Problem 14 - Algebra and Trigonometry

Question

In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$ \left(8 x^{2}+7 x-5\right)-\left(3 x^{2}-4 x\right)-\left(-6 x^{3}-5 x^{2}+3\right) $$

Step-by-Step Solution

Answer

The resulting polynomial in standard form is \(6x^{3} + 10x^{2} + 11x - 8\) and its degree is 3.

Step 1: Simplify the terms in brackets

Start by simplifying each bracket. We have three brackets here: 1. \(8x^{2}+7x-5\) 2. \(3x^{2}-4x\)3. \(-6x^{3}-5x^{2}+3\) These can be rewritten as is, since they are already simplified.

Step 2: Perform the operations across the brackets

Now, perform the operations. Subtract the second and third brackets from the first. First convert subtraction into addition by multiplying the second and third polynomials by -1 before adding to avoid confusion. So, \(8x^{2}+7x-5 - (3x^{2}-4x) - (-6x^{3}-5x^{2}+3)\) This becomes: \(8x^{2} + 7x - 5 - 3x^{2} + 4x + 6x^{3} + 5x^{2} - 3\)

Step 3: Combine like terms

Combine the like terms. In other words, add coefficients for equal powers of \(x\). This gives: \(6x^{3} + (8x^{2} - 3x^{2} + 5x^{2}) + (7x + 4x) - (5 + 3) = 6x^{3} + 10x^{2} + 11x - 8\)

Step 4: Determine the degree of the polynomial

The degree of a polynomial is the greatest exponent of that polynomial. Here, the largest exponent is 3 hence the degree of the polynomial is 3.

Key Concepts

Standard Form of Polynomial

A polynomial is an expression constructed from variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Writing a polynomial in its standard form means arranging terms in descending order of their exponents.

For example, consider the polynomial expression from the exercise: it initially appears as multiple polynomials to be subtracted:

• \(8x^{2}+7x-5\)
• \(3x^{2}-4x\)
• \(-6x^{3}-5x^{2}+3\)

To begin, simplify terms within each polynomial first. Then, by performing the indicated subtraction operations, you transform it step-by-step into a single polynomial expression.

The resultant polynomial is then written in standard form as \(6x^{3} + 10x^{2} + 11x - 8\), where each term is ordered from the highest to the lowest degree.

Combining Like Terms

Combining like terms is a fundamental part of simplifying polynomial expressions. Similar terms have the same variable raised to the same power and can be summed by adding their coefficients.

In our solved exercise, once separate terms are expanded, we look for like terms across the expression \(6x^{3} + 10x^{2} + 11x - 8\).

• Terms with \(x^{3}\) (the highest degree term): There was only \(6x^{3}\), so no other group operations occur here.
• Terms with \(x^{2}\): Combine \(8x^{2}\), \(-3x^{2}\), and \(5x^{2}\) to get \(10x^{2}\).
• Terms with \(x\): Combine \(7x\) and \(4x\) to get \(11x\).
• Constant terms: Combine \(-5\) and \(-3\) to get \(-8\).

After combining these terms, the expression becomes a simplified form: \(6x^{3} + 10x^{2} + 11x - 8\).

Degree of Polynomial

The degree of a polynomial is crucial because it provides information about the behavior of the polynomial function, especially at large values of the variable. By definition, the degree is the highest power of the variable within the polynomial.

In the expression \(6x^{3} + 10x^{2} + 11x - 8\), the highest degree term is \(6x^{3}\), which makes the degree of this polynomial 3.

• The degree of a single term is simply the power of the variable in that term.
• The degree of a constant (a term with no variable) is 0 because constants do not vary.
• A higher degree generally suggests more complexity in a graph such as increased turns or curvature.

Recognizing the degree helps understand not just the term structure but also affects solutions and graphical behaviors.