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Difficulty: MediumSimplifying Algebraic Expressions

An algebraic expression is given below:

−2p(3q−p)−[4pq−p(2p+5q)]-2p(3q - p) - [4pq - p(2p + 5q)]

When fully simplified, which of the following represents the equivalent polynomial?

  1. A
    −5pq-5pq
  2. B
    −p2q-p^2q
  3. 4p2−5pq4p^2 - 5pqAnswer
  4. D
    −7pq-7pq

Answer

4p2−5pq4p^2 - 5pq
The expression simplifies correctly by first distributing the multipliers: −2p(3q−p)=−6pq+2p2-2p(3q - p) = -6pq + 2p^2 and −p(2p+5q)=−2p2−5pq-p(2p + 5q) = -2p^2 - 5pq. The bracket becomes [4pq−2p2−5pq]=[−pq−2p2][4pq - 2p^2 - 5pq] = [-pq - 2p^2]. Distributing the negative sign outside the bracket gives +pq+2p2+pq + 2p^2. Finally, combining all terms: (−6pq+2p2)+(pq+2p2)=4p2−5pq(-6pq + 2p^2) + (pq + 2p^2) = 4p^2 - 5pq.

Step-by-Step Solution

1
Distribute the outer multiplier −2p-2p into the first binomial.
−6pq+2p2-6pq + 2p^2
Applying the distributive property: −2p(3q)=−6pq-2p(3q) = -6pq and −2p(−p)=+2p2-2p(-p) = +2p^2.
2
Distribute −p-p into the binomial within the square brackets.
4pq−2p2−5pq4pq - 2p^2 - 5pq
Applying the distributive property inside the grouping: −p(2p)=−2p2-p(2p) = -2p^2 and −p(5q)=−5pq-p(5q) = -5pq.
3
Combine like terms inside the square brackets.
−pq−2p2-pq - 2p^2
Combining the like terms 4pq4pq and −5pq-5pq yields −pq-pq.
4
Distribute the negative sign outside the square brackets to the simplified contents.
+pq+2p2+pq + 2p^2
Subtracting a quantity is equivalent to adding its opposite: −(−pq−2p2)=+pq+2p2-(-pq - 2p^2) = +pq + 2p^2.
5
Combine all remaining terms from the entire expression.
2p2−6pq+pq+2p2=4p2−5pq2p^2 - 6pq + pq + 2p^2 = 4p^2 - 5pq
Grouping like terms: (2p2+2p2)+(−6pq+pq)=4p2−5pq(2p^2 + 2p^2) + (-6pq + pq) = 4p^2 - 5pq.

Key Concept

Simplifying expressions involves correctly applying the distributive property, especially with negative multipliers, and combining like terms.
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Topics

Simplifying Algebraic Expressions

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