Rewrite each expression using the distributive property and simplify, if possible. See Example 7. -(2d - f)

Verified step by step guidance

1. Recognize that the distributive property allows you to multiply a factor outside the parentheses by each term inside the parentheses. Here, the factor is \(-1\) because of the negative sign before the parentheses.
2. Apply the distributive property: multiply \(-1\) by each term inside the parentheses \(2d - f\). This gives \(-1 \times 2d\) and \(-1 \times (-f)\).
3. Calculate each multiplication separately: \(-1 \times 2d = -2d\) and \(-1 \times (-f) = +f\) because multiplying two negatives results in a positive.
4. Rewrite the expression by combining the results from the previous step: \(-2d + f\).
5. Check if the expression can be simplified further. Since \(-2d\) and \(f\) are unlike terms, the expression \(-2d + f\) is already simplified.

Key Concepts

Distributive Property

The distributive property states that multiplying a number by a sum or difference is the same as multiplying each term inside the parentheses separately and then adding or subtracting the results. For example, a(b + c) = ab + ac. This property is essential for rewriting expressions like -(2d - f).

Handling Negative Signs

When a negative sign precedes parentheses, it acts as multiplying the entire expression inside by -1. This means each term inside the parentheses changes its sign when the parentheses are removed. For example, -(2d - f) becomes -2d + f.

Simplifying Algebraic Expressions

Simplifying involves combining like terms and reducing the expression to its simplest form. After applying the distributive property and handling signs, check if any terms can be combined to make the expression clearer and more concise.