Problem 18 - Precalculus: Functions and Graphs

Question

Factor the polynomial. $$9 x^{2}+24 x+16$$

Step-by-Step Solution

Answer

The polynomial is factored as \((3x + 4)^2\).

Step 1: Identify a Common Factor if any

Check if there is a greatest common factor for all terms in the polynomial. In this case, there is no common factor other than 1, so we proceed without factoring out any common factor.

Step 2: Recognize the Quadratic Form

Identify the polynomial as a quadratic of the form \( ax^2 + bx + c \). Here, \( a = 9 \), \( b = 24 \), and \( c = 16 \).

Step 3: Apply the Perfect Square Trinomial Formula

Notice that the polynomial resembles a perfect square trinomial \((ax)^2 + 2abx + b^2\). Thus, check if \((3x + 4)^2 = 9x^2 + 24x + 16\):- Square of \(3x\) is \(9x^2\).- The product of \(2 \times 3x \times 4\) is \(24x\).- Square of \(4\) is \(16\).Since all terms match, \(9x^2 + 24x + 16\) is a perfect square trinomial.

Step 4: Write the Perfect Square

Use the identity \((3x + 4)^2\) to express the polynomial. Therefore, the factorization of \(9x^2 + 24x + 16\) is \((3x + 4)^2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic polynomial, where you can express it as the square of a binomial. Recognizing this form can make factoring simpler and quicker.

A typical perfect square trinomial takes the form \[ (ax)^2 + 2abx + b^2 \] where:

• \(a\) corresponds to the coefficient of the squared term, \(x^2\).
• \(b\) is the constant term.
• \(2abx\) is the middle term coming from the product of the coefficients and twice the value of \(a\) and \(b\).

To determine if a quadratic polynomial is a perfect square trinomial, check if the square of the first term added to the square of the last term equals the middle term multiplied by twice the square root of the first and last terms. Thus, identifying these components correctly in \(9x^2 + 24x + 16\), you determine that it is indeed a perfect square trinomial because:

• The square of \(3x\) is \(9x^2\), matching the first term.
• The product \(2 \times 3x \times 4\) is \(24x\), matching the middle term.
• The square of \(4\) is \(16\), matching the last term.

Quadratic Polynomial

A quadratic polynomial is an expression of degree two, having the standard form \( ax^2 + bx + c \). Understanding its structure is crucial because the approach to solving or factoring it depends on how these terms interact.

Here’s a breakdown of its components:

• \(a\) is the leading coefficient influencing the parabola's opening – up or down.
• \(b\) affects the parabola's position along the x-axis.
• \(c\) locates the y-intercept, or where the curve crosses the y-axis.

In the provided exercise, \(9x^2 + 24x + 16\) is our quadratic polynomial. Each of these terms plays a part in both the shape and location of its corresponding parabola. The leading coefficient \(9\) suggests a steep curve. The middle term substitutes into factoring techniques to simplify into a recognizable pattern, like in our perfect square trinomial example.

Factoring Techniques

Factoring techniques are methods used to rewrite polynomials as a product of simpler ones. Learning these techniques is essential for simplifying expressions and solving polynomial equations efficiently.

Here are some common factoring methods:

• **Factoring by grouping:** Used when terms can be arranged into groups that have a common factor.
• **Factoring perfect trinomials:** As applied in this exercise, when the polynomial conforms to the square of a binomial form.
• **Using the quadratic formula:** When polynomials do not factor neatly, the quadratic formula provides solutions for \( ax^2 + bx + c \).

In our exercise, recognizing the polynomial \(9x^2 + 24x + 16\) as a perfect square trinomial led to using the formula to factor it into \((3x + 4)^2\). Accurate identification ensures you apply the correct technique, leading to simplified results and further insights into the polynomial's roots.