Question

Perform the indicated operation(s) and write the resulting polynomial in standard form.\(\left(15 x^{2}-6\right)-\left(-8 x^{3}-14 x^{2}-17\right)\)

Short answer

The resulting polynomial is \(8x^3 + 29x^2 +11\).

Step-by-step solution

Simplify the polynomials

The polynomials \(\left(15 x^{2}-6\right)\) and \(\left(-8 x^{3}-14 x^{2}-17\right)\) are already simplified.

Subtract the second polynomial from the first

Following the order of operations, start by subtracting each term of \(\left(-8 x^{3}-14 x^{2}-17\right)\) from \(\left(15 x^{2}-6\right)\): \((15x^2 - 6) - (-8x^3 -14x^2 -17) = 15x^2 - 6 + 8x^3 +14x^2 +17.\) Result will be \(8x^3 + 29x^2 +11\). Here, the parentheses around both polynomials have been removed, and the property of subtraction that a minus a negative equals a positive is used (subtracting a negative is the same as adding a positive).

Rearrange the polynomial in a standard form

The standard form for writing a polynomial is to put the terms with the highest degree first. Therefore, re-arrange \(8x^3 + 29x^2 +11\) to match this format. The polynomial already is in standard form.

Solution

The resulting polynomial in standard form after the operation is completed is \(8x^3 + 29x^2 +11\).

Key concepts

Standard Form of Polynomials

When working with polynomials, it's essential to understand the concept of writing them in standard form. The standard form of a polynomial means arranging the terms in order from the highest degree to the lowest degree. The degree of a term is determined by the exponent on its variable. For example, in the polynomial \(8x^3 + 29x^2 + 11\), each term should appear in order based on the exponent or power of \(x\). This organization helps in easily identifying the leading term the coefficient with the highest power, which in this case is \(8x^3\).

• Ensure each term is written in descending order of exponents.
• Identify the leading term which has the highest exponent.
• The constant term (without a variable) is typically placed last.

Knowing the standard form makes it easier to work with polynomials since operations like addition, subtraction, and multiplication will result in polynomials that align with this structure.

Subtraction of Polynomials

Subtracting polynomials might appear complicated at first, but it's a straightforward process once you break it down. To subtract one polynomial from another, each term of the polynomial being subtracted is changed from a positive to negative (or vice versa). This process is often simplified by removing parentheses and changing the signs. In our example, we have the polynomials \((15x^2 - 6)\) and \((-8x^3 -14x^2 -17)\). When subtracting these, you change the sign of each term in the second polynomial:

• \(15x^2 - 6 - (-8x^3 -14x^2 -17)\)
• Becomes \(15x^2 - 6 + 8x^3 + 14x^2 + 17\)
• Which results in \(8x^3 + 29x^2 + 11\)

The minus turns any negative into a positive, which can simplify calculations significantly. Subtraction can best be viewed as adding the opposite of the second polynomial.

Polynomial Simplification

Simplifying polynomials involves combining like terms to reduce the polynomial to its simplest form. This is especially useful after performing operations such as addition or subtraction. In the subtraction process shown earlier, once the signs have been changed and all terms are aligned, you simply add terms that are similar.

• Terms are alike if they have the same variables raised to the same power.
• In the example, \(15x^2 + 14x^2\) are combined because they have the same \(x^2\).
• The resulting polynomial is \(8x^3 + 29x^2 + 11\) after simplification.

By ensuring each type of term is combined, you end up with the polynomial in its most efficient expression. Simplifying makes it easier to identify the nature of the polynomial and perform further algebraic operations if needed.