Question

Simplify and write the resulting polynomial in descending order of degree. $$4 y+15-y^{2}-16+5 y+6 y^{2}$$

Short answer

Question: Simplify the polynomial expression and write it in descending order of degree: \(-y^2 + 4y - 16 + 6y^2 + 5y\). Answer: The simplified polynomial expression is \(5y^2 + 9y - 16\).

Step-by-step solution

Combine like terms

To combine like terms, we will group the terms that have the same degree of the variable. Here, we need to combine the constant terms, the terms with the variable \(y\), and the terms with the variable \(y^2\). Constant terms: \(-16\). Terms with the variable \(y\): \(4y + 5y\). Terms with the variable \(y^2\): \(-y^2 + 6y^2\). Now, we will combine these like terms: Constant terms: \(-16\). Terms with the variable \(y\): \(4y + 5y = 9y\). Terms with the variable \(y^2\): \(-y^2 + 6y^2 = 5y^2\).

Rearrange the terms in descending order of degree

Now that we have combined the like terms, we will rearrange the resulting polynomial expression in descending order of the degree of the variable \(y\): $$5y^2+9y-16$$ The simplified polynomial expression in descending order of degree is: $$5y^2 + 9y - 16$$

Key concepts

Combining Like Terms

When simplifying polynomial expressions, one of the most fundamental steps is combining like terms. This process involves grouping and adding or subtracting terms that have the same variable raised to the same power. For example, separate terms with a solitary variable y, such as 4y and 5y, are considered like terms because they both contain the same variable y to the first degree.

Similarly, if you encounter constants or variables raised to the second power, these too can be combined with their respective like terms. In our exercise, 15 and -16 are constants, hence they can be combined. Furthermore, -y^2 and 6y^2 are also like terms as they both have the variable y squared. It's crucial to follow the proper arithmetic operations when combining, ensuring addition with addition and subtraction with subtraction, to obtain the correct coefficients for like terms.

Once grouped correctly, simply perform the addition or subtraction of the coefficients, and maintain the common variable as is. This method synthesizes the polynomial, making it easier and more straightforward for further manipulation or evaluation.

Descending Order of Degree

After combining like terms in a polynomial, the next step to simplify it is arranging the terms in descending order of degree, which means starting with the term with the highest exponent and progressing to the one with the lowest. The degree of a term is determined by the sum of the exponents of the variables it contains. For instance, in the term 5y^2, y is squared, which means this term has a degree of two.

Organizing terms this way provides a standardized form that's often easier to read and use for further operations like polynomial division or graphing. In our exercise, the term 5y^2 has the highest degree (two), followed by 9y (degree one), and lastly, the constant term -16 (degree zero). As a result, the polynomial arranged in descending order of degree is 5y^2 + 9y - 16. This hierarchy is crucial not only for the visibility of the polynomial's degree but also for understanding its potential behavior graphically.

Polynomial Expressions

Polynomial expressions are mathematical expressions consisting of variables, coefficients, and the operations of addition, subtraction, and non-negative integer exponents of variables. They're a substantial component in various fields of algebra and calculus. The properties of a polynomial expression include its degree, leading coefficient, and if it's arranged correctly, its standard form.

For beginners, becoming familiar with the structure of these expressions is essential. A single-term polynomial is known as a monomial (e.g., 3x^2), more than one term is called a binomial (e.g., 3x^2 + 4x) or a trinomial (e.g., 3x^2 + 4x + 7), and beyond these, it's generally referred to simply as a polynomial.

Understanding how to manipulate and simplify these expressions is key to working through algebraic equations effectively. In essence, polynomials must often be simplified through combining like terms and ordering by degrees to ensure they are in a form that's useable for problem-solving, providing clear insights into their attributes and potential solutions.