The correct option is \(\boxed{{\mathbf{Option D}}}\).
Further explanation:
The binomial algebraic expression is an algebraic expression that consists two terms and it is separated by plus or minus.
Binomial expression can be mathematically expressed as,
\(a + b\)
The trinomial algebraic expression is an algebraic expression that consists three terms and it is separated by plus or minus.
Trinomial expression can be mathematically expressed as,
\(a + b + c\)
Here, \(a,b{\text{ and }}c\) are the real numbers.
The square of the binomial \(a + b\) can be written as,
\({\left( {a + b} \right)^2} = \left( {a + b} \right)\left( {a + b} \right)\)
Given:
The given algebraic expression is \({\left( {4xy - 3z} \right)^2}\).
Step by step explanation:
Step 1:
The square of the binomial \(a + b\) can be written as,
\({\left( {a + b} \right)^2} = \left( {a + b} \right)\left( {a + b} \right)\)
Similarly, the expression \({\left( {4xy - 3z} \right)^2}\) can be written as,
\(\begin{aligned}{\left( {4xy - 3z} \right)^2} &= \left( {4xy - 3z} \right)\left( {4xy - 3z} \right) \\&= 4xy\left( {4xy - 3z} \right) - 3z\left( {4xy - 3z} \right) \\\end{aligned}\)
Step 2:
The distributive property can be used to solve the square of the binomial.
The distributive property can be expressed as,
\(a\left( {b + c} \right) = ab + ac\)
Now apply the distributive property to solve the expression \({\left( {4xy - 3z} \right)^2}\).
\(\begin{aligned}{\left( {4xy - 3z} \right)^2} &= 4xy\left( {4xy - 3z} \right) - 3z\left( {4xy - 3z} \right) \\&= 16{x^2}{y^2} - 12xyz - 12xyz + 9{z^2} \\&= 16{x^2}{y^2} - 24xyz + 9{z^2} \\\end{aligned}\)
Therefore, the expression \(16{x^2}{y^2} - 24xyz + 9{z^2}\) is the perfect square of the binomial \(\left( {4xy - 3z} \right)\).
The expression \(16{x^2}{y^2} - 24xyz + 9{z^2}\) is the trinomial.
Thus, option D a perfect square trinomial \(16{x^2}{y^2} - 24xyz + 9{z^2}\) is correct.
Learn more:
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Answer details:
Grade: Middle school
Subject: Mathematics
Chapter: Algebraic expression
Keywords: binomial, polynomial, algebraic expression, difference, product, trinomial, distributive property, equivalent, expression, terms, plus, separated, multiply, minus, addition
