Problem 13 - Algebra for College Students

Question

Add the given polynomials. \(-5 t-4\) and \(-6 t+9\)

Step-by-Step Solution

Answer

-11t + 5

Step 1: Identify Like Terms

First, identify like terms in both polynomials. In -5t - 4 and -6t + 9, the terms involving t are -5t and -6t, and the constant terms are -4 and 9.

Step 2: Combine the Coefficients of Like Terms

Add the coefficients of the like terms separately. For the terms involving t, add -5 and -6 to get -11. For the constant terms, add -4 and 9 to get 5.

Step 3: Write the Resulting Polynomial

Combine the results from Step 2 to form the new polynomial. Thus, the sum of the polynomials is -11t + 5.

Key Concepts

Like Terms

In the world of polynomials, "like terms" play a crucial role in simplifying and solving expressions. Like terms are terms that contain the same variable raised to the same power. For example, in the polynomial \(-5t-4\) and \(-6t+9\), the terms \(-5t\) and \(-6t\) are like terms because they both contain the variable \(t\) raised to the first power. Similarly, the numbers \(-4\) and \(9\) are considered like terms since they do not contain any variables and are constants.
Recognizing like terms is essential because they can be combined, significantly simplifying polynomial expressions.

• To identify like terms, ensure that terms have the same variable with the same exponent.
• In simpler terms, focus on matching both the letters and the exponents.
• Avoid combining terms with different variables or exponents, as they are not like terms.

Finding and grouping like terms is the first step towards adding polynomials, allowing for a much clearer expression.

Coefficients

When dealing with polynomials, coefficients are the numerical part of terms, and they tell us how much of a like term we have. Think of them as the number of times a particular term is counted. For example, in the expression \(-5t\), the coefficient is \(-5\), and it indicates the term \(t\) is being multiplied by \(-5\). Similarly, in \(-6t\), \(-6\) is the coefficient.
Coefficients help in combining like terms effectively:

• When adding polynomials, add the coefficients of like terms together. This gives the new coefficient for the combined like terms.
• In our scenario, adding the coefficients – \(-5\) and \(-6\) – results in a sum of \(-11\), combining the terms to \(-11t\).
• Remember, constants also act as coefficients when they stand alone in an expression.

Understanding coefficients helps in grasping how polynomials are structured and how they simplify in computations.

Polynomial Expressions

A polynomial is an expression made up of variables, coefficients, and exponents, connected by addition, subtraction, and multiplication. They form a foundational part of algebra, acting like building blocks for more complex operations. In our exercise, the polynomials \(-5t-4\) and \(-6t+9\) are expressions that need to be added.
A few characteristics of polynomial expressions are:

• They consist of one or more terms, where each term includes a variable and its coefficient.
• There are no variable exponents that are negative or fractional, ensuring a simplified and manageable structure.
• Operations on polynomial expressions often involve identifying and combining like terms.

By combining like terms and their coefficients, the sum of the two polynomials results in a new polynomial: \(-11t+5\). Understanding how to manipulate polynomial expressions helps in solving various algebraic equations effectively.