Problem 60 - Elementary Algebra

Question

The height of a baseball \(t\) seconds after it is hit is given by the expression \(-16 t^{2}+80 t+4\). Factor the greatest common factor from the polynomial.

Step-by-Step Solution

Answer

The GCF of \(-16 t^{2}+80 t+4\) is 4, so the factored form is \4(-4 t^{2} + 20 t + 1)\.

Step 1: Identify the Greatest Common Factor (GCF)

Look at the coefficients and variables in each term of the polynomial \(-16 t^{2}+80 t+4\). Identify the largest factor that is common to all terms.

Step 2: Factor out the GCF

Once the GCF is identified, rewrite each term in the polynomial as a product of the GCF and the remaining factor.

Step 3: Simplify the Expression

Combine the factors to express the polynomial in its factored form.

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, often abbreviated as GCF, is an essential concept when dealing with polynomials. It is the largest factor that can evenly divide all terms in a given polynomial. For example, in the polynomial \(-16t^{2} + 80t + 4\), identifying the GCF helps simplify the polynomial. To find the GCF:
- List the factors of each coefficient (number in front of the variable).
- For \(-16t^{2}\), the factors are -1, -2, -4, -8, -16 and their positive counterparts.
- For \(+80t\), the factors are 1, 2, 4, 5, 8, 10, 16, 20, 40, 80.
- For \(+4\), the factors are 1, 2, 4.
- Identify the largest factor common to all terms, which in this example is 4.

Polynomial Expressions

A polynomial is an algebraic expression with more than one term. It comprises variables, coefficients, and exponents combined using addition, subtraction, and multiplication. For instance, \(-16t^{2} + 80t + 4\) is a polynomial expression. Here:
- \(-16t^{2}\) is the quadratic term.
- \(+80t\) is the linear term.
- \(+4\) is the constant term.
Understanding polynomial expressions is vital as it allows us to manipulate and solve various algebraic problems. When factoring polynomials, the goal is to break down the expression into simpler factors that, when multiplied, give back the original polynomial.

Factoring Techniques

Factoring is breaking down a polynomial into simpler expressions or factors that, when multiplied together, yield the original polynomial. Several techniques can be employed to factor polynomials effectively:
- **Factoring out the GCF:** As seen in the example \(-16t^{2} + 80t + 4\), start by identifying and factoring out the GCF, which simplifies the polynomial.
- **Using Special Formulas:** Recognize and apply formulas like Difference of Squares, Perfect Square Trinomials, or the Sum and Difference of Cubes.
- **Grouping:** For polynomials with four or more terms, grouping can help create factorable pairs.
Practicing these techniques enables more efficient problem-solving and simplifies complex polynomial expressions for further mathematical operations.