Answer :
Let's go step by step to solve each part of the algebra problem.
### 2.1 Algebraic Expression:
Given the expression:
[tex]\[ 4x^4 - x^3 + 5x^2 - 2x + 9 \][/tex]
#### 2.1.1. Write down the number of terms in the expression.
The expression \(4x^4 - x^3 + 5x^2 - 2x + 9\) has 5 terms.
#### 2.1.2. What is the coefficient of \(x^3\)?
The coefficient of \(x^3\) in the expression \(4x^4 - x^3 + 5x^2 - 2x + 9\) is \(-1\).
#### 2.1.3. Write down the constant term.
The constant term in the expression \(4x^4 - x^3 + 5x^2 - 2x + 9\) is 9.
#### 2.1.4. What is the degree of the expression?
The degree of the expression \(4x^4 - x^3 + 5x^2 - 2x + 9\) is determined by the highest power of \(x\), which is 4.
#### 2.1.5. What is the value of the expression when \(x=2\)?
Substitute \(x = 2\) into the expression:
[tex]\[ 4(2)^4 - (2)^3 + 5(2)^2 - 2(2) + 9 = 81 \][/tex]
So, the value of the expression when \(x=2\) is 81.
### 2.2. Solve for \(x\) in the equations below.
#### 2.2.1. \(2x = 88\)
To solve for \(x\):
[tex]\[ x = \frac{88}{2} = 44 \][/tex]
So, \(x = 44\).
#### 2.2.2. \(3x + 4 = 34\)
To isolate \(x\):
[tex]\[ 3x + 4 = 34 \implies 3x = 34 - 4 \implies 3x = 30 \implies x = \frac{30}{3} = 10 \][/tex]
So, \(x = 10\).
#### 2.2.3. \(4(x + 2) = 3(x + 2)\)
Let's simplify and solve for \(x\):
[tex]\[ 4(x + 2) = 3(x + 2) \implies 4 = 3 \][/tex]
There is no value of \(x\) that makes this equation true, so there is no solution.
#### 2.2.4. \(\frac{x}{3} + 2 = 5\)
To solve for \(x\):
[tex]\[ \frac{x}{3} + 2 = 5 \implies \frac{x}{3} = 5 - 2 \implies \frac{x}{3} = 3 \implies x = 3 \times 3 = 9 \][/tex]
So, \(x = 9\).
#### 2.2.5. \(3^x = 81\)
To solve for \(x\), note that \(81 = 3^4\):
[tex]\[ 3^x = 3^4 \implies x = 4 \][/tex]
So, \(x = 4\).
### 23. The sum of a number and 7 is 76. What is the number?
Let the number be \(y\). According to the problem:
[tex]\[ y + 7 = 76 \implies y = 76 - 7 = 69 \][/tex]
So, the number is 69.
This completes the step-by-step solution to the given algebra problem.
### 2.1 Algebraic Expression:
Given the expression:
[tex]\[ 4x^4 - x^3 + 5x^2 - 2x + 9 \][/tex]
#### 2.1.1. Write down the number of terms in the expression.
The expression \(4x^4 - x^3 + 5x^2 - 2x + 9\) has 5 terms.
#### 2.1.2. What is the coefficient of \(x^3\)?
The coefficient of \(x^3\) in the expression \(4x^4 - x^3 + 5x^2 - 2x + 9\) is \(-1\).
#### 2.1.3. Write down the constant term.
The constant term in the expression \(4x^4 - x^3 + 5x^2 - 2x + 9\) is 9.
#### 2.1.4. What is the degree of the expression?
The degree of the expression \(4x^4 - x^3 + 5x^2 - 2x + 9\) is determined by the highest power of \(x\), which is 4.
#### 2.1.5. What is the value of the expression when \(x=2\)?
Substitute \(x = 2\) into the expression:
[tex]\[ 4(2)^4 - (2)^3 + 5(2)^2 - 2(2) + 9 = 81 \][/tex]
So, the value of the expression when \(x=2\) is 81.
### 2.2. Solve for \(x\) in the equations below.
#### 2.2.1. \(2x = 88\)
To solve for \(x\):
[tex]\[ x = \frac{88}{2} = 44 \][/tex]
So, \(x = 44\).
#### 2.2.2. \(3x + 4 = 34\)
To isolate \(x\):
[tex]\[ 3x + 4 = 34 \implies 3x = 34 - 4 \implies 3x = 30 \implies x = \frac{30}{3} = 10 \][/tex]
So, \(x = 10\).
#### 2.2.3. \(4(x + 2) = 3(x + 2)\)
Let's simplify and solve for \(x\):
[tex]\[ 4(x + 2) = 3(x + 2) \implies 4 = 3 \][/tex]
There is no value of \(x\) that makes this equation true, so there is no solution.
#### 2.2.4. \(\frac{x}{3} + 2 = 5\)
To solve for \(x\):
[tex]\[ \frac{x}{3} + 2 = 5 \implies \frac{x}{3} = 5 - 2 \implies \frac{x}{3} = 3 \implies x = 3 \times 3 = 9 \][/tex]
So, \(x = 9\).
#### 2.2.5. \(3^x = 81\)
To solve for \(x\), note that \(81 = 3^4\):
[tex]\[ 3^x = 3^4 \implies x = 4 \][/tex]
So, \(x = 4\).
### 23. The sum of a number and 7 is 76. What is the number?
Let the number be \(y\). According to the problem:
[tex]\[ y + 7 = 76 \implies y = 76 - 7 = 69 \][/tex]
So, the number is 69.
This completes the step-by-step solution to the given algebra problem.
