Distributing with Numbers

Simplify: $4(5 + 3)$

Identify the factor outside: The number $4$ is outside the parentheses = Factor: $4$

Distribute to each term: $4 \times 5$ and $4 \times 3$ = $20 + 12$

Add the products: $20 + 12 = 32$ = $32$

Answer: $4(5 + 3) = 20 + 12 = 32$

Distributing with Variables

Expand: $3(x + 7)$

Identify the factor outside: The number $3$ will multiply each term inside = Factor: $3$

Distribute to the first term: $3 \times x = 3x$ = $3x$

Distribute to the second term: $3 \times 7 = 21$ = $21$

Write the expanded form: Combine both products with addition = $3x + 21$

Answer: $3(x + 7) = 3x + 21$

Distributing with Subtraction

Expand: $5(2a - 4)$

Identify the factor outside: The number $5$ will multiply each term = Factor: $5$

Distribute to the first term: $5 \times 2a = 10a$ = $10a$

Distribute to the second term: $5 \times 4 = 20$ = $20$

Keep the subtraction sign: The minus sign stays between the terms = $10a - 20$

Answer: $5(2a - 4) = 10a - 20$

Distributing a Negative

Expand: $-2(x + 6)$

Identify the negative factor: The factor $-2$ is negative = Factor: $-2$

Distribute to the first term: $-2 \times x = -2x$ = $-2x$

Distribute to the second term: $-2 \times 6 = -12$ = $-12$

Write the result: Negative times positive gives negative = $-2x - 12$

Answer: $-2(x + 6) = -2x - 12$

More Complex Distribution

Expand: $-3(4y - 5)$

Identify the negative factor: The factor $-3$ is negative = Factor: $-3$

Distribute to the first term: $-3 \times 4y = -12y$ = $-12y$

Distribute to the second term: $-3 \times (-5) = +15$ = $+15$

Write the final result: Negative times negative gives positive = $-12y + 15$

Answer: $-3(4y - 5) = -12y + 15$

Mistake: Only distributing to the first term: $3(x + 4) = 3x + 4$

Why: The factor must multiply EVERY term inside the parentheses, not just the first one.

Correct: $3(x + 4) = 3x + 12$ (multiply $3$ by both $x$ AND $4$)

Mistake: Forgetting to change signs when distributing a negative: $-2(x + 3) = -2x + 3$

Why: A negative factor changes the sign of EACH product.

Correct: $-2(x + 3) = -2x - 6$ (negative times positive equals negative)

Mistake: Double negative error: $-4(x - 2) = -4x - 8$

Why: Negative times negative equals positive, not negative.

Correct: $-4(x - 2) = -4x + 8$ (because $-4 \times -2 = +8$)

Shopping Calculations

The distributive property helps calculate totals when buying multiples of items with different components.

You buy 3 meal deals that each include a burger for 5 euros and fries for 2 euros. Total: $3(5 + 2) = 3 \times 5 + 3 \times 2 = 15 + 6 = 21$ euros.

Mental Math Tricks

Use the distributive property to make difficult calculations easier in your head.

To calculate $6 \times 52$, think of it as $6(50 + 2) = 300 + 12 = 312$.

Q: Why is it called the distributive property?

A: Because the factor outside the parentheses gets 'distributed' (shared) to each term inside. It's like distributing cookies equally to everyone!

Q: Does the order matter? Is $a(b + c)$ the same as $(b + c)a$?

A: Yes, they're the same! Multiplication is commutative, so $a(b + c) = (b + c)a = ab + ac$.

Q: Can I use the distributive property with three or more terms?

A: Absolutely! For example, $2(x + y + z) = 2x + 2y + 2z$. Distribute to every term.