Question

Determine whether each trinomial is a perfect square or not.. 1.A2+2a+1 2.4+4a+a2 3.A2-3a+9 4.4a2+24a+36 5.A4-10a+25 6.25a2+20a+9 7.A2b2-14ab+49 8.A2-2a-1 9.16a2+24a+36 10.10+6a+a2

Answer 1

1) Perfect square, 2) Perfect square, 3) Not a perfect square, 4) Perfect square, 5) Perfect square, 6) Not a perfect square, 7) Perfect square, 8) Not a perfect square, 9) Not a perfect square, 10) Not a perfect square.

Explanation:

From Algebra we must remember that a trinomial is a perfect square if and only if:

`(a+b)^(2) = a^(2)+2\cdot a\cdot b + b^(2)`,
`\forall \,a,b\in\mathbb{R}` (Eq. 1)

Now we proceed to prove each trinomial:

1)
`a^(2)+2\cdot a +1`:

If
`a^2 =a^2` and
`b^(2) = 1`, then
`a = \pm a` and
`b =\pm 1`. Therefore,
`2\cdot a\cdot b = 2\cdot (\pm a)\cdot (\pm 1) = \pm 2\cdot a`. In a nutshell, we conclude that this polynomial is a perfect square.

2)
`4+4\cdot a +a^(2)`

If
`a^(2) = 4` and
`b^(2) = a^(2)`, then
`a = \pm 2` and
`b = \pm a`. Therefore,
`2\cdot a \cdot b = 2\cdot (\pm 2)\cdot (\pm a) = \pm 4\cdot a`. In a nutshell, we conclude that this polynomial is a perfect square.

3)
`a^(2)-3\cdot a +9`

If
`a^(2) = a^(2)` and
`b^(2) = 9`, then
`a = \pm a` and
`b = \pm 3`. Therefore,
`2\cdot a \cdot b = 2\cdot (\pm a)\cdot (\pm 3) = \pm 6\cdot a`. In a nutshell, we conclude that this polynomial is not a perfect square.

4)
`4\cdot a^(2)+24\cdot a + 36`

If
`a^(2) = 4\cdot a^(2)` and
`b^(2) = 36`, then
`a = \pm 2\cdot a` and
`b = \pm 6`. Therefore,
`2\cdot a \cdot b = 2\cdot (\pm 2\cdot a)\cdot (\pm 6) = 24\cdot a`. In a nutshell, we conclude that this polynomial is a perfect square.

5)
`a^(4)-10\cdot a +25`

If
`a^(2) = a^(4)` and
`b^(2) = 25`, then
`a = \pm a^(2)` and
`b = \pm 5`. Therefore,
`2\cdot a \cdot b = 2\cdot (\pm a^(2))\cdot (\pm 5) = \pm 10\cdot a^(2)`. In a nutshell, we conclude that this polynomial is a perfect square.

6)
`25\cdot a^(2)+20\cdot a + 9`

If
`a^(2) = 25\cdot a^(2)` and
`b = 9`, then
`a = \pm 5\cdot a` and
`b = \pm 3`. Therefore,
`2\cdot a \cdot b = 2\cdot (\pm 5\cdot a)\cdot (\pm 3) = \pm 30\cdot a`. In a nutshell, we conclude that this polynomial is not a perfect square.

7)
`a^(2)\cdot b^(2)-14\cdot a \cdot b + 49`

If
`a^(2) = a^(2)\cdot b^(2)` and
`b^(2) = 49`, then
`a = \pm a\cdot b` and
`b = \pm 7`. Therefore,
`2\cdot a\cdot b = 2\cdot (\pm a\cdot b)\cdot (\pm 7) = \pm 14\cdot a \cdot b`. In a nutshell, we conclude that this polynomial is a perfect square.

8)
`a^(2)-2\cdot a -1`

If
`a^(2) = a^(2)` and
`b^(2) = -1`, then
`a = \pm a` and
`b = \pm i`. Therefore,
`2\cdot a \cdot b = \pm 2\cdot a \cdot i`. In a nutshell, we conclude that this polynomial is not a perfect square.

9)
`16\cdot a^(2) + 24\cdot a +36`

If
`a^(2) = 16\cdot a^(2)` and
`b^(2) = \pm 36`, then
`a = \pm 4\cdot a` and
`b = \pm 6`. Therefore,
`2\cdot a \cdot b = 2\cdot (\pm 4\cdot a)\cdot (\pm 6) = \pm 48\cdot a`. In a nutshell, we conclude that this polynomial is not a perfect square.

10)
`10+6\cdot a +a^(2)`

If
`a^(2) = 10` and
`b^(2) = a^(2)`, then
`a = \pm√(10)` and
`b = \pm a`. Therefore,
`2\cdot a \cdot b = \pm 2\cdot √(10)\cdot a`. In a nutshell, we conclude that this polynomial is not a perfect square.