Problem 12 - College Algebra

Question

Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$\left(18 x^{4}-2 x^{3}-7 x+8\right)-\left(9 x^{4}-6 x^{3}-5 x+7\right)$$

Step-by-Step Solution

Answer

The resulting polynomial is \(9x^{4} + 4x^{3} -2x +1\) and its degree is 4.

Step 1: Identify Like Terms

The like terms in the polynomials are \(18x^{4}\) and \(9x^{4}\), \(-2x^{3}\) and \(-6x^{3}\), \(-7x\) and \(-5x\), and the constant terms \(8\) and \(7\).

Step 2: Subtract Like Terms

Subtract the like terms from each other to get: \((18x^{4} - 9x^{4}) + (-2x^{3} -(-6x^{3})) + (-7x -(-5x)) + (8 -7)\). This will simplify to: \(9x^{4} + 4x^{3} -2x +1\).

Step 3: Write in Standard Form and Identify Degree

The standard form of a polynomial starts with the term of the highest degree. The polynomial is already arranged in standard form. The degree of a polynomial is the highest power in the polynomial. Thus the degree of the polynomial \(9x^{4} + 4x^{3} -2x +1\) is 4 because that's the highest power of \(x\) in the polynomial.

Key Concepts

Standard Form of Polynomial

The standard form of a polynomial is a way of writing the expression where the terms are arranged in descending order of their degrees, from the highest degree to the lowest. This makes it easy to identify the polynomial's degree just by looking at it. In a polynomial, each term consists of a coefficient (numerical part) and a variable raised to a power (the degree).
For example, if you have a polynomial expression such as \(9x^{4} + 4x^{3} - 2x + 1\), starting with the highest power of \(x\), which is \(x^4\), signals that this polynomial is in standard form.

• Keep the terms in order of their decreasing degree.
• Ensure each term is written clearly with its respective sign.
• Standard form assists in performing operations like addition, subtraction, multiplication, or division of polynomials efficiently.

Once in standard form, polynomials can be compared more easily, making it simpler to observe properties like the leading coefficient and the degree.

Polynomial Degree

The degree of a polynomial is an important concept as it determines many of the polynomial's properties, including its overall shape if graphed. The degree is defined as the highest power of the variable in the polynomial.
In the polynomial \(9x^{4} + 4x^{3} -2x + 1\), the term with the highest degree is \(9x^{4}\), indicating the degree is 4. This matters because

• The degree of the polynomial indicates its overall complexity.
• A polynomial of degree 4, for instance, is known as a quartic polynomial, and its graph could have up to 4 - 1 = 3 turning points.
• The degree helps in predicting the number of roots the polynomial equation might have.

Understanding the degree can aid in learning about expected behavior of the polynomial as it tends towards positive or negative infinity on a graph.

Like Terms in Polynomials

Like terms are terms within a polynomial that have the exact same variables raised to the same powers. Identifying like terms is vital for simplifying and solving polynomial expressions.
In the expression \(\left(18x^{4} - 2x^{3} - 7x + 8\right) - \left(9x^{4} - 6x^{3} - 5x + 7\right)\), like terms include:

• \(18x^{4}\) and \(9x^{4}\) — both have the variable raised to the power of 4.
• \(-2x^{3}\) and \(-6x^{3}\) — both involve \(x^{3}\).
• \(-7x\) and \(-5x\) — each contains \(x\) with a power of 1.
• The constant terms \(8\) and \(7\), which have no variables associated.

When performing operations on polynomials, like terms are combined to simplify the expression. Subtract or add coefficients of like terms while keeping the common variable and power unchanged. This step results in the polynomial becoming more simplified and easier to work with in additional operations.