Problem 22 - Algebra for College Students

Question

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$20 x^{2}-11 x-3$$

Step-by-Step Solution

Answer

The polynomial factors to \((5x + 1)(4x - 3)\).

Step 1: Identify the Polynomial

The given polynomial is \( 20x^2 - 11x - 3 \). We need to factor it completely.

Step 2: Multiply Leading Coefficient and Constant

Multiply the leading coefficient (20) by the constant term (-3). This gives us \( 20 imes (-3) = -60 \).

Step 3: Find Factor Pair of -60

Find two integers whose product is -60 and whose sum is the middle coefficient, -11. The factor pair -15 and 4 can be used because \((-15) + 4 = -11\).

Step 4: Rewrite Middle Term Using Factor Pair

Rewrite the middle term -11x using the factor pair: \( -11x = -15x + 4x \). Thus, the expression becomes \( 20x^2 - 15x + 4x - 3 \).

Step 5: Group and Factor by Grouping

Group the terms: \( (20x^2 - 15x) + (4x - 3) \). Factor out the greatest common factor from each group: \( 5x(4x - 3) + 1(4x - 3) \).

Step 6: Factor Out Common Binomial Factor

Since \( 4x - 3 \) is common in both groups, factor it out: \( (5x + 1)(4x - 3) \).

Step 7: Verify Factors

Expand \( (5x + 1)(4x - 3) \) to check if it simplifies back to the original polynomial: \( 5x \cdot 4x - 5x \cdot 3 + 1 \cdot 4x - 3 = 20x^2 - 15x + 4x - 3 = 20x^2 - 11x - 3 \).

Key Concepts

Quadratic Polynomials

A quadratic polynomial is a polynomial of degree two. This means the highest power of the variable you see is squared. It's usually expressed in the general form:

• \( ax^2 + bx + c \)

Where:

• \( a \), \( b \), and \( c \) are constants
• \( x \) is the variable

Quadratic polynomials can have three distinct forms based on their graph:

• Two real and distinct roots
• One real and repeated root
• No real roots (complex roots)

Every quadratic polynomial can be tested and either factored using integer factorization methods or it can be described as not factorable using integers. Factoring helps simplify expressions and solve equations by breaking them into their simpler components.

Integer Factorization

Factorization is the process of breaking down numbers into their integral components or factors. When applied to polynomials, integer factorization involves expressing the polynomial as a product of simpler polynomials with integer coefficients. For quadratic polynomials, this often comes down to finding two numbers:

• Whose product equals the product of the leading coefficient \( a \) and the constant \( c \)
• Whose sum equals the middle coefficient \( b \)

This strategy relies on identifying numbers that satisfy these conditions and using them to rewrite the polynomial in a factorizable form. For example, in the problem of factoring \( 20x^2 - 11x - 3 \), we seek integers that multiply to \(-60\) and sum to \(-11\). Once identified, these numbers can be used to split the middle term and group terms into factorable pairs. If no such pair exists, the polynomial may be upgraded with other methods, or stated as non-factorable over the integers.

Complete Factorization

Complete factorization is the process of breaking down a polynomial into a product of prime polynomials—meaning it's factored fully, and all factors are irreducible over the integers. For many quadratic polynomials, complete factorization uses factoring by grouping or trial and error with factor pairs. Here's a simplified approach:

• Identify the leading coefficient and constant term
• Find factor pairs that meet the necessary sum and product conditions
• Rewrite the polynomial to include the new factor pair
• Group terms to make common factors evident
• Extract the common factor to achieve the factored form

This ultimately advances students' understanding of how algebraic expressions can be simplified. In the example of \( 20x^2 - 11x - 3 \), complete factorization resulted in \( (5x + 1)(4x - 3) \). To verify, expanding these factors should yield the original expression, confirming it has been correctly factored.