Problem 29 - Elementary Algebra

Question

For the following problems, use the grouping method to factor the polynomials. Some polynomials may not. be factorable using the grouping method. $$ 5 m^{10} n^{17} p^{3}-m^{6} n^{7} p^{4}-40 m^{4} n^{10} q t^{2}+8 p q t^{2} $$

Step-by-Step Solution

Answer

Question: Factor the given polynomial using the grouping method: $$5 m^{10} n^{17} p^{3}-m^{6} n^{7} p^{4}-40 m^{4} n^{10} q t^{2}+8 p q t^{2}$$ Answer: The factored polynomial is $$m^{6} n^{7} p^{3}(5 m^{4} n^{10} - p) + 8 q t^{2}(-5 m^{4} n^{10} + p)$$

Step 1: Group the terms in pairs

We have the polynomial: $$ 5 m^{10} n^{17} p^{3}-m^{6} n^{7} p^{4}-40 m^{4} n^{10} q t^{2}+8 p q t^{2} $$ Group the terms in pairs: $$ (5 m^{10} n^{17} p^{3}-m^{6} n^{7} p^{4}) + (-40 m^{4} n^{10} q t^{2}+8 p q t^{2}) $$

Step 2: Find the GCF of each pair

Now, we will find the GCF of each pair of terms. For the first pair $$5 m^{10} n^{17} p^{3}-m^{6} n^{7} p^{4}$$, the GCF is $$m^{6} n^{7} p^{3}$$. For the second pair $$-40 m^{4} n^{10} q t^{2}+8 p q t^{2}$$, the GCF is $$8 q t^{2}$$.

Step 3: Factor the polynomial

Now we will factor the polynomial by combining the GCFs. $$ (5 m^{10} n^{17} p^{3}-m^{6} n^{7} p^{4}) + (-40 m^{4} n^{10} q t^{2}+8 p q t^{2}) = m^{6} n^{7} p^{3}(5 m^{4} n^{10} - p) + 8 q t^{2}(-5 m^{4} n^{10} + p) $$ We are left with the factored polynomial: $$ m^{6} n^{7} p^{3}(5 m^{4} n^{10} - p) + 8 q t^{2}(-5 m^{4} n^{10} + p) $$

Key Concepts

Grouping Method

The grouping method is a powerful tool to factor complex polynomials. It is particularly useful for polynomials with several terms. Instead of factoring the entire polynomial at once, we break it into smaller, more manageable pairs of terms. This allows us to find common factors more easily and makes the factoring process simpler.

To apply the grouping method, we start by grouping together terms that might share common factors. For example, in the polynomial \( 5m^{10}n^{17}p^{3}-m^{6}n^{7}p^{4}-40m^{4}n^{10}qt^{2}+8pqt^{2} \), we pair the terms into two groups: \((5m^{10}n^{17}p^{3}-m^{6}n^{7}p^{4})\) and \((-40m^{4}n^{10}qt^{2}+8pqt^{2})\).

Next, we look for the greatest common factor (GCF) in each of these pairs and factor them out. This step simplifies each group, and eventually, the entire polynomial. The key is identifying pairs that allow you to extract similar expressions, making it easier to factor completely in subsequent steps.

GCF (Greatest Common Factor)

The GCF is the largest expression that can evenly divide terms in a polynomial. Finding the GCF is a crucial step in the factoring process, especially when using the grouping method. It helps simplify expressions by reducing complex terms into simpler ones.

To determine the GCF of each group of terms, identify common elements such as coefficients, variables, and their powers. For instance, in the group \(5m^{10}n^{17}p^{3}-m^{6}n^{7}p^{4}\), the GCF is \(m^{6}n^{7}p^{3}\) because it is the highest power of each variable that appears in all terms.

Similarly, for the pair \(-40m^{4}n^{10}qt^{2}+8pqt^{2}\), we find that the GCF is \(8qt^{2}\). By factoring out these GCFs, the polynomial becomes more workable, as we separate out common factors from each group of terms.

• Look for the highest power of each variable present in all terms.
• Identify the largest number that can divide all coefficients of the terms.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and operation symbols. In the context of factoring, we're often dealing with multiple algebraic expressions combined into a single polynomial. This makes understanding and manipulating these expressions essential for successful factoring.

The expressions are made up of terms, where each term is a product of coefficients (numbers) and variables raised to powers. In our problem, the terms in the polynomial, such as \(5m^{10}n^{17}p^{3}\), \(-m^{6}n^{7}p^{4}\), \(-40m^{4}n^{10}qt^{2}\), and \(8pqt^{2}\), are rearranged to group them effectively for easier factoring.

By recognizing the structure of these algebraic expressions, and using methods like the grouping technique, we can systematically reduce complex problems into simpler parts. This approach not only makes solving polynomials easier but also helps build a stronger understanding of how algebraic expressions work and interact.

• Understand the role of each term in the expression.
• Practice manipulating algebraic expressions to build familiarity.
• Use factoring to simplify and solve problems.