91Ó°ÊÓ

A quadratic polynomial is an expression of the form
\(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This type of polynomial results in a parabolic curve when graphed.

In the given exercise, the quadratic polynomial is \(x^2 - 6x + 9\). The general form helps us identify the quadratic, linear, and constant terms:

• \(ax^2\) = quadratic term (here, \(a = 1\))
• \(bx\) = linear term (here, \(b = -6\))
• \(c\) = constant term (here, \(c = 9\))

Understanding these terms is crucial as they guide us in the methods for factoring and solving the polynomial.

This specific quadratic polynomial is a perfect square trinomial because it can be written as a square of a binomial.

Factoring binomials involves expressing the polynomial as a product of two simpler expressions. For our exercise, we need to factor \(x^2 - 6x + 9\) into binomials. The steps are:

1. Identifying the form: We recognize that the given quadratic can potentially be factored as \((x - a)(x - b)\).

2. Solving for \(a\) and \(b\): We use the relationships originating from \( (x - a)(x - b) = x^2 - (a + b)x + ab \). For our polynomial \( x^2 - 6x + 9 \):

• The sum \( a + b = 6 \)
• The product \( ab = 9 \)

We solve these equations and find that both \(a\) and \(b\) must be 3 since \(3 + 3 = 6\) and \(3 \times 3 = 9\). Thus, the factorization is \( (x - 3)(x - 3) \).

3. Verifying the result: To ensure correctness, we expand \( (x - 3)^2 \):
] (x - 3)(x - 3) = x^2 - 3x - 3x + 9 = x^2 - 6x + 9 ] which matches our original polynomial.

Solving quadratic equations often begins with factoring. If we set the original polynomial to zero, we get:
\( x^2 - 6x + 9 = 0 \).

Using our factored form, we then set each binomial factor to zero:
\( (x - 3)^2 = 0 \).

Each factor yields: \( x - 3 = 0 \).

Solving this simple linear equation gives:
\( x = 3 \).

In this case, the quadratic equation has a single solution, which is a double root. Because our quadratic is a perfect square trinomial, it crosses the x-axis at a single point, \( x = 3 \).

To sum up, breaking down quadratic polynomials through factoring and solving equations involves:

• Identifying the form
• Finding suitable binomial factors
• Setting the polynomial to zero
• Solving the resulting simpler equations

With practice, these steps become intuitive and easier to apply across different quadratic equations.