Question

7. All of the following are perfect square trinomials EXCEPT one. Which is it?

A. 4x2 + 12xy + 9y2
C. xy2 - 4x2y2 + 4x2y2
B. 9a? - 36a + 36
D. a²b2 + 4a3b + 4a?​

Answer 1

All answers EXCEPT answer C. are perfect square trinomials.

Explanation:

A perfect square trinomial is polynomial that satisfies the following condition:

`(a+b)^(2) = a^(2)+2\cdot a \cdot b + b^(2)`,
`\forall\,a,b\in\mathbb{R}`

Let prove if each option observe this:

a)
`4\cdot x^(2) + 12\cdot x\cdot y + 9\cdot y^(2)`

1)
`4\cdot x^(2) + 12\cdot x\cdot y + 9\cdot y^(2)` Given

2)
`(2\cdot x)^(2)+ 2\cdot (2\cdot x)\cdot (3\cdot y)+(3\cdot y)^(2)` Definition of power/Distributive, associative and commutative properties.

3)
`a = 2\cdot x`,
`b = 3\cdot y` Definition of perfect square trinomial/Result.

b)
`9\cdot a^(2)-36\cdot a + 36`

1)
`9\cdot a^(2)-36\cdot a + 36` Given.

2)
`(3\cdot a)^(2)-2\cdot (3\cdot a)\cdot 6 + 6^(2)` Definition of power/Distributive, associative and commutative properties.

3)
`a = 3\cdot a`,
`b = 6` Definition of perfect square trinomial/Result.

c)
`x\cdot y^(2)-4\cdot x^(2)\cdot y^(2)+4\cdot x^(2)\cdot y^(2)`

1)
`x\cdot y^(2)-4\cdot x^(2)\cdot y^(2)+4\cdot x^(2)\cdot y^(2)` Given

2)
`x\cdot y\cdot (1-4\cdot x+4\cdot x)` Distributive property.

3)
`x\cdot y \cdot 1` Existence of the additive inverse/Modulative property.

4)
`x\cdot y` Modulative property/Result.

d)
`a^(2)\cdot b^(2)+4\cdot a^(3)\cdot b + 4\cdot a^(4)`

1)
`a^(2)\cdot b^(2)+4\cdot a^(3)\cdot b + 4\cdot a^(4)` Given

2)
`(a\cdot b)^(2)+2\cdot (a\cdot b)\cdot (2\cdot a^(2))+(2\cdot a^(2))` Definition of power/Distributive, associative and commutative properties.

3)
`a = a\cdot b`,
`b = 2\cdot a^(2)` Definition of perfect square trinomial.

All answers EXCEPT answer C. are perfect square trinomials.