Problem 68 - Introductory Algebra for College Students

Question

Use the negative of the greatest common factor to factor completely. $$-16 t^{2}+80 t+96$$

Step-by-Step Solution

Answer

The completely factorized form of the polynomial is \(-16(t−6)(t+1)\).

Step 1: Find the GCF

Identify the greatest common factor (GCF) among the coefficients -16, 80, and 96. The GCF is 16.

Step 2: Apply negative of GCF

Use the negative of GCF as a factor. So, apply -16 as a factor and divide each term of the polynomial by -16. This gives us: \(-16(t^{2} - 5t - 6)\).

Step 3: Factorize the polynomial

Now factorize the polynomial part \(t^{2} - 5t - 6\). Find two numbers when multiplied give -6 and when added give -5. The numbers are -6 and 1. So, this can be written as \((t−6)(t+1)\).

Step 4: Write the final expression

Multiplying -16 with the factorized polynomial, the final expression will be \(-16(t−6)(t+1)\). So, the polynomial is completely factorized.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is an essential concept in simplifying mathematical expressions. To find the GCF of a set of numbers, you need to determine the largest integer that divides each of the numbers without leaving a remainder. In the context of factoring polynomials, the GCF is the largest factor common to all terms within the polynomial.When dealing with the polynomial \(-16t^2 + 80t + 96\),we first need to identify the GCF among the coefficients -16, 80, and 96.

• -16 can be divided by 16
• 80 can be divided by 16
• 96 can also be divided by 16

Thus, the GCF is 16. However, since the exercise asks to use the negative of the GCF, we utilize -16 instead. Applying the negative GCF helps in setting the polynomial up for further simplification, especially when factoring trinomials.

Factoring Trinomials

Factoring trinomials involves breaking down a polynomial with three terms (a trinomial) into simpler binomial factors. Trinomials usually have the form \(ax^2 + bx + c\). To factor these, you want to find two numbers that - multiply to give the product of the coefficient of \(x^2\) term (the leading coefficient) \(a\)and the constant \(c\)- add or subtract to give the middle coefficient \(b\).In our example:\(t^2 - 5t - 6\),we look for two numbers that multiply to -6 and add to -5. The numbers -6 and 1 fit this requirement, so we rewrite the trinomial as\((t-6)(t+1)\).This process converts a complex expression into simpler, more understandable parts.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like \(t\)in our case), and operations.Understanding how to manipulate these expressions is crucial for solving various algebra problems. In this exercise, \(-16(t^2 - 5t - 6)\)serves as the algebraic expression that was simplified by factoring.Each term within an algebraic expression can be composed of:

• constants (like numbers)
• variables (such as \(t\))
• coefficients (numbers in front of variables, like -16 in front of \(t^2\))
• operators (addition, subtraction, multiplication)

By recognizing patterns and applying the correct factoring methods, we convert a complex algebraic expression into a product of simpler binomial expressions like \(-16(t-6)(t+1)\).This approach helps solve equations and gain deeper insights into the behavior of mathematical functions.