https://file.aiquickdraw.com/mathbot/file/1726114274685zcs5us5h.jpg

Let's go through the problems one by one.

### **Question 8**

#### a) Write a simplified expression for the **perimeter** of this rectangle.

The formula for the perimeter of a rectangle is:

\[
P = 2 \times (\text{length} + \text{width})
\]

From the diagram, the **length** of the rectangle is \( x + 7 \), and the **width** is \( x \).

Substitute these into the perimeter formula:

\[
P = 2 \times ((x + 7) + x) = 2 \times (2x + 7) = 4x + 14
\]

So, the simplified expression for the perimeter is:

\[
\boxed{4x + 14}
\]

---

#### b) The length \( (x + 7) \) of this rectangle is tripled. Write a simplified expression for the **new perimeter** of the rectangle.

If the length is tripled, the new length becomes \( 3(x + 7) = 3x + 21 \). The width remains \( x \).

Now, applying the perimeter formula again:

\[
P = 2 \times ((3x + 21) + x) = 2 \times (4x + 21) = 8x + 42
\]

The new perimeter expression is:

\[
\boxed{8x + 42}
\]

---

### **Question 9**

#### a) Expand and simplify: \( (x + 3)(x + 6) \)

Using the distributive property (FOIL method):

\[
(x + 3)(x + 6) = x(x + 6) + 3(x + 6) = x^2 + 6x + 3x + 18 = x^2 + 9x + 18
\]

So, the expanded form is:

\[
\boxed{x^2 + 9x + 18}
\]

---

#### b) Expand and simplify: \( (2x + 5)(3x + 10) \)

Again, use the distributive property:

\[
(2x + 5)(3x + 10) = 2x(3x + 10) + 5(3x + 10) = 6x^2 + 20x + 15x + 50 = 6x^2 + 35x + 50
\]

So, the expanded form is:

\[
\boxed{6x^2 + 35x + 50}
\]

---

#### c) Expand and simplify: \( (x + 5)(x - 6) \)

Using the distributive property:

\[
(x + 5)(x - 6) = x(x - 6) + 5(x - 6) = x^2 - 6x + 5x - 30 = x^2 - x - 30
\]

So, the expanded form is:

\[
\boxed{x^2 - x - 30}
\]

---

Would you like more details on any of these steps, or do you have any questions? Here are some related questions to further explore the concepts:

1. What happens to the perimeter if both the length and width are doubled?
2. How does the area of a rectangle change if the length is tripled?
3. Can you find the perimeter of a square using a similar approach?
4. What other methods can be used to simplify polynomials like in Question 9?
5. How does factoring help simplify expanded polynomial expressions?

**Tip:** When expanding two binomials, always remember the FOIL method: First, Outer, Inner, Last to ensure each term is correctly multiplied.

Consider the following rectangle. a) Write a simplified expression for the perimeter of this rectangle. b) The length (x + 7) of this rectangle is tripled. Write a simplified expression for the new perimeter of the rectangle. Expand and simplify: a) (x + 3)(x + 6), b) (2x + 5)(3x + 10), c) (x + 5)(x - 6).

This problem involves finding the perimeter of a rectangle, including a case where the length is tripled. It also covers expanding and simplifying binomials using algebraic methods. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation