Problem 54 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 7 x^{2}+10 x-8 $$

Step-by-Step Solution

Answer

The polynomial factors to \((x + 2)(7x - 4)\).

Step 1: Identify the A, B, and C terms

In the polynomial equation, the general form is \( ax^2 + bx + c \). For the polynomial \( 7x^2 + 10x - 8 \), the coefficients are identified as: \( a = 7 \), \( b = 10 \), and \( c = -8 \).

Step 2: Use the AC method

Multiply \( a \) and \( c \): \( 7 \times (-8) = -56 \).Next, identify two numbers that multiply to \(-56\) and add to \(10\). These numbers are \(14\) and \(-4\).

Step 3: Rewrite the middle term

Use the numbers found to split the middle term \(10x\):\[ 7x^2 + 14x - 4x - 8 \].

Step 4: Group terms

Group the terms in pairs and factor out the common factor:\[ (7x^2 + 14x) + (-4x - 8) \].

Step 5: Factor each pair

Factor the common factor from each group:- From \(7x^2 + 14x\), factor out \(7x\): \( 7x(x + 2) \).- From \(-4x - 8\), factor out \(-4\): \( -4(x + 2) \).

Step 6: Factor out the common binomial

Notice the common binomial \((x + 2)\) in each term. Factor it out:\( (x + 2)(7x - 4) \).

Key Concepts

Quadratic Equations

Quadratic equations are an essential part of algebra and mathematics. They are equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. These equations are called 'quadratic' because the highest degree is two, meaning they involve terms with \( x^2 \). Quadratic equations come up in various contexts, such as physics and engineering, where they can model real-world phenomena. Understanding how to solve and factor them opens the door to tackling more complex mathematical challenges.

AC Method

The AC method is a technique used in factoring polynomials, particularly when dealing with quadratic expressions. It focuses on identifying a pair of numbers that relate to the coefficients from the quadratic equation. Here's how the method works:

• First, identify the coefficients \( a \) and \( c \) from the quadratic expression \( ax^2 + bx + c \).
• Next, calculate the product \( ac \).
• Find two numbers that multiply to \( ac \) and add up to one more term: \( b \).

In our example with \( 7x^2 + 10x - 8 \), multiplying the \( a \) and \( c \) values gives \(-56\). We look for two numbers that multiply to \(-56\) and sum to \(10\). These numbers are \(14\) and \(-4\), which help us rewrite the middle term for easier factoring.

Polynomial Factoring Steps

Factoring polynomials involves breaking them down into simpler expressions that multiply to the original polynomial. The process may seem complex at first, but with practice, it becomes straightforward:

• Identify the terms \( a \), \( b \), and \( c \) as seen in the polynomial.
• Use the AC method to find two numbers that satisfy the multiplication and addition conditions.
• Rewrite the polynomial by splitting the middle term using these two numbers.
• Group the terms in pairs to prepare them for factoring.
• Factor out common factors in each group.
• Look for a common binomial factor in the grouped terms and factor this out.

In our example, after rewriting and grouping \( 7x^2 + 14x - 4x - 8 \), we factor out common parts: \( 7x(x+2) \) and \(-4(x+2)\), revealing a common binomial \((x+2)\) which is then factored alongside \((7x-4)\).

Mathematical Problem Solving

Mathematical problem solving is a vital skill, enabling you to approach polynomial equations methodically. Here are some general strategies:

• Understand the problem: Start by comprehending what the equation involves and identify key terms.
• Divide and conquer: Break the equation into smaller, more manageable parts, similar to how we split the middle term in factoring.
• Verify your solution: Always check your work by multiplying the factors back to ensure you retrieve the original polynomial.

Solving problems like factoring requires practice and an investigative mindset. Embrace mistakes, as they are an integral part of learning mathematics and improving your problem-solving abilities. Approaching problems systematically makes mathematical challenges less daunting, and enhances your overall learning experience.