Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 9 x^{2}-24 x+16 $$

Short answer

(3x - 4)^2

Step-by-step solution

Identify the polynomial

Recognize that the polynomial given is a quadratic polynomial: \[9x^2 - 24x + 16\]

Check for perfect square terms

Identify whether each term is a perfect square. Notice: \[9x^2 = (3x)^2\] \[16 = 4^2\]

Apply the perfect square trinomial rule

The middle term \[-24x\] should be twice the product of the square roots of the first and last terms: \[-24x = 2 \times 3x \times 4\] Hence, \[-24x = -24x\] This matches, so the polynomial \[9x^2 - 24x + 16\] can be written as a perfect square trinomial: \[(3x - 4)^2\]

Verify the factorization

To ensure correctness, expand \[(3x - 4)^2\] back into its original form: \[ (3x - 4)(3x - 4) = 9x^2 - 12x - 12x + 16 = 9x^2 - 24x + 16\] This matches the given polynomial.

Key concepts

quadratic polynomial

A quadratic polynomial is a type of polynomial that involves a variable raised to the second power as its highest exponent.
It takes the general form: \(a x^{2} +bx +c\).
Here, 'a', 'b', and 'c' are constants, and \(x\) is the variable.
Quadratic polynomials are fundamental in algebra because they frequently appear in various mathematical contexts, including equations and function analysis.
For example, in the exercise, the polynomial: \(9 x^{2}-24 x+16\) is a quadratic polynomial because the highest power of the variable \(x\) is 2.

perfect square trinomial

A perfect square trinomial is a special type of quadratic polynomial. It can be expressed as the square of a binomial term.
The general form is: \(a^2 x^2 + 2abx + b^{2} = (ax + b)^{2}\).
Identifying a perfect square trinomial involves checking if both the first and the last terms are perfect squares and if the middle term is twice the product of the square roots of the first and last terms.
In the example \(9 x^2 - 24 x + 16\), we have:

• \(9x^2 = (3x)^2\)
• \(16 = 4^2\)
• Middle term: \(-24x = 2 \times 3x \times 4\)
This means the entire polynomial can be written as \((3x - 4)^2\).

factoring

Factoring is the process of breaking down an expression into a product of simpler expressions.
This is especially useful in solving equations, simplifying expressions, and finding roots.
In the context of quadratic polynomials, factoring often involves converting the polynomial into a product of binomials.
For instance, in the given exercise, we started with \(9 x^{2}-24 x+16\).

• First, we identified it as a quadratic polynomial.
• Next, we recognized it as a perfect square trinomial.
• Finally, we factored it as \((3x - 4)^2\) by writing it in the form of a binomial squared.
Factoring makes it easier to solve equations by setting each factor equal to zero and solving for the variable.