Perfect Square Trinomial: Perfect square trinomials can be formulated as squares of binomials. When we square a binomial, we add twice the product of the two terms and the square of the last term.
Perfect Square Trinomial
For example, consider the following trinomial:
x^2 + 4x + 4
This trinomial can be rewritten as:
(x + 2)^2 = x^2 + 4x + 4
The parentheses around x+2 tell us to treat this as a single term. When we simplify this equation, we get:
x^2 + 6x + 4
This trinomial is a perfect square. This process can be repeated as long as the binomial squared results in a perfect square (equivalent to repeating the process without the parentheses).
However, perfect squares are not limited to trinomials with only two terms.
For example, consider this Perfect Square Trinomial
16^2 = 16*16 = 256
A perfect cube trinomial is created when given three terms of any degree; a binomial exists whose cube results in that exact term.
A perfect fourth-power trinomial is created when given four terms of any degree; there exists a binomial whose fourth power results in that same term.
A perfect fifth-power trinomial is created when given five terms of any degree; there exists a binomial whose fifth power results in that exact term.
Similarly, perfect sixth-power trinomials exist for six terms and perfect nth-powers for n terms of any degree.
Simplifying perfect square trinomials can be time-consuming, depending on the number of terms.
Also Read: Volume Of Cube
For perfect squares with only two terms, the result will always have four units as one term and an exponent of 2 as another.
In this situation, sometimes it may be easier to look at the
Perfect square as two separate factors:
x^2 = x(x)
4^2 = 4(2)
Notice that each perfect square is the perfect square of a binomial. The perfect squares are not perfect squares within perfect squares.
Therefore, it makes perfect sense to treat perfect squares as quadratic factors when simplifying perfect squares with only two terms.
A perfect cube trinomial can be thought of as similar to perfect squares in that its result is formed by multiplying each term of the given trinomial by the same number.
This means that when writing out perfect cubes, if any exponent on an x is odd (for example, 3), then there will always be a “1” at the end of this term’s factor because no number raised to an odd power can ever result in anything other than 1 for its last digit.
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