Problem 51 - Elementary Algebra

Question

(a) factor by grouping. Identify any prime polynomials. (b) check. $$ 3 m^{2}+3 m v-m-v $$

Step-by-Step Solution

Answer

(m + v)(3m - 1)

Step 1: Group the terms

Group the terms in pairs to facilitate factoring:groups: group 1: (3m^{2} + 3mv)group 2: (-m - v)

Step 2: Factor out the Greatest Common Factor (GCF) in each group

Factor out the GCF from each group:Group 1: 3m(m + v) Group 2: -1(m + v)

Step 3: Factor out the common binomial factor

Both terms now have a common factor of (m + v). Factor this out:(m + v)(3m - 1)

Step 4: Check the factorization

Expand the factored form to verify the result:(m + v)(3m - 1)=m(3m - 1) + v(3m - 1)=3m^{2} - m + 3mv - v= 3m^{2} + 3mv - m - v, which matches the original polynomial.

Key Concepts

prime polynomials

Before diving into the details of factoring by grouping, it's important to know what a prime polynomial is. A polynomial is considered prime when it cannot be factored into simpler polynomials using integer coefficients. This is similar to prime numbers in arithmetic. For example, the polynomial \(x^2 + 1\) is prime because there are no two polynomials with integer coefficients that multiply together to give \(x^2 + 1\). Recognizing prime polynomials will help you understand when you've fully factored a polynomial, such as the one in our exercise.

greatest common factor

To start factoring by grouping, you need to be familiar with the concept of the greatest common factor (GCF). The GCF of two or more terms is the highest degree of any polynomial that divides each of the terms without leaving a remainder. For instance, in the polynomial \(3m^2 + 3mv - m - v\), we initially group it into pairs: \( (3m^2 + 3mv) \) and \( (-m - v) \).
The GCF of \( 3m^2 \) and \( 3mv \) is \(3m\) because \(3m\) is the largest polynomial that divides both terms. Similarly, the GCF of \(-m\) and \(-v\) is \(-1\). Finding these GCFs helps break down the polynomial into simpler factors, making the factorization process easier and clearer.

polynomial factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. It's like breaking down a number into its prime factors. To factor our polynomial by grouping:
1. Group the terms: \((3m^2 + 3mv)\) and \((-m - v)\)
2. Extract the GCF from each group: \(3m(m + v)\) and \(-1(m + v)\)
3. Factor out the common binomial factor from both groups: \((m + v)(3m - 1)\)
This factorization process turns a complex polynomial into simpler, more manageable factors. Always remember to check your work by expanding the factored form to see if you get the original polynomial back. This ensures your factorization is correct and complete.