Problem 21 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomial. \(12 t^{2}+t-13\)

Step-by-Step Solution

Answer

The polynomial factors to \((t - 1)(12t + 13)\).

Step 1: Identify Components

First, we need to identify the coefficients in the quadratic polynomial. The polynomial is given as \(12t^2 + t - 13\), which can be rewritten in the general form \(ax^2 + bx + c\) where \(a = 12\), \(b = 1\), and \(c = -13\).

Step 2: Calculate the Product of a and c

Calculate the product of \(a\) and \(c\). Here, \(a = 12\) and \(c = -13\), so their product is \(12 imes (-13) = -156\).

Step 3: Find Two Numbers that Multiply to -156 and Add to 1

We need two numbers that multiply to \(-156\) and add up to \(b = 1\). After testing factors, we find that \(13\) and \(-12\) satisfy these conditions because \(13 imes (-12) = -156\) and \(13 + (-12) = 1\).

Step 4: Rewrite the Middle Term

Rewrite the middle term \(t\) using the numbers found: \(12t^2 + 13t - 12t - 13\).

Step 5: Factor by Grouping

Group the terms into two pairs: \((12t^2 + 13t) + (-12t - 13)\). Factor each group separately. The first group gives \(t(12t + 13)\) and the second group gives \(-1(12t + 13)\).

Step 6: Factor out the Common Binomial

Both groups now have a common binomial factor \((12t + 13)\). Factor this out to get \((t - 1)(12t + 13)\).

Step 7: Verify the Solution

Verify the factorization by expanding \((t - 1)(12t + 13)\) back to the original expression. This should yield \(12t^2 + t - 13\), confirming that the solution is correct.

Key Concepts

Quadratic Polynomial

A quadratic polynomial is a type of polynomial that has the form of \[ ax^2 + bx + c \] where:

• a is the coefficient of the square term, making the polynomial quadratic.
• b is the coefficient of the linear term.
• c is the constant term, with no variable attached.

The polynomial you'll often encounter in exercises is characterized by the highest power of the variable, which is always two—the defining feature of a quadratic polynomial. This exercise involved the polynomial \(12t^2 + t - 13\), which means \(a = 12\), \(b = 1\), and \(c = -13\). Recognizing these components is pivotal, as they dictate the steps for factorization.

Coefficients in Polynomials

Understanding the coefficients in a polynomial is essential because they are the building blocks of the expression. For quadratic polynomials like the one in our exercise, knowing the coefficients helps determine how to proceed with operations like factorization.

• Coefficient of the quadratic term (\(a\)): Controls the parabola's shape when graphed.
• Coefficient of the linear term (\(b\)): Affects the parabola's position and opening direction.
• Constant term (\(c\)): Shifts the graph up or down without affecting its shape.

Grasping these coefficients can often simplify complex algebraic procedures, as seen when we determined that the product of \(a = 12\) and \(c = -13\) resulted in \(-156\), which is crucial for effective factorization.

Factor by Grouping

Factoring by grouping is a practical technique used to decompose polynomials, particularly when direct factoring is not straightforward. This technique divides the polynomial into manageable sections.Here's how it's applied with our quadratic polynomial \(12t^2 + t - 13\):

• First, rewrite the middle term using numbers identified from the previous steps (like \(13t - 12t\) in this case).
• Split the polynomial into two groups: \((12t^2 + 13t)\) and \((-12t - 13)\).
• Factor out the greatest common factor from each group. Here, we got \(t(12t + 13)\) and \(-1(12t + 13)\).

The final step is to factor out the common factor, \((12t + 13)\), resulting in the product \((t - 1)(12t + 13)\). This group factoring transforms a seemingly complex polynomial into a pair of simple expressions.

Verify Factorization

Verification is a critical step in factorization. It ensures that the factorized form can be multiplied back to retrieve the original polynomial.To verify, take the factorized form \((t - 1)(12t + 13)\) and expand it:

• Apply the distributive property (also known as "FOIL" method for binomials):
  • First: Multiply \(t \) by \(12t\): \(12t^2\)
  • Outer: Multiply \(t\) by \(13\): \(13t\)
  • Inner: Multiply \(-1\) by \(12t\): \(-12t\)
  • Last: Multiply \(-1\) by \(13\): \(-13\)
• Add all the terms obtained: \(12t^2 + 13t - 12t - 13\).
• Simplify to get back the original polynomial: \(12t^2 + t - 13\).

This final step reassures that the factorization is correct and helps solidify your understanding of the process.