Problem 103 - Algebra for College Students

Question

Factor each trinomial and assume that all variables that appear as exponents represent positive integers. $$20 x^{2 n}+21 x^{n}-5$$

Step-by-Step Solution

Answer

The trinomial factors as \((5x^n - 1)(4x^n + 5)\).

Step 1: Identify the Trinomial Form

We need to factor the trinomial \( 20x^{2n} + 21x^n - 5 \). Notice that it is in the form \( ax^{2n} + bx^n + c \), where \( a = 20 \), \( b = 21 \), and \( c = -5 \).

Step 2: Multiply 'a' and 'c'

Calculate the product of the coefficients \( a \times c = 20 \times (-5) = -100 \). This product will be used to find two numbers that add up to \( b \) and multiply to this result.

Step 3: Find Two Numbers

To factor the trinomial, we need two numbers that multiply to \(-100\) and add to \(21\). These numbers are \(25\) and \(-4\), since \( 25 \times (-4) = -100 \) and \( 25 + (-4) = 21 \).

Step 4: Rewrite the Middle Term

Rewrite the original trinomial by splitting the middle term based on the two numbers found: \( 20x^{2n} + 25x^n - 4x^n - 5 \).

Step 5: Group and Factor by Grouping

Group the terms into pairs and factor each pair:- First pair: \( (20x^{2n} + 25x^n) = 5x^n(4x^n + 5) \)- Second pair: \( (-4x^n - 5) = -1(4x^n + 5) \)

Step 6: Factor Out Common Binomial Factor

Factor out the common binomial \( (4x^n + 5) \):\[ 5x^n(4x^n + 5) - 1(4x^n + 5) = (5x^n - 1)(4x^n + 5) \]

Step 7: Verify the Factorization

Expand the factors to ensure the factorization is correct:\( (5x^n - 1)(4x^n + 5) = 5x^n imes 4x^n + 5x^n imes 5 - 1 imes 4x^n - 1 imes 5 = 20x^{2n} + 25x^n - 4x^n - 5 \).This simplifies to \( 20x^{2n} + 21x^n - 5 \), confirming the factorization is correct.

Key Concepts

Polynomial Factorization

Polynomial factorization is an essential algebraic skill that involves breaking down a polynomial into simpler factors that, when multiplied together, produce the original polynomial. This is much like breaking a composite number into its prime factors. Consider the trinomial given: \(20x^{2n} + 21x^n - 5\). Here, our goal is to find two binomials that multiply together to yield this expression.

The factorization process often follows a systematic approach. First, identify the standard form of a trinomial, which typically resembles \(ax^2 + bx + c\). In our case:

• \(a = 20\)
• \(b = 21\)
• \(c = -5\)

Multiply \(a\) and \(c\) to find possible factors to help split the middle term. Search for two numbers that multiply to \(-100\) (product of 20 and -5) and add to 21. These numbers are 25 and -4, crucial for rewriting the trinomial.

• Split the middle term using these numbers.
• Group terms and factor each group.
• Identify a common factor in both groups to factor out.

Ultimately, factorization allows solving polynomial equations and simplifying expressions, playing a key role in algebraic problem-solving.

Exponents in Algebra

Exponents in algebra are used to denote repeated multiplication of a number or a variable. The expression \(x^n\) means \(x\) is multiplied by itself \(n\) times. They are crucial in algebra because they simplify the representation of large numbers and help in solving exponential equations.

In the trinomial \(20x^{2n} + 21x^n - 5\), exponents appear in terms of \(x^{2n}\) and \(x^n\). This indicates that these terms are related by powers of a variable. Here’s how exponents work in this context:

• \(x^{2n}\) is the square of \(x^n\), indicating it is raised to the power of two.
• Exponents can drastically change the value of expressions, depending on their positive, negative, or fractional nature.
• In polynomial factorization, considering the highest power is crucial, as it determines the degree of the polynomial and guides the factorization strategy.

Understanding these principles of exponents helps in grasping how polynomial expressions and factorizations work, especially in dealing with higher-degree polynomials.

Algebraic Expressions

Algebraic expressions are combinations of variables, coefficients, and constants entwined with arithmetic operations. They are the building blocks of algebra, allowing us to create equations, inequalities, and polynomial expressions.

In the expression \(20x^{2n} + 21x^n - 5\), each term contributes to the overall algebraic expression:

• \(20x^{2n}\) is a term where 20 is the coefficient, \(x\) is the variable, and \(2n\) is the exponent.
• \(21x^n\) is another term, displaying a different relationship between its coefficient, variable, and exponent.
• \(-5\) is a constant term that affects the polynomial's shape and intercept.

Algebraic expressions can be simple or complex, depending on the operations that involve these elements. Simplifying such expressions through factorizations not only helps in solving equations but also in evaluating mathematical models more efficiently. It's essential to practice manipulating these expressions to become adept in algebra.