Problem 74 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 16 p^{3}-8 p^{2}+p $$

Step-by-Step Solution

Answer

The completely factored form is \( p(4p - 1)^{2} \).

Step 1: Identify the Greatest Common Factor (GCF)

First, factor out the greatest common factor (GCF) from all terms in the polynomial. The GCF of the terms in the polynomial \(16p^{3} - 8p^{2} + p\) is \(p\). Factor \(p\) out of each term:\[ p(16p^{2} - 8p + 1) \]

Step 2: Factor the Quadratic Expression

Next, factor the quadratic expression \(16p^{2} - 8p + 1\). Since this is a trinomial, look for two binomials that multiply to give this quadratic expression. We need to find two numbers that multiply to \(16 \times 1 = 16\) and add up to \(-8\). These numbers are \(-4\) and \(-4\):\[ 16p^{2} - 8p + 1 = (4p - 1)(4p - 1) = (4p - 1)^{2} \]

Step 3: Combine All Factors

Combine the factors from Steps 1 and 2 to get the completely factored form of the polynomial:\[ p(4p - 1)^{2} \]

Step 4: Identify Prime Polynomials

A prime polynomial cannot be factored further over the set of integers. The factors in the expression, \(4p - 1\) and \(p\), are both prime polynomials because they cannot be factored further over the set of integers.

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest factor that divides all the terms in a given polynomial without leaving a remainder. Finding the GCF is the first step in factoring any polynomial because it simplifies the expression, making further factoring easier. To find the GCF, identify the highest power of each common variable and the largest number that divides all coefficients.

Let's take a look at our polynomial:

• Terms: 16p^3, -8p^2, and p.
• GCF: Each term has at least one

Quadratic Expression

A quadratic expression is a polynomial of degree 2, typically written in the form:

• ax^2 + bx + c

where 'a', 'b', and 'c' are constants. To factor the quadratic expression, we look for two binomials that multiply together to give the original quadratic. In our exercise, the quadratic expression after factoring out the GCF is:

• 16p^2 - 8p + 1.

We need two numbers that multiply to 16 (the product of 'a' and 'c') and add up to -8 (the 'b' coefficient). These numbers are

• -4 and -4.

Factorizing it further, we get: (4p - 1)(4p - 1) or (4p - 1)^2.

Prime Polynomial

A prime polynomial is a polynomial that cannot be factored further over the set of integers. In other words, it is already in its simplest form. Identifying prime polynomials is crucial because it tells us that we have factored the expression as much as possible.

In our problem, after factoring out the GCF and the quadratic expression, we got:

• p(4p - 1)^2. <

The factors 'p' and '(4p - 1)' are both prime polynomials because neither can be reduced further. Recognizing these prime factors helps confirm that our solution is complete and accurate.