Problem 65 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ \begin{aligned} &-x^{2}+12 x-11\\\ &\text { (Factor out }-1 \text { first.) } \end{aligned} $$

Step-by-Step Solution

Answer

The factorization of \(-x^2 + 12x - 11\) is \(-1(x - 1)(x - 11)\).

Step 1: Identify the Greatest Common Factor (GCF)

The trinomial is \[-x^2 + 12x - 11\]. Observe that all terms can be divided by \(-1\). Therefore, the GCF of the trinomial is \(-1\). We factor this out first.

Step 2: Factor out the GCF

Factor out \(-1\) from the trinomial:\[-x^2 + 12x - 11 = -1(x^2 - 12x + 11)\]. This simplifies the trinomial inside the parentheses to \(x^2 - 12x + 11\).

Step 3: Look for Other Factorable Components

Now, we need to factor the trinomial \(x^2 - 12x + 11\). In this case, we are looking for two numbers that multiply to \(11\) and add to \(-12\).

Step 4: Determine Potential Factors

The factors of \(11\) are \(1\) and \(11\). To get a sum of \(-12\), we use \(-1\) and \(-11\), since \(-1 + (-11) = -12\).

Step 5: Factor the Trinomial Completely

Use the factors determined in the previous step to factor the trinomial: \[x^2 - 12x + 11 = (x - 1)(x - 11)\].

Step 6: Include the GCF in the Final Answer

Remember to include the \(-1\) factored out initially. Thus, the completely factored form of the original trinomial is: \[-1(x - 1)(x - 11)\].

Key Concepts

Greatest Common Factor

In algebra, the Greatest Common Factor, or GCF, plays a crucial role, especially when simplifying expressions like trinomials. The GCF is the largest number or expression that divides each term in the polynomial without leaving a remainder.
Finding the GCF is often the first step in the process of factoring a polynomial, as it reduces the complexity of the expression.When you have a polynomial such as \[-x^2 + 12x - 11,\]look at the coefficients and terms. Here, you notice that each term can be divided by \(-1\).

• This means the GCF is \(-1\),since it simplifies the equation while maintaining the integrity of the expression.
• Always factor out the GCF first, as it makes subsequent factoring steps easier.

Factoring Polynomials

Factoring polynomials is a key concept in algebra, enabling simplification and solving. Once you have factored out the GCF, you focus on breaking down the remaining expression into products of simpler polynomials.
For the trinomial \[x^2 - 12x + 11,\]you will seek to express it as a product of two binomials.

• Identify numbers that multiply to the constant term (\(11\)) and add to the middle coefficient (\(-12\)). In this case, the numbers are \(-1\)and \(-11\).
• This allows you to factor \[x^2 - 12x + 11\]into\[(x - 1)(x - 11).\]

Thus, fully factoring this trinomial gives a neat expression that can be used for further calculations or graphing.

Algebra Problem Solving

Algebra problem solving often requires breaking down complex expressions into manageable parts. Factoring is one technique used extensively, particularly when dealing with quadratic equations.
With the expression\[-x^2 + 12x - 11,\]you begin by factoring out the common factor of \(-1\),transforming it into a more straightforward trinomial:\[x^2 - 12x + 11.\]This simplification is crucial for problem-solving as it makes identifying further factors easier.

• Problem-solving involves deep analysis of coefficients and constant terms to predict their impact on the overall expression.
• Utilize strategies like seeking factor pairs for effective breakdown.
• Combining the initial GCF factor with the products of the simpler expression leads us to the completely factored expression,\[-1(x - 1)(x - 11).\]

These steps highlight the systematic approach required in algebra, turning potentially complicated expressions into simpler, factorable forms.