Question

Simplify and write the resulting polynomial in descending order of degree. $$-10 b^{3}+7 b^{4}+1+b^{3}-2 b^{2}-12+b^{4}$$

Short answer

Question: Simplify the following polynomial and write it in descending order of degree: $$-10b^3 + 7b^4 + b^3 + b^4 - 2b^2 + 1 - 12$$ Answer: $$8b^4 - 9b^3 - 2b^2 - 11$$

Step-by-step solution

Identify and Combine Like Terms

Identify terms with the same degree of b and add/subtract them: $$(-10b^3 + b^3) + (7b^4 + b^4) + (-2b^2) + (1 - 12)$$ Now combine the like terms: $$-9b^3 + 8b^4 - 2b^2 - 11$$

Write the Resulting Polynomial in Descending Order

To write the polynomial in descending order, rearrange the terms from the highest to the lowest degree of b: $$8b^4 - 9b^3 - 2b^2 - 11$$ Now we have the simplified polynomial in descending order of degree: $$8b^4 - 9b^3 - 2b^2 - 11$$

Key concepts

Combining Like Terms

When simplifying polynomials, one of the most crucial steps is to combine like terms. Like terms are two or more terms that have the exact same variable component raised to the same power. For instance, in the expression \( -10b^3 + b^3 \) both terms are like terms because they have the variable \( b \) raised to the third power. To combine them, you simply add or subtract their numerical coefficients. In our example, \( -10 + 1 = -9 \) so \( -10b^3 + b^3 \) simplifies to \( -9b^3 \).

Similarly, when \( 7b^4 \) and \( b^4 \) are combined, the resulting term is \( 8b^4 \), since \( 7 + 1 = 8 \). This combining step is essential for reducing the polynomial to its simplest form. It's also important to remember to include constants in this process; in the given exercise, \( 1 - 12 \) are also like terms, combining to \( -11 \). This simplification leads to a cleaner and more manageable expression.

Polynomial in Descending Order

Organizing polynomial terms in descending order of their degree helps in understanding the hierarchy of terms based on their power. The degree of a term is the sum of the exponents of the variables within that term. In descending order, you start with the term that has the highest degree and progress to the term with the lowest. For the given expression, \( 8b^4 \) is the term with the highest degree, followed by \( -9b^3 \) and then \( -2b^2 \).

The constant term, which in our case is \( -11 \) and has a degree of 0, always comes last. Writing the polynomial in this order not only streamlines the expression, making it follow a predictable pattern, but also prepares the expression for further algebraic operations, such as long division or factoring, where the leading term (the term with the highest degree) often plays a key role.

Degree of a Polynomial

The degree of a polynomial is the highest degree of its terms when the polynomial is expressed in its standard form, meaning it is fully expanded and like terms have been combined. The degree indicates the highest exponent of the variable within the polynomial. For example, in the simplified polynomial \( 8b^4 - 9b^3 - 2b^2 - 11 \), the highest exponent is 4, which appears in the term \( 8b^4 \). Therefore, the degree of this polynomial is 4.

Knowing the degree of a polynomial is not just a trivial fact; it gives insights into the polynomial's behavior. Graphically, the degree predicts the maximum number of turns the graph of the polynomial can have and it also determines the end behavior of the function as the variable approaches positive or negative infinity. In applied contexts, the degree can inform you of the maximum number of solutions, or roots, that the polynomial equation might have.