Problem 60 - Elementary and Intermediate Algebra

Question

Factor out \(-1\) from each polynomial. $$ -t^{2}-9 t+1 $$

Step-by-Step Solution

Answer

Factor out \(-1\): \[-1(t^2 + 9t - 1)\].

Step 1: Identifying the Leading Term

Look at the polynomial \[-t^2 - 9t + 1\].The leading term here is \(-t^2\), which is a negative term, making it a candidate to factor out \(-1\) from the entire polynomial.

Step 2: Factoring Out \(-1\)

To factor out \(-1\), multiply each term in the polynomial by \(-1\) and reverse their signs:\[-1(t^2 + 9t - 1)\].Here, the \(-t^2\) becomes \(t^2\), \(-9t\) becomes \(9t\), and \(+1\) becomes \(-1\).

Step 3: Verification

Multiply back each term inside the parentheses by \(-1\) to verify:\[-1(t^2 + 9t - 1) = -t^2 - 9t + 1\].The expression matches the original polynomial, confirming the correctness of factoring \(-1\) out.

Key Concepts

Negative Leading Term

In polynomial expressions, the leading term is the term with the highest power. It is crucial to identify this term as it often determines how we approach factoring. In our exercise, the polynomial is given as \(-t^2 - 9t + 1\). Here, \(-t^2\) is the leading term, and it's negative. Having a negative leading term can alter the way we manipulate the expression, particularly in factoring techniques.

When the leading term is negative, it can be beneficial to factor \(-1\) out from the entire polynomial. This process simplifies the expression and avoids negative signs that can complicate further algebraic operations. It makes the polynomial easier to handle, especially if it is part of a larger equation.

Factoring Techniques

Factoring is breaking down a complex expression into simpler components that are multiplied together to give the original expression. In this case, factoring \(-1\) involves switching every term's sign in the polynomial. Let's go through this:

• The original expression is \(-t^2 - 9t + 1\).
• When we factor out \(-1\), each term within the polynomial changes sign. Such as:
• \(-t^2\) becomes \(t^2\)
• \(-9t\) becomes \(9t\)
• \(+1\) becomes \(-1\)

After factoring, the polynomial becomes \(-1(t^2 + 9t - 1)\).This method significantly clarifies the polynomial, making further algebraic operations more straightforward.

Verification of Factoring

Verification in factoring ensures that the factor we applied correctly simplifies or modifies the polynomial. It's a very important step as it confirms that no mistakes were made during the process of factoring. Let’s verify the step we completed:

After factoring out \(-1\), the expression inside the parentheses was \(t^2 + 9t - 1\).

By distributing the \(-1\) back through the parenthesis:

• It multiplies to give \(-t^2\)
• and \(-9t\)
• and \(+1\)

The result is \(-t^2 - 9t + 1\), which exactly matches the original polynomial. This shows that the \(-1\) was factored correctly.Verification acts as a double-check. It provides assurance that the process has been executed correctly and that the simplified version truly reflects the original expression.