^{5}$
- For $-3a^{2}\times(-7a)$: We get $21a^{3}$
- For $-3a^{2}\times9$: We get $-27a^{2}$

So the expanded form is $-6a^{5}+21a^{3}-27a^{2}$

# Answer: $-6a^{5}+21a^{3}-27a^{2}$

### 3. (b)
# Explanation:
## Step1: Use the FOIL Method $(a + b)(c + d)=ac+ad+bc+bd$
$(2x + 4)(x - 2)=2x\times x+2x\times(-2)+4\times x+4\times(-2)$
## Step2: Simplify Each Term
- $2x\times x = 2x^{2}$
- $2x\times(-2)=-4x$
- $4\times x = 4x$
- $4\times(-2)=-8$
## Step3: Combine Like Terms
$-4x + 4x=0$, so the simplified form is $2x^{2}-8$

# Answer: $2x^{2}-8$

### 3. (c)
# Explanation:
## Step1: Recall the Formula $(a - b)^{2}=a^{2}-2ab + b^{2}$
Here $a = 3x$ and $b = 2$
## Step2: Apply the Formula
$(3x - 2)^{2}=(3x)^{2}-2\times(3x)\times2+2^{2}$
## Step3: Simplify Each Term
- $(3x)^{2}=9x^{2}$
- $2\times(3x)\times2 = 12x$
- $2^{2}=4$

So $(3x - 2)^{2}=9x^{2}-12x + 4$

# Answer: $9x^{2}-12x + 4$

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QUESTION IMAGE

Question

Explanation:

1. (a)

Step1: Recall Polynomial Definition

Step2: Identify the Polynomial

Step1: Recall Monomial, Binomial, Trinomial Definitions

Step1: Find the Degree of the Polynomial

Step2: Find the Constant Term

Answer:

1. (b)