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Chapter 6: Problem 11

Factor completely. If a polynomial is prime, state this. $$ 3 x^{2}-17 x-6 $$

Short Answer

Expert verified
The completely factored form is \(x(x - 9)^2\).

Step by step solution

01

Identify the common factor

Look for the greatest common factor (GCF) in each term of the polynomial. The terms of the polynomial are \(x^3\), \(-18x^2\), and \(81x\). Each term has the common factor \(x\).
02

Factor out the GCF

Factor out the common factor \(x\) from each term:\[ x(x^2 - 18x + 81) \]
03

Factor the quadratic expression

Now, focus on the quadratic expression inside the parenthesis, \(x^2 - 18x + 81\). Check if it can be factored further. Notice that it is a perfect square trinomial, since \(81 = 9^2\) and \(-2(9) = -18\). So it can be rewritten as:\[ x^2 - 18x + 81 = (x - 9)^2 \]
04

Write the completely factored form

Substitute the factored form of the quadratic expression back into the polynomial:\[ x(x^2 - 18x + 81) = x(x - 9)^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

greatest common factor
To start factoring a polynomial, it's efficient to first identify the Greatest Common Factor (GCF). The GCF is the largest factor that can be evenly divided into each term of the polynomial.
In the example, the terms are: \(x^3\), \(-18x^2\), and \(81x\).
All these terms share a common factor: \(x\).
By factoring out the GCF, we simplify the polynomial from \(x^3 - 18x^2 + 81x\) to \(x(x^2 - 18x + 81)\). This makes the expression easier to handle and paves the way for further simplification.
perfect square trinomials
A perfect square trinomial is a special type of quadratic expression that can be factored into a binomial squared. It takes the form \(a^2 - 2ab + b^2\) which can be written as \((a - b)^2\).
In our example, \(x^2 - 18x + 81\) is a perfect square trinomial.
To confirm this, we check two conditions:
  • Firstly, the first and last terms must be perfect squares (here, \(x^2\) and \(81\) or \(9^2\) are perfect squares).
  • Secondly, the middle term must equal twice the product of the two square roots (\(-2 * 9 * x = -18x\) in this case).
Both conditions are satisfied, so \(x^2 - 18x + 81\) can be written as \((x - 9)^2\). This recognition immensely simplifies the process of factoring.
quadratic expressions
Quadratic expressions are polynomials of the form \(ax^2 + bx + c\).
Factoring quadratics is a common problem in algebra, and it's crucial to recognize different patterns they might exhibit.
In the polynomial \(x^2 - 18x + 81\), we identified it as a perfect square trinomial. Hence, we factored it into \((x - 9)^2\).
This step comes after identifying and factoring out the GCF initially. The overall completely factored form from our original polynomial \(x^3 - 18x^2 + 81x\) thus becomes \(x(x - 9)^2\). Understanding these foundational patterns can make handling more complex polynomials manageable and less intimidating.

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Most popular questions from this chapter

Akio concludes that since \(x^{2}-9=(x-3)(x+3)\) it must follow that \(x^{2}+9=(x+3)(x-3) .\) What mistake(s) is he making?

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