Simplifying And Classifying Polynomials By Degree And Terms

Polynomial 1: (3x - 1/4)(4x + 8)

To simplify the first polynomial, $(3x - \frac{1}{4})(4x + 8)$, we'll use the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that each term in the first binomial is multiplied by each term in the second binomial. Let's break down the steps:

1. First: Multiply the first terms of each binomial: $(3x)(4x) = 12x^2$.
2. Outer: Multiply the outer terms of the expression: $(3x)(8) = 24x$.
3. Inner: Multiply the inner terms: $(-\frac{1}{4})(4x) = -x$.
4. Last: Multiply the last terms of each binomial: $(-\frac{1}{4})(8) = -2$.

Now, we combine these products:

$$12x^2 + 24x - x - 2$$

Next, we combine like terms, which are the terms with the same variable and exponent. In this case, $24x$ and $-x$ are like terms:

$$12x^2 + (24x - x) - 2$$

$$12x^2 + 23x - 2$$

Thus, the simplified form of the polynomial is $12x^2 + 23x - 2$. Now, let's classify this simplified polynomial. The degree of the polynomial is the highest power of the variable, which in this case is 2. A polynomial with a degree of 2 is called a quadratic polynomial. The number of terms in the simplified polynomial is 3: $12x^2$, $23x$, and $-2$. A polynomial with three terms is called a trinomial. Therefore, the polynomial $(3x - \frac{1}{4})(4x + 8)$ is a quadratic trinomial.

Classifying the polynomial as a quadratic trinomial provides valuable information about its behavior and characteristics. For example, we know that a quadratic polynomial will have a parabolic shape when graphed, and the trinomial form indicates that it is composed of three distinct terms, each contributing to the overall shape and position of the parabola. The process of simplification, using the distributive property and combining like terms, is a fundamental skill in algebra and is essential for further polynomial operations such as factoring and solving equations. Understanding these classifications helps in predicting the behavior of the polynomial and applying appropriate techniques for solving related problems.

Polynomial 2: (5x² + 7x) - 1/2(10x² - 4)

To simplify the second polynomial, $(5x^2 + 7x) - \frac{1}{2}(10x^2 - 4)$, we again use the distributive property and combine like terms. First, distribute the $-\frac{1}{2}$ across the terms inside the parentheses:

$$5x^2 + 7x - \frac{1}{2}(10x^2) - \frac{1}{2}(-4)$$

$$5x^2 + 7x - 5x^2 + 2$$

Now, we combine like terms. Here, $5x^2$ and $-5x^2$ are like terms, and they cancel each other out:

$$(5x^2 - 5x^2) + 7x + 2$$

$$0 + 7x + 2$$

Thus, the simplified form of the polynomial is $7x + 2$. To classify this polynomial, we look at its degree and the number of terms. The highest power of the variable $x$ is 1 (since $7x$ can be written as $7x^1$). A polynomial with a degree of 1 is called a linear polynomial. The number of terms in the simplified polynomial is 2: $7x$ and $2$. A polynomial with two terms is called a binomial. Therefore, the polynomial $(5x^2 + 7x) - \frac{1}{2}(10x^2 - 4)$ is a linear binomial.

Identifying this polynomial as a linear binomial provides key insights into its properties. Linear polynomials, when graphed, produce a straight line, and the binomial structure indicates that the line's equation consists of two terms—a term involving the variable x and a constant term. This classification is essential for understanding how the polynomial behaves graphically and algebraically. The process of distributing and combining like terms is a fundamental algebraic technique that simplifies the expression and reveals its underlying structure. By simplifying the polynomial, we can easily determine its degree and the number of terms, which are critical for further mathematical operations and applications.

Polynomial 3: 3(8 - x + 5x²)

The third polynomial given is $3(8 - x + 5x^2)$. To simplify this polynomial, we will apply the distributive property. This involves multiplying each term inside the parentheses by the constant outside, which in this case is 3. Let's perform the distribution:

$$3(8) - 3(x) + 3(5x^2)$$

$$24 - 3x + 15x^2$$

It's conventional to write polynomials in descending order of degree, so we rearrange the terms:

$$15x^2 - 3x + 24$$

Now that the polynomial is simplified, we can classify it based on its degree and the number of terms it contains. The degree of the polynomial is the highest power of the variable, which in this case is 2 (from the term $15x^2$). A polynomial with a degree of 2 is called a quadratic polynomial. The number of terms in the simplified polynomial is 3: $15x^2$, $-3x$, and $24$. A polynomial with three terms is called a trinomial. Therefore, the polynomial $3(8 - x + 5x^2)$ is a quadratic trinomial.

Classifying this polynomial as a quadratic trinomial provides important information about its characteristics and behavior. As a quadratic polynomial, its graph will be a parabola. The trinomial form indicates that the polynomial consists of three distinct terms, each contributing to the overall shape and position of the parabola. The process of distributing the constant across the terms within the parentheses and then rearranging the terms in descending order of degree is a crucial step in simplifying and understanding polynomials. This simplified form allows for easier analysis and manipulation of the polynomial in various mathematical contexts, such as graphing, solving equations, and performing calculus operations.

Conclusion

In conclusion, simplifying and classifying polynomials is a fundamental skill in algebra. By using the distributive property and combining like terms, we can reduce polynomials to their simplest form. Classifying polynomials by degree and the number of terms provides valuable insights into their behavior and characteristics. Polynomial $(3x - \frac{1}{4})(4x + 8)$ simplifies to a quadratic trinomial ($12x^2 + 23x - 2$), polynomial $(5x^2 + 7x) - \frac{1}{2}(10x^2 - 4)$ simplifies to a linear binomial ($7x + 2$), and polynomial $3(8 - x + 5x^2)$ simplifies to a quadratic trinomial ($15x^2 - 3x + 24$). These classifications help in predicting the behavior of the polynomials and applying appropriate techniques for solving related problems. This process is essential for various algebraic operations and applications in mathematics and beyond. Mastering these techniques provides a solid foundation for more advanced mathematical concepts and problem-solving scenarios. The ability to simplify and classify polynomials not only enhances algebraic proficiency but also deepens understanding of the structure and properties of mathematical expressions.