Answer:
All answers EXCEPT answer C. are perfect square trinomials.
Step-by-step explanation:
A perfect square trinomial is polynomial that satisfies the following condition:
[tex](a+b)^{2} = a^{2}+2\cdot a \cdot b + b^{2}[/tex], [tex]\forall\,a,b\in\mathbb{R}[/tex]
Let prove if each option observe this:
a) [tex]4\cdot x^{2} + 12\cdot x\cdot y + 9\cdot y^{2}[/tex]
1) [tex]4\cdot x^{2} + 12\cdot x\cdot y + 9\cdot y^{2}[/tex] Given
2) [tex](2\cdot x)^{2}+ 2\cdot (2\cdot x)\cdot (3\cdot y)+(3\cdot y)^{2}[/tex] Definition of power/Distributive, associative and commutative properties.
3) [tex]a = 2\cdot x[/tex], [tex]b = 3\cdot y[/tex] Definition of perfect square trinomial/Result.
b) [tex]9\cdot a^{2}-36\cdot a + 36[/tex]
1) [tex]9\cdot a^{2}-36\cdot a + 36[/tex] Given.
2) [tex](3\cdot a)^{2}-2\cdot (3\cdot a)\cdot 6 + 6^{2}[/tex] Definition of power/Distributive, associative and commutative properties.
3) [tex]a = 3\cdot a[/tex], [tex]b = 6[/tex] Definition of perfect square trinomial/Result.
c) [tex]x\cdot y^{2}-4\cdot x^{2}\cdot y^{2}+4\cdot x^{2}\cdot y^{2}[/tex]
1) [tex]x\cdot y^{2}-4\cdot x^{2}\cdot y^{2}+4\cdot x^{2}\cdot y^{2}[/tex] Given
2) [tex]x\cdot y\cdot (1-4\cdot x+4\cdot x)[/tex] Distributive property.
3) [tex]x\cdot y \cdot 1[/tex] Existence of the additive inverse/Modulative property.
4) [tex]x\cdot y[/tex] Modulative property/Result.
d) [tex]a^{2}\cdot b^{2}+4\cdot a^{3}\cdot b + 4\cdot a^{4}[/tex]
1) [tex]a^{2}\cdot b^{2}+4\cdot a^{3}\cdot b + 4\cdot a^{4}[/tex] Given
2) [tex](a\cdot b)^{2}+2\cdot (a\cdot b)\cdot (2\cdot a^{2})+(2\cdot a^{2})[/tex] Definition of power/Distributive, associative and commutative properties.
3) [tex]a = a\cdot b[/tex], [tex]b = 2\cdot a^{2}[/tex] Definition of perfect square trinomial.
All answers EXCEPT answer C. are perfect square trinomials.
