Problem 73 - Elementary Algebra

Question

Use any of the factoring methods to factor. Identify any prime polynomials. $$ 42 p^{5}-28 p^{4}+56 p^{3}-70 p^{2}+21 p $$

Step-by-Step Solution

Answer

The factored form is \(7p(6p^4 - 4p^3 + 8p^2 - 10p + 3)\), where \(6p^4 - 4p^3 + 8p^2 - 10p + 3\) is a prime polynomial.

Step 1: Identify the Greatest Common Factor (GCF)

The first step in factoring is to identify the Greatest Common Factor (GCF) that can be factored out from all terms. In this case, all terms are divisible by 7 and contain at least one factor of p, giving us a GCF of 7p.

Step 2: Factor out the GCF

Factor out the GCF from the polynomial:\[ 42p^5 - 28p^4 + 56p^3 - 70p^2 + 21p = 7p(6p^4 - 4p^3 + 8p^2 - 10p + 3) \]

Step 3: Evaluate the remaining polynomial

After factoring out the GCF, we are left with the polynomial \(6p^4 - 4p^3 + 8p^2 - 10p + 3\). Check if this polynomial can be factored further. In this case, it cannot be factored further using common factoring methods and is therefore considered to be a prime polynomial.

Key Concepts

Greatest Common Factor

In the context of factoring polynomials, identifying the Greatest Common Factor (GCF) is a crucial first step. The GCF is the largest factor that is common to all terms in the polynomial.
To find the GCF:

• List the factors for each term in the polynomial
• Identify the common factors
• Select the greatest one

In the given exercise, the polynomial is
\( 42 p^{5} - 28 p^{4} + 56 p^{3} - 70 p^{2} + 21 p \) After examining each term, we see that 7 and \( p \) are common factors in all terms. Hence, the GCF is \( 7p \). Factoring out the GCF simplifies the polynomial and makes further factoring easier.

Prime Polynomials

A prime polynomial is a polynomial that cannot be factored into terms of lower degrees other than 1 and itself. After factoring out the GCF from a polynomial, it's important to determine if the resulting polynomial can be factored further.
In the problem, after removing the GCF \( 7p \), we are left with \( 6p^4 - 4p^3 + 8p^2 - 10p + 3 \). If this polynomial cannot be factored by usual methods such as grouping or using special identities (like difference of squares or sum and difference of cubes), it is deemed prime.
Since \( 6p^4 - 4p^3 + 8p^2 - 10p + 3 \) does not factor further, it is a prime polynomial.

Factoring Methods

There are several methods to factor polynomials, each useful in different situations:

• Finding the GCF: The first method where you factor out the greatest common factor from all terms. For instance, factoring \( 7p \) from \( 42 p^{5}-28 p^{4}+56 p^{3}-70 p^{2}+21 p \).


• Factoring by grouping: This involves regrouping terms with common factors. While not applicable in the given example, it’s useful when terms share common variables or coefficients.


• Special Products: Recognize and factor special forms such as \( a^2 - b^2 = (a-b)(a+b) \) or \( a^3 \) forms.


• Trial and error: Test possible factor pairs for simple polynomials.

Depending on the polynomial, you may need to use one or multiple methods to achieve complete factorization.