Problem 62 - Algebra A Combined Function

Question

Factor each polynomial by grouping. $$ x^{2 n}+6 x^{n}+10 x^{n}+60 $$

Step-by-Step Solution

Answer

The polynomial factors to \((x^n + 6)(x^n + 10)\).

Step 1: Identify the Polynomial Terms

The given polynomial is \(x^{2n} + 6x^n + 10x^n + 60\). Notice that it has four terms which suggests using factoring by grouping.

Step 2: Group the Terms

Group the polynomial into two pairs: \((x^{2n} + 6x^n)\) and \((10x^n + 60)\).

Step 3: Factor the First Group

In the first group \((x^{2n} + 6x^n)\), factor out the common term \(x^n\), which gives \(x^n(x^n + 6)\).

Step 4: Factor the Second Group

In the second group \((10x^n + 60)\), factor out the common term \(10\), resulting in \(10(x^n + 6)\).

Step 5: Combine the Factored Terms

Notice that both factored groups have a common factor \((x^n + 6)\). Factor this out to get \((x^n + 6)(x^n + 10)\).

Key Concepts

Polynomials

Polynomials play a vital role in algebra and mathematics as a whole. They are algebraic expressions that consist of several terms combined through addition, subtraction, or multiplication. Each term in a polynomial is composed of a coefficient and one or more variables raised to a power, typically expressed in the form of \(ax^n\). Here, \(a\) is the coefficient, \(x\) is the variable, and \(n\) is the non-negative integer exponent.
A common representation of a polynomial might look like \(4x^3 + 3x^2 - 2x + 5\).

• The degree of a polynomial is determined by the highest power of the variable in the expression.
• Polynomials can have any number of terms, including monomials (one term), binomials (two terms), and trinomials (three terms).

Understanding polynomials is crucial, as they appear in various branches of mathematics and help in modeling real-world situations.

Factoring by Grouping

Factoring by grouping is an effective method used to simplify polynomials, especially those with four or more terms. The essential idea is to rearrange and combine terms in a way that makes factoring possible.
In the process:

• First, we identify groups of terms that may have common factors.
• We then factor out the greatest common factor (GCF) from each group.
• After factoring, we observe if there is a common binomial factor in the groups.
• If there is a common binomial, we can factor it out to simplify the polynomial further.

This method is especially useful when polynomials are not easily simplifiable by other factoring techniques, like when they aren't standard binomials or trinomials.

Algebraic Expressions

Algebraic expressions are combinations of numbers and variables connected by operations such as addition, subtraction, multiplication, and division. They're the foundation of algebra, facilitating the description of mathematical relationships.
Key components of an algebraic expression include:

• **Variables:** Symbols like \(x, y, z\) that can represent unknown numbers.
• **Coefficients:** Numbers multiplied by the variables, e.g., in \(3x\), 3 is the coefficient.
• **Constants:** Numbers on their own within the expression, such as \(7\) in \(2x + 7\).

Algebraic expressions form the basis for equations and inequalities, from which we seek to find the values of the variables that make the expressions true. Simplifying these using factoring techniques like grouping allows for easier handling of complex polynomials.