The correct option is [tex]\boxed{{\mathbf{Option D}}}[/tex].
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,
[tex]a + b[/tex]
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,
[tex]a + b + c[/tex]
Here, [tex]a,b{\text{ and }}c[/tex] are the real numbers.
The square of the binomial [tex]a + b[/tex] can be written as,
[tex]{\left( {a + b} \right)^2} = \left( {a + b} \right)\left( {a + b} \right)[/tex]
Given:
The given algebraic expression is [tex]{\left( {4xy - 3z} \right)^2}[/tex].
Step by step explanation:
Step 1:
The square of the binomial [tex]a + b[/tex] can be written as,
[tex]{\left( {a + b} \right)^2} = \left( {a + b} \right)\left( {a + b} \right)[/tex]
Similarly, the expression [tex]{\left( {4xy - 3z} \right)^2}[/tex] can be written as,
[tex]\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}[/tex]
Step 2:
The distributive property can be used to solve the square of the binomial.
The distributive property can be expressed as,
[tex]a\left( {b + c} \right) = ab + ac[/tex]
Now apply the distributive property to solve the expression [tex]{\left( {4xy - 3z} \right)^2}[/tex].
[tex]\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}[/tex]
Therefore, the expression [tex]16{x^2}{y^2} - 24xyz + 9{z^2}[/tex] is the perfect square of the binomial [tex]\left( {4xy - 3z} \right)[/tex].
The expression [tex]16{x^2}{y^2} - 24xyz + 9{z^2}[/tex] is the trinomial.
Thus, option D a perfect square trinomial [tex]16{x^2}{y^2} - 24xyz + 9{z^2}[/tex] is correct.
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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
