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To factor a trinomial completely, start by identifying the Greatest Common Factor (GCF). The GCF is the largest term that can be divided evenly into each term of the polynomial.
In the given problem, the trinomial is: \(9r^3 - 6r^2 + 16r.\)
Here are the steps to find the GCF:

• List out the factors of each term:
  9r^3: 1, 3, 9, r, r^2, r^3
  -6r^2: 1, 2, 3, 6, r, r^2
  16r: 1, 2, 4, 8, 16, r
• Identify the common factors: r is the only factor present in each term.

So, the GCF of \(9r^3, -6r^2,\text{ and } 16r\) is \(r.\)

Algebraic expressions are mathematical phrases that include numbers, variables, and operation symbols. Trinomials are a type of algebraic expression with three terms.
In this problem, we are given the trinomial: \(9r^3 - 6r^2 + 16r.\)
To simplify an algebraic expression, follow these steps:

• Identify and factor out any common factors (as previously done).
• Combine like terms and perform any possible arithmetic operations.

For our problem, factoring out the GCF \(r\), we get:
\(9r^3 - 6r^2 + 16r = r(9r^2 - 6r + 16)\)
Factoring the GCF simplifies the original expression and may help us identify if it can be simplified further.

Polynomial factorization is the process of breaking down a polynomial into simpler 'factor' polynomials that, when multiplied together, give back the original polynomial.
Factoring can involve different techniques such as GCF, grouping, or special products.
For our trinomial, after factoring out the GCF, we have: \(r(9r^2 - 6r + 16)\).
Next, check if the quadratic polynomial \(9r^2 - 6r + 16\) can be factored further. Look for two numbers that multiply to the product \(9 \times 16 = 144\) and add up to \(-6\).
However, in this case, no such pairs of numbers exist. So, the polynomial \(9r^2 - 6r + 16\) cannot be factored further.
Therefore, the completely factored form of the original trinomial is:
\(9r^3 - 6r^2 + 16r = r(9r^2 - 6r + 16)\).
Learning to factor polynomials is crucial in algebra as it simplifies equations and helps solve for unknown variables.