Problem 79 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 2 h x-8 x+3 h p-15 p $$

Step-by-Step Solution

Answer

The polynomial \(2hx - 8x + 3hp - 15p\) is prime and cannot be factored further.

Step 1: Group terms

Group the terms in pairs to factor by grouping. Change the expression from \(2hx - 8x + 3hp - 15p\) to \((2hx - 8x) + (3hp - 15p)\).

Step 2: Factor out the greatest common factor (GCF) in each group

For the first group \(2hx - 8x\), the GCF is \(2x\). This gives: \[2x(h - 4)\]. For the second group \(3hp - 15p\), the GCF is \(3p\). This gives: \[3p(h - 5)\]. Now rewrite the expression: \(2x(h - 4) + 3p(h - 5)\).

Step 3: Check for a common binomial factor

In the factored expression, the binomial terms \((h - 4)\) and \((h - 5)\) are different. There is no common binomial factor.

Step 4: Conclude factorization

Since there are no common binomial factors and no further factoring is possible, the polynomial \(2hx - 8x + 3hp - 15p\) is already in its most factored form. It is a prime polynomial.

Key Concepts

greatest common factor

When factoring polynomials, a useful method is finding the greatest common factor (GCF). The GCF is the largest factor that each term in the polynomial shares. For example, consider the terms in our exercise:

• 2hx and 8x have a GCF of 2x. Both terms have a common multiple in 2 and the variable x.
• 3hp and 15p have a GCF of 3p. Here, both terms share 3 and the variable p as common multiples.

By factoring out the GCF from each pair, we simplify the equation, making it easier to handle. The equation changes from 2hx - 8x + 3hp - 15p to this intermediate simplified form: 2x(h - 4) + 3p(h - 5). This step sets the stage for further simplification by revealing shared patterns in the polynomial.

factoring by grouping

Factoring by grouping is a technique used to simplify complex polynomials. This method involves grouping terms that share a common factor and then factoring out the GCF of each group.
For our polynomial, 2hx - 8x + 3hp - 15p, we first rearrange it into pairs: (2hx - 8x) and (3hp - 15p).
Once grouped, we find the GCF for each group and factor it out:

• 2hx - 8x becomes 2x(h - 4)
• 3hp - 15p becomes 3p(h - 5)

This method transforms a more complex-looking polynomial into a simpler form, helping reveal any further common factors or aid in determining if the polynomial is prime.

prime polynomials

A prime polynomial is a polynomial that cannot be factored further using integer coefficients. Essentially, it is 'irreducible'.
Once we've factored out the GCF in our exercise and simplified it to 2x(h - 4) + 3p(h - 5), the next step is to check if we can factor it further. We look for any common binomial factors.
In this case, h - 4 and h - 5 are different and share no common factors. Since no further factoring is possible, we conclude that the original polynomial 2hx - 8x + 3hp - 15p is a prime polynomial. Recognizing this concept is important for ensuring that you have completely simplified the equation as much as possible.

binomial factors

Binomial factors are expressions containing two terms and are a key part of factored polynomials.
When factoring polynomials, it is common to arrive at expressions involving binomials. In the given exercise, once we factor out the GCF from each pair of terms, we see binomials in the expression.

• 2x(h - 4) + 3p(h - 5) shows us the binomial factors (h - 4) and (h - 5).

However, no common binomial factor exists in this particular case, so the polynomial can't be further simplified. Understanding binomial factors helps in recognizing the simpler, factored form of complex polynomials, or determining when a polynomial is prime.