Problem 14 - Algebra for College Students

Question

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$x^{2}+8 x-24$$

Step-by-Step Solution

Answer

The polynomial factors as \((x - 2)(x + 12)\).

Step 1: Identify the Polynomial

The given polynomial is \(x^2 + 8x - 24\). Our task is to factor this expression completely.

Step 2: Determine Factoring Approach

To factor the quadratic \(x^2+8x-24\), we need to find two numbers that multiply to -24 (the constant term) and add up to 8 (the coefficient of the x-term).

Step 3: Find Suitable Numbers

Consider pairs of numbers that multiply to -24: (-1,24), (1,-24), (-2,12), (2,-12), (-3,8), (3,-8), (-4,6), (4,-6). Among these, (12, -2) add up to 8.

Step 4: Factor the Polynomial

Using the suitable numbers from the previous step, express the middle term as two terms: \[x^2 + 12x - 2x - 24\]Then group and factor by grouping: \[x(x + 12) - 2(x + 12)\]This gives us the factors: \[(x - 2)(x + 12)\].

Step 5: Verify the Factorization

Expand \((x - 2)(x + 12)\) to ensure it equals the original polynomial:\[(x - 2)(x + 12) = x(x + 12) - 2(x + 12) = x^2 + 12x - 2x - 24 = x^2 + 8x - 24\].Since the expanded form matches the original polynomial, the factorization is correct.

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. Quadratics are unique because they form a parabola when graphed.Key characteristics of quadratic equations include:

• Nature of Roots: The solutions or roots of quadratic equations can be real or complex, depending on the discriminant \(b^2 - 4ac\). If the discriminant is positive, there are two distinct real roots; if zero, there is one real root (a repeated root); and if negative, there are two complex roots.
• Factoring: One common method to solve a quadratic equation is by factoring it into two binomial expressions. This is possible when the quadratic can be factored over the set of integers.

Recognizing the structure and characteristics of a quadratic equation is crucial in factoring and finding its roots efficiently.

Factoring by Grouping

Factoring by grouping is a technique used to simplify polynomials, often employed with quadratics or other expressions where simple factoring methods don't immediately apply. It involves rearranging and grouping terms to find a common factor.Here's how you can use factoring by grouping:

• Identify the Product and Sum: First, identify two numbers that multiply to give the product of the leading coefficient and the constant term, and add to the middle term.
• Rewrite the Middle Term: Once the right pair is found, rewrite the middle term of the quadratic using these two numbers.
• Group and Factor: Group the terms into two pairs and then factor out the common factors in each group.

In the exercise example, the expression \(x^2 + 8x - 24\) is rearranged by breaking the middle term into two parts, allowing it to be grouped effectively and factored. This results in the factors \((x - 2)(x + 12)\).

Integer Factorization

Integer factorization involves breaking down a number or expression into a product of integers that, when multiplied together, yield the original number or expression. With polynomials, our goal is often to express the polynomial in terms of simpler, integer-based factors.For a polynomial like \(x^2 + 8x - 24\), integer factorization helps to determine whether and how it can be rewritten using only integers:

• Finding Pairs: We look for pairs of integers that multiply to the constant term (in this case, -24) and add up to the coefficient of the middle term (which is 8).
• Testing Combinations: Each pair is tested until the correct combination that fits both conditions is found: (12, -2).

Using integer factorization, one can verify that the polynomial is factorable with integers, as demonstrated in the solution where \(x^2 + 8x - 24\) gets successfully factored into \((x - 2)(x + 12)\).

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations like addition, multiplication, and exponentiation. These expressions do not have an equality sign, distinguishing them from equations.When handling algebraic expressions, some crucial points to consider are:

• Simplification: This involves reducing the expression to its simplest form, which can include factoring.
• Variable Terms: Expressions consist of terms, which are separated by '+' or '-' signs. Terms with the same variables can often be combined to simplify an expression.
• Factoring: As seen in the exercise, factoring is a key technique to breaking down complex algebraic expressions into products of simpler expressions.

In the given exercise, recognizing \(x^2 + 8x - 24\) as an algebraic expression allows focusing on reducing it by employing factoring techniques, simplifying the expression into a product of binomials.