Question

Factor the trinomial. $$x^{2}+18 x+81$$

Short answer

The factored form of the trinomial \(x^{2}+18x+81\) is \((x + 9)^2\)

Step-by-step solution

Identify a Perfect Square Trinomial

Recognize the trinomial as a perfect square trinomial. This is done by comparing the given trinomial, \(x^{2}+18x+81\) with the perfect square trinomial pattern \(a^2 + 2ab + b^2\). In this case, \(a=x\), \(2ab=18x\), and \(b^2=81\).

Find the square root of first and last term

The square root of first term \(a^2\) is \(a\) which is \(x\). The square root of last term \(b^2\) is \(b\), which is \(9\), because \(9 * 9 = 81\). We use these values to verify if \(2ab = 18x\).

Confirm 2ab = the middle term

This verifies that \(2*a*b = 2*x*9 =18x\) which is indeed the middle term of the trinomial.

Express as a square of a binomial

Since all conditions for a perfect square trinomial are fulfilled, the trinomial can be expressed as a square of a binomial: \( (a + b) ^ 2 = (x + 9) ^ 2\)