Problem 148 - Elementary Algebra

Question

Factor completely. \(11 n^{3}-55 n^{2}+44 n\)

Step-by-Step Solution

Answer

11n(n-1)(n-4)

Step 1 - Identify the Greatest Common Factor (GCF)

Find the greatest common factor of the terms in the polynomial. The terms are: 11n³, 55n², and 44n. The GCF is 11n.

Step 2 - Factor Out the GCF

Divide each term by the GCF, 11n: \[ 11n( n^{2} - 5n + 4 ) \]

Step 3 - Factor the Quadratic Expression

Factor the quadratic expression inside the parentheses: \[ n^{2} - 5n + 4 \] Find two numbers whose product is 4 and whose sum is -5. These numbers are -1 and -4.

Step 4 - Write the Factored Form

Combine the factors: \[ 11n( n - 1 )( n - 4 ) \]

Key Concepts

greatest common factor

To factor a polynomial completely, start by finding the Greatest Common Factor (GCF) of all its terms. The GCF of terms is the highest factor that divides each term. In the polynomial [11n³, -55n², and 44n], we break them down to their prime factors along with variable powers:

• 11n³ = 11 * n * n * n
• -55n² = -1 * 11 * 5 * n * n
• 44n = 2 * 2 * 11 * n

Here, the GCF is 11n. We divide each term by 11n and factor it out. This simplifies the polynomial for further factoring steps. Breaking down complex expressions into their factors simplifies solving them.

factoring quadratic expressions

Next, we focus on the quadratic expression inside the parentheses. The expression ( n² - 5n + 4 ) is a standard quadratic trinomial. To factor this, find two numbers that multiply to the constant term (4) and add to the linear coefficient (-5). These numbers are -1 and -4 because:
( -1 * -4 = 4 ) and ( -1 + -4 = -5 ).
Therefore, the expression can be factored as: ( n - 1 )( n - 4 ).
Factoring into binomials makes complex expressions manageable and prepares them for solving equations or further simplification.

polynomial factoring

Polynomial factoring involves breaking down a polynomial into simpler 'factor' polynomials whose product returns the original polynomial. By factoring out the GCF first, we simplify the polynomial. Then, we look for patterns or use methods like grouping or applying the quadratic formula. For example:
( 11n³ - 55n² + 44n )
We start by finding the GCF, giving:( 11n( n² - 5n + 4 ) ).
Inside the parentheses is a quadratic expression, which we factor into binomials:
( 11n( n - 1 )( n - 4 ) ).
This step-by-step approach ensures that every polynomial is factored completely, making algebraic solutions easier.

elementary algebra

Elementary algebra involves understanding and manipulating expressions and equations at a fundamental level. Key skills include identifying the GCF, factoring quadratics, and simplifying complex expressions. To factor polynomials, follow these steps:

• Identify and factor out the GCF.
• Rewrite the polynomial in a simpler form inside parentheses.
• Factor any remaining quadratic expressions.

These techniques build a foundation for advanced algebra. By practicing polynomial factoring, you develop problem-solving skills and understand algebraic properties. Remember, breaking down complex expressions into simpler parts is at the heart of elementary algebra.