What is the expanded form of (4x + 3y + 5z)^2?

Question

Examoogle · Accepted Answer

The correct answer is Option A: $16x^2 + 9y^2 + 25z^2 + 24xy + 30yz + 40xz$. Use the standard algebraic identity for the square of a trinomial and compute the cross-terms quickly by multiplying coefficients pairwise.

Step-by-Step Solution & Concepts:
Shortcut Trick: Use the standard algebraic identity for the square of a trinomial and compute the cross-terms quickly by multiplying coefficients pairwise.

Formula Used: $$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$$

Given: Expression to expand: $(4x + 3y + 5z)^2$

Steps: Identify the standard identity: To expand the square of a trinomial with three terms $a$, $b$, and $c$, we use the identity for $(a + b + c)^2$.. Substitute the values: Here, let $a = 4x$, $b = 3y$, and $c = 5z$. Substitute these into the identity.. Simplify each term: Square each individual term and compute the product for each cross-term: $(4x)^2 = 16x^2$, $(3y)^2 = 9y^2$, $(5z)^2 = 25z^2$.. Final expansion: Multiply the cross-terms by $2$ to get the final expanded expression.

Answer: A is correct because expanding $(4x + 3y + 5z)^2$ using the algebraic identity yields $16x^2 + 9y^2 + 25z^2 + 24xy + 30yz + 40xz$.

Exam Tip: Trinomial expansions are frequently tested in algebraic simplification and polynomial factorization questions in CBT-1 exams.

Examoogle · Answer

Option B: $16x^2 + 9y^2 + 25z^2 + 12xy + 15yz + 20xz$

Examoogle · Answer

Option C: $8x^2 + 6y^2 + 10z^2 + 24xy + 30yz + 40xz$

Examoogle · Answer

Option D: $16x^2 + 9y^2 + 25z^2 + 24xy + 30yz + 20xz$

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