Problem 27 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$3 x^{3}-24 x^{2}+48 x$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(3 x^{3}-24 x^{2}+48 x\) is \(3x*(x-4)^{2}\).

Step 1: Finding Common Factors

Firstly, notice that each term in the polynomial, \(3 x^{3}-24 x^{2}+48 x\), has a factor of 3x. So, we'll factor 3x out from each term: \(3x * (x^{2}-8x+16)\).

Step 2: Factoring the Quadratic Polynomial

The quadratic polynomial inside the parentheses, \(x^{2}-8x+16\), is a perfect square trinomial. A perfect square trinomial can be factored into \((x-a)^{2}\), where 'a' is the square root of the third term. In our case, a = 4, so \((x^{2}-8x+16)\) factors to \((x-4)^{2}\).

Step 3: Checking Factorization by Multiplication

To make sure our factorization is correct, we'll multiply our factors back out. \(3x * (x-4)^{2}\) = \(3x*(x^{2}-8x+16)\) = \(3x^{3}-24x^{2}+48x\), which matches our original polynomial, confirming that our factorization is correct.

Key Concepts

Common Factor

When approaching the factorization of any polynomial, identifying a common factor is often the first crucial step. A common factor is a term or number that divides each term of the polynomial without leaving a remainder. In the polynomial \(3x^{3} - 24x^{2} + 48x\), each term shares a common factor of \(3x\).
By extracting \(3x\) from the polynomial, we simplify it to \(3x(x^{2} - 8x + 16)\). This not only reduces the complexity of the problem but is also a critical step that forms the foundation for further simplification.

• Start by listing the coefficients and terms.
• Find the greatest common factor by identifying the largest term and power present in all terms.
• Factor out the common factor from each term to simplify the polynomial.

Identifying the common factor right away helps streamline the process of factorization and makes managing the polynomial much easier.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be written as the square of a binomial. The general form of a perfect square trinomial is \((a - b)^2\), and expands to \(a^2 - 2ab + b^2\). Detecting these forms can significantly simplify factorization processes. When facing the quadratic polynomial \(x^{2} - 8x + 16\), note that this can be perceived as a perfect square trinomial.
To confirm this, check if \((\frac{b}{2})^2 = c\), where \(b\) is the coefficient of \(x\) and \(c\) is the constant term. Here, \(b = -8\) and \(c = 16\), illustrating that \((\frac{-8}{2})^2 = 16\). Thus, \(x^{2} - 8x + 16\) factors neatly into \((x-4)^2\).

• Identify the middle term and use it to find the square root of the constant term, confirming it forms a perfect square.
• Rewrite the trinomial as a square of a binomial.
• This perfect square factor is simpler to handle and verify.

The ability to spot perfect square trinomials can bring efficiency to solving many algebraic problems.

Checking Factorization

Once a polynomial is factored, it is essential to verify its correctness to ensure no mistakes were made in the process. Verification can be performed by multiplying the factors back and checking if you retrieve the original polynomial. In our case, the original polynomial \(3x^3 - 24x^2 + 48x\) can be rewritten as \(3x(x-4)^2\).
To check this, express \((x - 4)^2\) as \(x^2 - 8x + 16\) and multiply it by \(3x\):
1. Distribute the \(3x\) across each term of \(x^2 - 8x + 16\).2. This results in \(3x^3 - 24x^2 + 48x\), which matches the original polynomial.

• Ensure that each step in multiplication is accurate as even small mistakes can lead to incorrect conclusions.
• Use alternative methods, such as graphing utilities, if unsure, which offer visual verification.

Confirming factorization by multiplication helps identify and correct errors, making it an indispensable part of problem-solving in algebra.