Question

factor the polynomial. 25x² + 90x + 81 = □

Explanation:

Step1: Identify the form

The polynomial \(25x^{2}+90x + 81\) is a quadratic trinomial. We can check if it is a perfect square trinomial. A perfect square trinomial has the form \((ax + b)^{2}=a^{2}x^{2}+2abx + b^{2}\).

Step2: Check coefficients

For \(25x^{2}+90x + 81\), \(a^{2}=25\), so \(a = 5\) (since \(a>0\) for simplicity). \(b^{2}=81\), so \(b = 9\) (since the middle term is positive, \(b\) is positive). Now check the middle term: \(2ab=2\times5\times9 = 90\), which matches the middle term of the given polynomial.

Step3: Factor the polynomial

Since it is a perfect square trinomial, \(25x^{2}+90x + 81=(5x + 9)^{2}\)

Answer:

\((5x + 9)^{2}\)