Distributive Property worksheet with problems involving constant, variable, and mixed factors.

Let's solve each problem in the worksheet using the Distributive Property. The distributive property states:

> $ a(b + c) = ab + ac $

This means we multiply the term outside the bracket by each term inside the bracket and then add or subtract accordingly.

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✔ Section A: Constant Factors

We expand the brackets by multiplying the constant outside by each term inside.

1) $ 7(a + 4) = 7 \cdot a + 7 \cdot 4 = 7a + 28 $

2) $ 9(b + 5) = 9 \cdot b + 9 \cdot 5 = 9b + 45 $

3) $ 8(7 + c) = 8 \cdot 7 + 8 \cdot c = 56 + 8c $

4) $ 8(2 - d) = 8 \cdot 2 - 8 \cdot d = 16 - 8d $

5) $ 3(11 - 3e) = 3 \cdot 11 - 3 \cdot 3e = 33 - 9e $

6) $ 6(2f - 9) = 6 \cdot 2f - 6 \cdot 9 = 12f - 54 $

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✔ Section B: Variable Factors

Now the factor outside is a variable. We still distribute it to each term inside.

1) $ g(2 + g) = g \cdot 2 + g \cdot g = 2g + g^2 $

2) $ h(h - 6) = h \cdot h - h \cdot 6 = h^2 - 6h $

3) $ i(i + 2) = i \cdot i + i \cdot 2 = i^2 + 2i $

4) $ j(7 - j) = j \cdot 7 - j \cdot j = 7j - j^2 $

5) $ k(4 - k) = k \cdot 4 - k \cdot k = 4k - k^2 $

6) $ l(l + m) = l \cdot l + l \cdot m = l^2 + lm $

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✔ Section C: Constant and Variable Factors

Here, the factor outside is a product of a constant and a variable (like $ 2n $). We distribute this entire term to each part inside.

1) $ 2n(5 + n) = 2n \cdot 5 + 2n \cdot n = 10n + 2n^2 $

2) $ 8p(4 - p) = 8p \cdot 4 - 8p \cdot p = 32p - 8p^2 $

3) $ 4q(2q + 7) = 4q \cdot 2q + 4q \cdot 7 = 8q^2 + 28q $

4) $ 3r(7 - 2r) = 3r \cdot 7 - 3r \cdot 2r = 21r - 6r^2 $

5) $ 5s(5s - 2) = 5s \cdot 5s - 5s \cdot 2 = 25s^2 - 10s $

6) $ 3t(5t + 9) = 3t \cdot 5t + 3t \cdot 9 = 15t^2 + 27t $

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✔ Final Answers:

#### Section A
1) $ 7a + 28 $
2) $ 9b + 45 $
3) $ 56 + 8c $
4) $ 16 - 8d $
5) $ 33 - 9e $
6) $ 12f - 54 $

#### Section B
1) $ 2g + g^2 $
2) $ h^2 - 6h $
3) $ i^2 + 2i $
4) $ 7j - j^2 $
5) $ 4k - k^2 $
6) $ l^2 + lm $

#### Section C
1) $ 10n + 2n^2 $
2) $ 32p - 8p^2 $
3) $ 8q^2 + 28q $
4) $ 21r - 6r^2 $
5) $ 25s^2 - 10s $
6) $ 15t^2 + 27t $

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📌 Explanation Summary:

- Distributive Property: Multiply the term outside the bracket with each term inside.
- Always remember signs: if there’s a minus sign, it applies to both terms.
- When variables are multiplied, combine them using exponent rules (e.g., $ x \cdot x = x^2 $).
- Constants and variables are treated separately during distribution.

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