Problem 53 - Elementary Algebra

Question

Factor out -1 and then factor further. $$ 32+4 x-x 2 $$

Step-by-Step Solution

Answer

Factor the expression as \(-1(x - 8)(x + 4)\).

Step 1: Identify Terms and Rearrange

The given expression is \( 32 + 4x - x^2 \). To make factoring easier, rearrange the terms in decreasing powers of \( x \): \( -x^2 + 4x + 32 \).

Step 2: Factor Out -1

Notice that the leading term is negative. Factor out \(-1\) from every term to make the leading coefficient positive: \(-1(x^2 - 4x - 32)\).

Step 3: Factor the Trinomial

Now, focus on the trinomial \( x^2 - 4x - 32 \). To factor this, look for two numbers that multiply to \(-32\) and add to \(-4\). The numbers \(-8\) and \(4\) work: \( (x - 8)(x + 4) \).

Step 4: Combine Factors

Substitute the factored expression back, considering the factored \(-1\): \(-1(x - 8)(x + 4)\).

Key Concepts

Rearranging Terms

The skill of rearranging terms in a polynomial expression can significantly simplify the process of factoring. When given a polynomial like \(32 + 4x - x^2\), it's often helpful to order terms by their degree.

In polynomials, the degree of a term is determined by the power of the variable. By rearranging the terms in decreasing order of degrees, you create a standard form. This ordering not only aligns with common mathematical practice but also makes further operations, like factoring, more straightforward.

For the expression \(32 + 4x - x^2\), we rearrange it to \(-x^2 + 4x + 32\). This new order highlights the fact that the expression is a quadratic trinomial, paving the way for effective factoring.

Factoring Out

Factoring out is a technique where a common factor is extracted from each term in a polynomial. This is particularly useful when the leading term of a polynomial is negative, as it can make further operations simpler or reveal other factoring opportunities.

In our example, with the polynomial \(-x^2 + 4x + 32\), the leading term is negative. By factoring out \(-1\), we transform the expression into a form with a positive leading term: \(-1(x^2 - 4x - 32)\).

This step not only simplifies the polynomial but also makes it easier to see and perform subsequent factoring operations. Factoring out common elements early can save time and reduce errors in complex algebraic manipulations.

• Look for common factors in the expression.
• Factor them out uniformly from all terms.

Trinomial Factoring

Trinomial factoring is the process of breaking down a quadratic trinomial into two binomials. This is especially useful when dealing with expressions of the form \(ax^2 + bx + c\).

The trinomial we need to factor is \(x^2 - 4x - 32\). To begin trinomial factoring, look for two numbers that multiply to give the constant term \(-32\) and add to give the middle coefficient, \(-4\).

These numbers are \(-8\) and \(4\). Thus, the trinomial can be factored into \((x - 8)(x + 4)\). This method relies on the relationships between multiplication and addition, employing them to split a quadratic into simpler, solvable parts.

Using trinomial factoring effectively turns a polynomial into a product of linear binomials, making it much more manageable for solving equations or further simplification.

• Identify the product-sum relationship.
• Factor into two binomials that reflect it.