Problem 19 - Algebra 1: Concepts and Skills

Question

Choose the correct factorization. If neither choice is correct, find the correct factorization. $$ 3 x^{2}+2 x-8 $$ A. \((3 x-4)(x+2)\) B. \((3 x-4)(x-2)\)

Step-by-Step Solution

Answer

Neither choice A nor B offers the correct factorization. The correct factorization of the given expression \(3x^2 + 2x -8\) is \((3x+4)(x-2)\).

Step 1: Evaluate Choice A

Substitute the values from Choice A into the factor product and sum of product pairs formula. For the choice A \((3x-4)(x+2)\), product ac equals to 3 which doesn't match the coefficient of x^2 in the given expression which is 3, and sum ad+bc equals to 2*(-4) + 3*2 = -2 which is also not equal to the coefficient of x term in the given expression, which is 2. Therefore, Choice A is not correct.

Step 2: Evaluate Choice B

Substitute the values from Choice B into the factor product and sum of product pairs formula. For the choice B \((3x-4)(x-2)\), product ac equals to -6 which again does not match the coefficient of x^2 in the given expression which is 3, and sum - ad+bc equals to -8 + 12 = 4 which doesn't match the coefficient of x term in the given expression which is 2. Therefore, Choice B is not correct either.

Step 3: Find the Correct Factorization

Let's find the correct factorization of \(3x^2+2x-8\). We are looking for two numbers that multiply to (a*c = 3*-8 = -24) and add to b (2). These two numbers are -4 and 6.\n Now, we rewrite the middle term: \(3x^2+2x-8 = 3x^2 - 4x + 6x - 8\). Next, we factor by grouping: \(3x(x - 2) + 4(2x -4)\). Simplify to get \( (3x+4)(x -2)\) as the correct factorization.

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 \( a eq 0 \). These equations are fundamental with roots found using various methods, like factoring, using the quadratic formula, or completing the square.
Quadratic equations often appear in academic and real-world applications, from physics to everyday financial calculations. Solving these equations helps find where a parabola intersects the x-axis, known as its roots.
To solve or factor a quadratic equation like \( 3x^2 + 2x - 8 \), understanding the relationship between coefficients and factors is crucial. Notice how coefficients directly influence both the potential solutions and the factorization process.

Polynomial Factoring

Polynomial factoring breaks down a polynomial into simpler components or 'factors' that, when multiplied together, give the original polynomial. It’s similar to breaking down numbers into prime factors.
The quadratic polynomial \( 3x^2 + 2x - 8 \) involves finding two binomials whose product equals the quadratic expression.

• First, find two numbers that multiply to the product of \( a \) and \( c \) (\( 3 \times -8 = -24 \)), and sum to \( b \) (2).
• These numbers, -4 and 6 in this exercise, help to split the middle term: \( 3x^2 - 4x + 6x - 8 \).
• Next, apply factor by grouping: Treat pairs and factor out common terms.
• The factored form \((3x+4)(x-2)\) is the correct result after grouping and simplification.

This method makes solving polynomial equations faster and is a cornerstone of algebraic manipulation.

Algebraic Expressions

Algebraic expressions combine variables, numbers, and operations. They’re the building blocks of algebra and are used extensively in solving mathematical problems.
The original expression \( 3x^2 + 2x - 8 \) is a quadratic algebraic expression.
Manipulating algebraic expressions involves:

• Simplifying or expanding expressions.
• Substitution (replacing variables with values).
• Solving for variables (finding values that satisfy the equation).

In factorization, expressions are rewritten from a sum or difference into a product of simpler expressions. Recognizing patterns or using systematic approaches like grouping help in finding factorized forms.
Understanding these principles allows efficient manipulation and resolution of more complex algebraic problems, forming a solid foundation for higher mathematical learning.