Problem 35 - Algebra Form and Function

Question

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ 15 x-4 x^{3}+12 x-5 x^{4}+9 x-6 x^{5} $$

Step-by-Step Solution

Answer

Answer: The polynomial in standard form is \(-6x^5 - 5x^4 - 4x^3 + 36x\). The coefficients are: \(a_{5} = -6, a_{4} = -5, a_{3} = -4, a_{1} = 36\). The degree of the polynomial is \(5\).

Step 1: Rearrange the terms in descending order of exponents

Rearrange the terms so the highest exponent comes first, followed by the next highest, and so on: $$ -6x^5 - 5x^4 - 4x^3 + 15x + 12x + 9x $$

Step 2: Combine like terms

After rearranging the terms, we'll now combine the terms with the same exponents: $$ -6x^5 - 5x^4 - 4x^3 + 36x $$

Step 3: Identify the values of the coefficients

The polynomial in standard form is: $$ -6x^5 - 5x^4 - 4x^3 + 36x $$ The coefficients are as follows: $$ a_{5} = -6, a_{4} = -5, a_{3} = -4, a_{1} = 36 $$

Step 4: Determine the degree of the polynomial

The degree of a polynomial is the highest exponent. In this case, the highest exponent is \(5\). Therefore, the degree of the polynomial is \(5\).

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a fundamental concept that helps us understand its characteristics. In simple terms, it refers to the highest power or exponent of the variable in the polynomial.

• For example, in the polynomial \(-6x^5 - 5x^4 - 4x^3 + 36x\), the highest exponent is \(5\).
• This means the degree of the polynomial is \(5\).

The degree offers insight into the behaviour of the polynomial as the variable increases or decreases. Polynomials of higher degrees generally have more turning points and zeroes. Knowing the degree is also crucial when performing operations such as addition, subtraction or multiplication of polynomials, as it helps predict the degree of the resulting polynomial. Always remember: the degree of a polynomial is determined by the term with the highest exponent.

Coefficients of a Polynomial

Coefficients are the numbers in front of the variable terms in a polynomial. They give each term a different 'weight' and play a critical role in shaping the polynomial's graph. In a polynomial written in standard form, such as \(-6x^5 - 5x^4 - 4x^3 + 36x\), the coefficients are the numbers before each power of \(x\):

• \(-6\) is the coefficient of \(x^5\)
• \(-5\) is the coefficient of \(x^4\)
• \(-4\) is the coefficient of \(x^3\)
• \(36\) is the coefficient of \(x\)

Each coefficient affects how steep or flat the polynomial curve will be and impacts the solutions of the polynomial equation. In our example, the remaining coefficient not mentioned is \(a_0\), which is \(0\) because the term with \(x^0\) or the constant term is absent. Coefficients are crucial in functions as they determine the precise form of the polynomial expression.

Combining Like Terms

Combining like terms is a crucial activity in simplifying polynomial expressions. It involves merging terms that have the same variable raised to the same power. Here's how it works:

• For the polynomial \(15x - 4x^3 + 12x - 5x^4 + 9x - 6x^5\), you start by grouping the terms with the same power.
• In our case, the like terms with \(x\) are \(15x\), \(12x\), and \(9x\).
• They combine to make \(36x\) since \(15 + 12 + 9 = 36\).

By combining like terms, the polynomial becomes simpler and easier to handle, \(-6x^5 - 5x^4 - 4x^3 + 36x\). This step is essential in rewriting polynomials in a more readable form, often necessary before performing further operations and analyses. It ensures accuracy and efficiency in solving polynomial-related problems.