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Zorluk: ZorSimplifying Expressions and Combining Like Terms

When the expression −(p−2q)3−p(p−3q)(2p+q)+3q2(p−q)-(p - 2q)^3 - p(p - 3q)(2p + q) + 3q^2(p - q) is completely simplified by combining like terms, what is the coefficient of p2qp^2q?

Cevap: 11

Cevap

The coefficient of p2qp^2q in the fully simplified expression is 11.
Expanding the three components of the expression yields: −(p−2q)3=−p3+6p2q−12pq2+8q3-(p - 2q)^3 = -p^3 + 6p^2q - 12pq^2 + 8q^3; −p(p−3q)(2p+q)=−2p3+5p2q+3pq2-p(p - 3q)(2p + q) = -2p^3 + 5p^2q + 3pq^2; and 3q2(p−q)=3pq2−3q33q^2(p - q) = 3pq^2 - 3q^3. Combining these terms gives the simplified polynomial −3p3+11p2q−6pq2+5q3-3p^3 + 11p^2q - 6pq^2 + 5q^3. The coefficient of p2qp^2q is 11.

Adım Adım Çözüm

1
Expand and negate the term −(p−2q)3-(p - 2q)^3
−p3+6p2q−12pq2+8q3-p^3 + 6p^2q - 12pq^2 + 8q^3
Using the binomial theorem expansion for (a−b)3=a3−3a2b+3ab2−b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 and distributing the negative sign.
2
Multiply and distribute −p(p−3q)(2p+q)-p(p - 3q)(2p + q)
−2p3+5p2q+3pq2-2p^3 + 5p^2q + 3pq^2
First multiply the binomials (p−3q)(2p+q)=2p2−5pq−3q2(p - 3q)(2p + q) = 2p^2 - 5pq - 3q^2, and then multiply each term by −p-p.
3
Distribute 3q2(p−q)3q^2(p - q)
3pq2−3q33pq^2 - 3q^3
Multiply 3q23q^2 by both terms inside the binomial.
4
Combine the like terms of p2qp^2q
1111
Identify and sum all terms containing p2qp^2q: 6p2q+5p2q=11p2q6p^2q + 5p^2q = 11p^2q.

Anahtar Kavram

Simplifying algebraic expressions by expanding polynomials, applying the distributive property with negative signs, and combining like terms.
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Simplifying Expressions and Combining Like Terms

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