Question - Simplifying Algebraic Expression
Solution:
3p(p - q) - (2p - q)²3p² - 3pq - (4p² - 4pq + q²)3p² - 3pq - 4p² + 4pq - q²-p² + pq - q²
Tutor Solution:
To solve the given expression $$ 3p(p−q)−(2p−q)^2 $$, let's go through it step by step:1. **Distribute the first term:**Start by applying the distributive property to the first term $$ 3p(p−q) $$. The distributive property states that $$ a(b+c) = ab+ac $$. In your case, we need to multiply $$ 3p $$ with both $$ p $$ and $$ -q $$ respectively:\[ 3p(p−q) = 3p \times p - 3p \times q = 3p^2 - 3pq \]2. **Expand the second term:**We have a perfect square trinomial here: $$ (2p−q)^2 $$. This means you will apply the binomial square formula $$ (a−b)^2 = a^2−2ab+b^2 $$. \[ (2p−q)^2 = (2p)^2 - 2 \cdot (2p) \cdot q + q^2 \]This simplifies to:\[ (2p−q)^2 = 4p^2 - 4pq + q^2 \]3. **Combine like terms:**Now you can combine like terms from the expanded expressions:\[ 3p(p−q)−(2p−q)^2 = (3p^2 - 3pq) - (4p^2 - 4pq + q^2) \]This means we subtract every term in the second bracket from the corresponding term in the first.4. **Subtract corresponding terms:**When subtracting the terms in the perfect square trinomial from the initial terms:\[ (3p^2 - 3pq) - 4p^2 \rightarrow 3p^2 - 4p^2 = -p^2 \]\[ -(3p^2 - 3pq) + 4pq \rightarrow -3pq + 4pq = pq \]\[ -(-q^2) \rightarrow +q^2 \]5. **Combine the results:**Combine the results from step 4:\[ -p^2 + pq + q^2 \]So the final simplified form of the expression $$ 3p(p−q)−(2p−q)^2 $$ is:\[ -p^2 + pq + q^2 \]This final expression cannot be simplified further as there are no like terms remaining.
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