Problem 67 - Introductory Algebra for College Students

Question

Use the negative of the greatest common factor to factor completely. $$-16 t^{2}+64 t+80$$

Step-by-Step Solution

Answer

The factored form of the given quadratic is \(-16(t-5)(t+1)\).

Step 1: Identify the Greatest Common Factor (GCF)

The coefficients in the given quadratic are -16, 64, and 80. The greatest common factor among these is 16.

Step 2: Factor out the GCF

Now, factor out the greatest common factor (16) from the quadratic. However, since the instruction emphasized the use of the negative, we factor -16 out of each term, such that \(-16t^{2}+64t+80\) becomes \(-16(t^{2}-4t-5)\).

Step 3: Factor the Quadratic Expression Further

The quadratic \(t^{2}-4t-5\) can be factored further to (t-5)(t+1). Thus, our final answer becomes: \(-16(t-5)(t+1)\).

Key Concepts

Greatest Common Factor

The process of factoring any algebraic expression starts with identifying the Greatest Common Factor (GCF), which is the largest factor that divides all the terms. For numerical coefficients, this means finding the highest number that can evenly divide into each coefficient. In our exercise, the GCF for the coefficients -16, 64, and 80 is 16.

When a quadratic expression's leading coefficient is negative, as in \( -16t^2 + 64t + 80 \), it's often useful to factor out the negative of the GCF. This step simplifies the factoring process and helps to avoid mistakes with signs later on.

Mastering the skill of finding the GCF is crucial because it's the first step in simplifying expressions and solving equations in algebra.

• Always start by listing factors of each term.
• Determine the largest common factor.
• Remember to consider the sign of the GCF when it's factored out.

Algebraic Expression Simplification

Simplifying algebraic expressions is like tidying up a room — it makes everything neater and easier to understand. This process often involves factoring to reduce an expression to its simplest form.

In the given exercise, simplification started by factoring out the GCF. Following this, the inner quadratic \(t^2 - 4t - 5\) needed further attention. To simplify it, we look for two numbers that multiply to give the constant term, -5, and add up to the middle coefficient, -4, which are -5 and +1 in this case.

Remember, simplification might involve:

• Factoring out common factors.
• Finding pairs of numbers that help to break down complex quadratics.
• Checking for special factoring patterns like difference of squares.

Always strive for the most reduced form without changing the value of the initial expression.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, generally taking the form \( ax^2 + bx + c \), where \( a \), \( b \) and \( c \) are constants and \( a \eq 0\). The exercise we are working with had a quadratic expression in its midst after factoring out the GCF.

The key to factoring quadratic expressions lies in finding two binomials that will multiply to give the original quadratic. This can be challenging, so it's helpful to understand certain strategies and patterns, such as the use of the “AC method” or recognizing perfect square trinomials.

In our original problem, the quadratic expression inside the parentheses, \(t^2 - 4t - 5\), was successfully broken down into \(t - 5\) and \(t + 1\), making the expression cleaner and more useful for further mathematical processes like solving for zeros or sketching graphs.