Problem 20 - College Algebra with Modeling and Visualization

Question

Add the polynomials. $$\left(5 y^{2}+y^{3}\right)+\left(12 y^{2}-5 y^{3}\right)$$

Step-by-Step Solution

Answer

The sum is \(-4y^3 + 17y^2\).

Step 1: Arrange Like Terms

First, identify and pair up like terms from both polynomials. The given polynomials are \(5y^2 + y^3\) and \(12y^2 - 5y^3\). "Like terms" have the same variable raised to the same power. In this case, group the \(y^3\) terms and the \(y^2\) terms.

Step 2: Add \(y^3\) Terms

Now, add the coefficients of the \(y^3\) terms from both polynomials: \(y^3 + (-5y^3)\). The result is \(-4y^3\).

Step 3: Add \(y^2\) Terms

Next, add the coefficients of the \(y^2\) terms from both polynomials: \(5y^2 + 12y^2\). The result is \(17y^2\).

Step 4: Write the Result

Combine the results from Step 2 and Step 3 to form the simplified resultant polynomial. Thus, the sum of the polynomials is \(-4y^3 + 17y^2\).

Key Concepts

Like Terms in Polynomials

When adding or subtracting polynomials, identifying like terms plays a crucial role. Like terms are terms that have identical variables raised to the same power. For instance, in the expression \(5y^2 + y^3\) and \(12y^2 - 5y^3\), the like terms are \(y^2\) and \(y^3\) terms, which should be processed separately.

Like terms allow you to simplify expressions by adding their coefficients. Mixing unlike terms, such as \(y^2\) and \(y^3\), would result in an incorrect expression, as they represent entirely different "amounts" or concepts in mathematics. Therefore, always ensure that you are grouping and calculating like terms, paving the way for simplified mathematical equations.

Coefficients in Polynomial Addition

Coefficients are the numerical parts of terms with variables. They are vital in polynomial operations as they indicate the quantity of the variable term present. Take the polynomial expression \(5y^2\); here, 5 is the coefficient.

In polynomial addition, coefficients of like terms are added together. For example, in the expression \(5y^2 + 12y^2\), we add the coefficients 5 and 12, resulting in \(17y^2\). This addition only applies to like terms because only such terms, having the same base and exponent, can have their coefficients combined to give a sum. Variables remain the same, as exhibited in the result of the polynomial addition seen in the original exercise.

Understanding Polynomials

Polynomials are expressions consisting of multiple terms, combined through addition or subtraction, and each term may include variables, coefficients, and exponents. For example, in the polynomial \(5y^2 + y^3\), there are two distinct terms: \(5y^2\) and \(y^3\).

These expressions are constructed to represent various algebraic forms and can vary in complexity from simple binomials like \(a + b\) to more intricate forms containing several terms and variable powers. Polynomials are essential in describing and solving algebraic problems in mathematics. Their operations, including addition, subtraction, and multiplication, are foundational skills in algebra.

Role of Exponents

Exponents are numerical values that indicate how many times a base is multiplied by itself. In polynomials, they determine the degree of each term. In the expression \(y^3\), the 3 is the exponent and signifies that \(y\) is multiplied by itself three times.

Exponents provide the power in terms of forming distinct like terms. Terms in a polynomial, which vary by exponents, cannot be added directly. They ensure each term's distinct presence and influence how terms are grouped, as seen when adding \(y^3\) terms separately from \(y^2\) terms in polynomial addition. The role of exponents is pivotal in preserving each term's unique contribution in a polynomial expression.