Problem 70 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$5 x^{2}+42 x-27$$

Step-by-Step Solution

Answer

The polynomial is completely factored as \((5x - 3)(x + 9)\).

Step 1: Examine the Polynomial

The polynomial we are given is of the form \( ax^2 + bx + c \), where \( a = 5 \), \( b = 42 \), and \( c = -27 \). We need to factor this quadratic polynomial completely or determine that it is not factorable with integers.

Step 2: Check for a Common Factor

First, let's check if there is a common factor for all terms in the polynomial. The coefficients 5, 42, and -27 have no common factor other than 1, so we cannot factor any common numbers out of the polynomial.

Step 3: Identify Factorization Technique

Since there is no common factor, we can use factoring techniques such as the AC method to factor the quadratic. The AC method involves finding two numbers that multiply to \( a \times c = 5 \times (-27) = -135 \) and add up to \( b = 42 \).

Step 4: Use AC Method

We need two numbers, \( m \) and \( n \), such that \( m \times n = -135 \) and \( m + n = 42 \). After testing several factor pairs of -135, we find that \( m = 45 \) and \( n = -3 \) satisfy both conditions.

Step 5: Rewrite the Middle Term

Now, we can rewrite the polynomial as \( 5x^2 + 45x - 3x - 27 \). This helps us to factor by grouping.

Step 6: Factor by Grouping

Group the terms: \( (5x^2 + 45x) + (-3x - 27) \). Factor out the greatest common factor from each group: \( 5x(x + 9) - 3(x + 9) \).

Step 7: Complete the Factorization

Notice \( (x + 9) \) is a common factor, so factor it out: \( (5x - 3)(x + 9) \).

Step 8: Check the Factorization

Expand \( (5x - 3)(x + 9) \) to ensure it gives the original polynomial: \( 5x^2 + 45x - 3x - 27 = 5x^2 + 42x - 27 \), confirming the factorization is correct.

Key Concepts

Quadratic Polynomials

Quadratic polynomials are expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These polynomials represent parabolic curves when graphed on a coordinate plane. The highest power of the variable \( x \) in such expressions is 2, hence the term "quadratic."
The standard form \( ax^2 + bx + c \) is essential because it forms the basis for many operations in algebra, especially factoring.

Understanding the components of a quadratic polynomial helps in determining its roots, analyzing its graph, and solving equations. Each term has a specific function:

• The \( ax^2 \) term dictates the parabola's direction and width.
• The \( bx \) term affects the orientation and axis of symmetry.
• The constant \( c \) shows where the parabola crosses the y-axis.

Working with quadratic polynomials often involves factoring them to find solutions or simplify expressions. Factoring reveals the roots or zeroes, providing important insight into the polynomial's properties.

Factoring Techniques

Factoring techniques for polynomials involve breaking down a complicated expression into factors. These factors are expressions that, when multiplied together, give the original polynomial.

Factoring is useful because it simplifies calculations and reveals polynomial roots.

• Common Factor: Check for a number or variable that divides all terms. If present, factor it out first. This simplifies the polynomial and can make further factoring easier.
• Factoring Quadratics: Often involves finding two numbers that multiply to \( ac \) (when the polynomial is written as \( ax^2 + bx + c \)) and add to \( b \). This is vital for techniques like the AC method.
• Group Factoring: Break the expression into groups that share a common factor. Then, factor each group and find remaining common factors.

Understanding these techniques is essential for solving polynomial equations efficiently without the guesswork or trial and error.

AC Method

The AC method is a systematic approach to factoring quadratics by considering the product of the coefficient of \( x^2 \), \( a \), and the constant term, \( c \). The method is particularly useful when \( a eq 1 \) and there is no common factor.

The idea is to multiply \( a \) and \( c \) to get the product \( ac \), find two numbers whose product equals \( ac \) and whose sum equals the middle coefficient \( b \).

Steps for the AC method include:

• Compute \( ac \) by multiplying the leading coefficient \( a \) and the constant \( c \).
• Identify two numbers, \( m \) and \( n \), such that \( m \times n = ac \) and \( m + n = b \).
• Rewrite the middle term \( bx \) as \( mx + nx \) based on the values found.
• Factor by grouping, isolating the common factor in each part.
• Extract the common binomial factor to complete the factorization.

This structured approach helps avoid errors, outlining clear steps to confirm each factor is correct. If executed properly, it results in a well-factored polynomial showing its roots.