Questions: Write each polynomial as the square of a bisemial or as an expression opposite to the square of a binomial. (C) (9/16) a^4 + a^3 + (4/9) a^2

Transcript text: Write each polynomial as the square of a bisemial or as an expression opposite to the square of a binomial. (C) $\frac{9}{16} a^{4}+a^{3}+\frac{4}{9} a^{2}$

Solution

Solution Steps

Solution Approach

To write the given polynomial as the square of a binomial or as an expression opposite to the square of a binomial, we need to identify if it can be factored into a perfect square trinomial. A perfect square trinomial takes the form $(x + y)^2$ or $(x - y)^2$. We will check if the given polynomial can be expressed in this form.

Step 1: Identify the Polynomial

The given polynomial is

\[ \frac{9}{16} a^{4} + a^{3} + \frac{4}{9} a^{2} \]

Step 2: Factor the Polynomial

We can factor the polynomial as follows:

\[ \frac{9}{16} a^{4} + a^{3} + \frac{4}{9} a^{2} = a^{2} \left( \frac{9}{16} a^{2} + a + \frac{4}{9} \right) \]

Step 3: Analyze the Quadratic Expression

Now we need to analyze the quadratic expression

\[ \frac{9}{16} a^{2} + a + \frac{4}{9} \]

to determine if it can be expressed as a perfect square trinomial.