Problem 67 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 35 x^{8}-2 x^{7}-x^{6} $$

Step-by-Step Solution

Answer

The expression factors as \(x^6(7x + 1)(5x - 1)\).

Step 1: Identify common factors

First, look at the expression \(35x^8 - 2x^7 - x^6\) and identify the greatest common factor (GCF) of all terms. Each term contains the variable \(x\), so the smallest power of \(x\) is \(x^6\). Thus, we factor out \(x^6\) from each term.

Step 2: Factor out the GCF

Factor \(x^6\) from each term in the expression:\[35x^8 - 2x^7 - x^6 = x^6(35x^2 - 2x - 1)\]

Step 3: Recognize quadratic trinomial

Now, focus on the quadratic trinomial \(35x^2 - 2x - 1\). We will try to factor this further if possible.

Step 4: Determine factors of quadratic trinomial

For the trinomial \(35x^2 - 2x - 1\), use the method of trial and error or structured factoring techniques to split the middle term. We are looking for factors of \(35 \times (-1) = -35\) that add up to the middle term coefficient, which is \(-2\). The numbers \(-7\) and \(5\) satisfy this because \(-7 \times 5 = -35\) and \(-7 + 5 = -2\).

Step 5: Restructure and factor the quadratic

Rewrite the quadratic expression as:\[35x^2 - 2x - 1 = 35x^2 - 7x + 5x - 1\]Now, factor by grouping. Group the first two terms and the last two terms:\[(35x^2 - 7x) + (5x - 1)\]From the first group, factor out \(7x\):\[7x(5x - 1)\]From the second group, factor out \(1\):\[+ 1(5x - 1)\]Combine using the common factor \((5x - 1)\):\[(7x + 1)(5x - 1)\]

Step 6: Final expression

Combine all the factors together with the original GCF:\[ 35x^8 - 2x^7 - x^6 = x^6(7x + 1)(5x - 1)\]

Key Concepts

Greatest Common Factor

The concept of the Greatest Common Factor (GCF) is essential when you're learning to factor polynomial expressions. The GCF is the largest factor that is common to all terms in the expression. Identifying the GCF is the first step in simplifying and factoring a polynomial.

When a polynomial includes terms with variable expressions like in the example, the GCF can include both numerical coefficients and variables. To find the GCF of a polynomial:

• Look at each term separately. Identify the numerical factor that is common to all coefficients.
• For the variable part, select the smallest power of the variable present in all terms.

For instance, in the polynomial expression \(35x^8 - 2x^7 - x^6\), start by noticing that each term contains the variable \(x\). The smallest power of \(x\) across all terms is \(x^6\), which becomes the GCF for the variable part of this expression.

The numerical coefficients (35, 2, and 1) do not share a common factor other than 1, so the GCF is purely based on the smallest power of the variable. By factoring out \(x^6\), the expression simplifies to \(x^6(35x^2 - 2x - 1)\). Recognizing and factoring out the GCF significantly simplifies the polynomial and sets the stage for further factorization.

Quadratic Trinomials

Quadratic trinomials are expressions of the form \(ax^2 + bx + c\), where the highest degree is 2. Factoring these types of expressions can often feel tricky at first, but with practice, it becomes more intuitive. The purpose is to rewrite the expression as a product of two or more simpler expressions.

For the quadratic trinomial to be factored, you're essentially looking for two numbers that multiply to give the product of \(a\) and \(c\) (the first and last coefficients) and at the same time add up to \(b\), the middle coefficient.

• Identify the product of the leading coefficient (\(a\)) and the constant term (\(c\)).
• Find two numbers that multiply to give this product and add up to the middle term \(b\).

In the case of \(35x^2 - 2x - 1\), the product is \(35 \times -1 = -35\). We search for two numbers that multiply to \(-35\) and add to the middle term, \(-2\). The numbers \(-7\) and \(5\) work perfectly because \(-7 + 5 = -2\).

Once these numbers are identified, you can rewrite the quadratic trinomial in a way that sets up factorization by grouping. This process of searching for the right pair of numbers is sometimes called trial and error, but with strategic thinking, it becomes a truth-seeking process to unlock the structure of the expression.

Factoring by Grouping

Factoring by grouping is a technique used when an expression cannot be immediately factored using basic methods. This approach is particularly useful in cases where polynomial expressions are more than three terms or when dealing with a four-term polynomial result from breaking down a trinomial.

The factoring by grouping method involves rewriting the middle term of a quadratic trinomial as two terms, based on the numbers we identified in the previous section, and then grouping terms strategically:

• Rewrite the expression, breaking the middle term into two terms that work with the identified numbers.
• Group terms with a common factor into pairs.
• Factor out the common factors from each group.

In our example of the expression \(35x^2 - 7x + 5x - 1\), group firstly \(35x^2 - 7x\) and secondly \(5x - 1\). From the first group, factor out the greatest common factor, which is \(7x\), to get \(7x(5x - 1)\). For the second group, extract a \(1\) to obtain \(1(5x - 1)\).

Finally, since both groups contain \(5x - 1\) as a factor, the whole expression organizes into a product: \((7x + 1)(5x - 1)\). This approach allows you to turn a complex expression into products of simpler expressions, which is especially useful for solving equations or simplifying expressions.