Problem 21 - Algebra for College Students

Question

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$12 x^{2}-x-6$$

Step-by-Step Solution

Answer

The polynomial factors to \((4x - 3)(3x + 2)\).

Step 1: Find the Product and Sum

For the quadratic polynomial \(12x^2 - x - 6\), we want to factor it into the form \((ax + b)(cx + d)\). To do this, first identify the product of the coefficients of \(x^2\) and the constant term: \(12 \times (-6) = -72\). Next, identify the middle coefficient as the sum: \(-1\). We need to find two numbers that multiply to \(-72\) and add to \(-1\).

Step 2: Identify the Pair of Numbers

The numbers that multiply to \(-72\) and add up to \(-1\) are 8 and -9. This means we can decompose the middle term, \(-x\), as \(8x - 9x\).

Step 3: Rewrite the Polynomial

Rewrite \(12x^2 - x - 6\) as \(12x^2 + 8x - 9x - 6\).

Step 4: Group Terms and Factor by Grouping

Next, group the terms: \((12x^2 + 8x) + (-9x - 6)\). Factor each group separately. From \(12x^2 + 8x\), factor out \(4x\) to get: \(4x(3x + 2)\). From \(-9x - 6\), factor out \(-3\) to get: \(-3(3x + 2)\).

Step 5: Factor Out the Common Binomial Factor

Both terms now contain the factor \((3x + 2)\). Factoring \((3x + 2)\) out of the expression gives us \((4x - 3)(3x + 2)\).

Step 6: Verify the Factors

Expand \((4x - 3)(3x + 2)\) to check if you obtain the original polynomial. \((4x - 3)(3x + 2) = 12x^2 + 8x - 9x - 6 = 12x^2 - x - 6\). The factors are correct.

Key Concepts

Quadratic Polynomials

Quadratic polynomials are algebraic expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients and \(a eq 0\). These expressions can often be factored into two binomials, significantly simplifying equations or systems they are part of.
Understanding quadratics is important as they are the foundation for many mathematical concepts in algebra and beyond.

• They graph as parabolas.
• They're used to find the roots, or solutions, of quadratic equations.

When we factor quadratic polynomials, we're looking for two binomials that multiply together to give the original quadratic. The ability to do this is not only crucial in solving equations but also helps in understanding the properties of the expressions.

Factor by Grouping

Factor by grouping is a method used to simplify polynomial expressions, particularly useful for quadratic polynomials when they cannot be directly factored using simple factoring techniques.
This method breaks a polynomial into parts, allowing a more straightforward factorization of complex expressions. Here's how it works in simple steps:

• Group terms that have a common factor.
• Factor out the greatest common factor from each group.
• Look for and factor out a common binomial factor from the entire expression.

In the given exercise, the expression \(12x^2 - x - 6\) uses factor by grouping to first rewrite the polynomial, allowing for a shared factor extraction. It involves finding numbers that add up to the middle term and produce a product equal to the product of the leading coefficient and constant term, allowing the factor by grouping process to work smoothly.

Binomial Factors

Binomial factors are the building blocks when factoring polynomials. A binomial is an expression with two terms, like \((ax + b)\). Recognizing common binomial factors can greatly simplify complex algebraic expressions.

• They are crucial in transforming quadratic polynomials into simpler forms.
• Each binomial factor will typically contain both variable and constant terms.

In the process of factoring by grouping, the two groups typically share a common binomial. Once this factor is identified, we can rewrite the polynomial expression, significantly simplifying it.
This technique is validated by verifying the factors through expansion, ensuring the factorization is correct, as shown in the final step of the exercise where the factors of \((4x - 3)(3x + 2)\) when expanded, provide the original quadratic polynomial, confirming their correctness.