$$ =2 x-3 y $$ b. Factor to write an equivalent expression: $$ 36 a-16 $$ Lin missed math class on the day they worked on expanding and factoring. Kiran is helping Lin catch up. a. Lin understands that expanding is using the distributive property, but she doesn't understand what factoring is or why it works. How can Kiran explain factoring to Lin? b. Lin asks Kiran how the diagrams with boxes help with factoring. What should Kiran tell Lin about the boxes? c. Lin asks Kiran to help her factor the expression $$ -4 x y-12 x z+20 x u $$. How can Kiran use this example to help Lin understand factoring? Complete the equation with numbers that make the expression on the right side of the equal sign equivalent to the expression on the left side. $$ 75 a+25 b=\ldots(\ldots a+b) $$

Question

UpStudy AI · Accepted Answer

1. **Factor the expression $$36a-16$$:**

- Both terms have a common factor of 4.
   - Factor out 4:
     $$
     36a-16 = 4(9a-4)
     $$

2. **Explaining Factoring:**

a. **Explaining what factoring is and why it works:**

- Factoring is like working the distributive property in reverse.  
      - When you expand an expression using the distributive property, you write a product as a sum. For example,  
        $$
        a(b+c) = ab+ac.
        $$
      - Factoring takes an expression like $$ab+ac$$ and writes it in the product form $$a(b+c)$$.  
      - This works because the distributive property guarantees that multiplication over addition is reversible.

b. **Using diagrams with boxes to help with factoring:**

- The box method (or area model) is a visual tool that organizes the terms of an expression into a grid.  
      - Each box holds a part of the expression, and by arranging the terms, you can see common factors along the rows or columns.  
      - The boxes help you "group" terms visually, making it easier to factor by identifying shared factors in each group.

c. **Factoring the expression $$ -4xy-12xz+20xu $$:**

- First, look for a common factor in all three terms.
      - Each term contains a factor of $$ -4x $$ (factoring out the negative helps keep the signs inside tidy):
        - $$ -4xy = -4x \cdot y $$
        - $$ -12xz = -4x \cdot 3z $$
        - $$ 20xu = -4x \cdot (-5u) $$
      - Factor out $$ -4x $$:
        $$
        -4xy-12xz+20xu = -4x \,(y+3z-5u)
        $$
      - This example shows that by identifying and removing the common factor $$ -4x $$, we simplify the expression into a product of the common factor and a simpler expression.

3. **Completing the equation:**

- We need to express:
     $$
     75a+25b = \ldots (\ldots a+b)
     $$
   - Notice that both terms on the left side are divisible by 25.
   - Factor out $$ 25 $$:
     $$
     75a+25b = 25(3a+b)
     $$
   - Hence, the complete equation is:
     $$
     75a+25b = 25(3a+b)
     $$
**Factoring Explained:** - **What is Factoring?** - Factoring is the process of breaking down an expression into a product of simpler expressions or factors. It's like reversing the distributive property. - **Why Does Factoring Work?** - Factoring works because the distributive property allows us to multiply and add terms in a reversible way. By factoring, we're essentially grouping common factors to simplify expressions. - **Using Diagrams with Boxes:** - Diagrams, or box methods, help visualize the grouping of terms. By organizing terms into boxes, it's easier to identify common factors in each group. - **Factoring Example:** - For the expression $$ -4xy - 12xz + 20xu $$, we factor out $$ -4x $$ to get $$ -4x(y + 3z - 5u) $$. - **Completing the Equation:** - $$ 75a + 25b = 25(3a + b) $$

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions