Problem 15 - Precalculus: Functions and Graphs

Question

Express as a polynomial. $$\left(3 x^{3}-4 x^{2}+x-7\right)+\left(x^{4}-2 x^{3}+3 x^{2}+5\right)$$

Step-by-Step Solution

Answer

The polynomial is \(x^4 + x^3 - x^2 + x - 2\).

Step 1: Distribute the Addition

The expression is composed of two polynomials that are to be added together. There are no parentheses indicating multiplication, so we simply distribute the addition across each term in the polynomials: \((3x^3 - 4x^2 + x - 7) + (x^4 - 2x^3 + 3x^2 + 5).\)

Step 2: Group Like Terms

Group the terms with similar degrees from each polynomial. This means grouping constants, \(x\), \(x^2\), \(x^3\), and \(x^4\) terms together:\[x^4 + (3x^3 - 2x^3) + (-4x^2 + 3x^2) + x + (-7 + 5).\]

Step 3: Simplify Each Group

Simplify the expression by combining the like terms you identified:- \(x^4\) stays as it is.- Combine the \(x^3\) terms: \(3x^3 - 2x^3 = x^3\).- Combine the \(x^2\) terms: \(-4x^2 + 3x^2 = -x^2\).- The linear term \(x\) stays the same as there’s no other matching term.- Combine the constants: \(-7 + 5 = -2\).So, the expression becomes:\[x^4 + x^3 - x^2 + x - 2.\]

Step 4: Write the Final Polynomial

After combining all like terms, the polynomial is: \(x^4 + x^3 - x^2 + x - 2\). Make sure to write the polynomial in descending order of the powers.

Key Concepts

Combining Like Terms

Combining like terms is an essential skill when working with polynomials. To master this concept, it is important to identify terms that share the same variable and exponent. For instance, in the expression

• \(3x^3 - 2x^3 + 5x^2\),

all terms that have the variable raised to the same power are considered like terms. These terms can be added or subtracted by simply addressing their coefficients. This makes it easier to simplify expressions. In our original exercise, we group terms such as those involving \(x^3\) and \(x^2\) from the two polynomials so that they can be combined. We add their coefficients, resulting in expressions like \(x^3\) (from \(3x^3 - 2x^3\)) and \(-x^2\) (from \(-4x^2 + 3x^2\)).

Remember to include all coefficients, even if they equal zero, to get an accurate final expression. This reduces complexity and makes it easier to see the structure of the polynomial.

Polynomial Expression

A polynomial expression is a mathematical expression involving a sum of powers of one or more variables multiplied by coefficients. Polynomials are used widely in mathematics to model various situations and processes. They can look intimidating at first but are actually quite structured. For example, a polynomial like

• \(3x^3 - 4x^2 + x - 7 + x^4 - 2x^3 + 3x^2 + 5\)

contains terms with different powers of \(x\). Understanding that a polynomial is simply a sum of terms helps to break down the process of simplification.

Each polynomial can be written in standard form, which means listing the terms in order from highest to lowest degree. For our consolidated polynomial after combining like terms, it results in

• \(x^4 + x^3 - x^2 + x - 2\).

With each operation, focus on aligning terms with similar exponents and carefully alter the coefficients so that the result accurately reflects the original expression.

Simplifying Expressions

Simplifying expressions is a key step in handling polynomial mathematics, where the main goal is to present the expression in its simplest form with no like terms left to combine. Starting with the given expression, the task involves grouping and simplifying by performing addition or subtraction of coefficients. Here’s a quick checklist for simplifying:

• Identify terms with the same variable and power.
• Combine these terms by adding or subtracting their coefficients.
• A quick tidy-up by rewriting the polynomial in order of descending powers if not already.

For example, from the exercise, combining

• \(3x^3 - 2x^3 \)
• and \(-4x^2 + 3x^2\)

results in \(x^3\) and \(-x^2\), leading to the simplified polynomial,
\(x^4 + x^3 - x^2 + x - 2\).

Remember, the more organized your approach in simplifying, such as writing neatly and grouping similar terms immediately, the easier the process becomes. This gives a clearer understanding and ensures accuracy in handling polynomial equations.