Problem 73 - Elementary Algebra

Question

Factor using the \(A C\) method. $$ 4 x 2+28 x+40 $$

Step-by-Step Solution

Answer

The expression factors to \(4(x + 2)(x + 5)\).

Step 1: Identify A, B, and C

In the quadratic expression \(4x^2 + 28x + 40\), identify the coefficients: \(A = 4\), \(B = 28\), and \(C = 40\).

Step 2: Compute A*C

Multiply \(A\) and \(C\) together. This gives us \(4 \times 40 = 160\).

Step 3: Find two numbers that multiply to A*C and add to B

We need two numbers that multiply to \(160\) and add up to \(28\). After examining the factors of \(160\), the numbers \(8\) and \(20\) satisfy both conditions: \(8 \times 20 = 160\) and \(8 + 20 = 28\).

Step 4: Rewrite the middle term using the two numbers

Rewrite the expression \(4x^2 + 28x + 40\) as \(4x^2 + 8x + 20x + 40\).

Step 5: Factor by grouping

Group the terms: \((4x^2 + 8x) + (20x + 40)\). Factor out the common factors in each group: \(4x(x + 2) + 20(x + 2)\).

Step 6: Factor out the common binomial

Notice \((x + 2)\) is a common factor: \((4x + 20)(x + 2)\).

Step 7: Simplify the expression

Factor out the greatest common factor from the first part: \(4(x + 5)(x + 2)\). So the expression is factored as \(4(x + 2)(x + 5)\).

Key Concepts

AC method

The AC method is a popular approach for factoring quadratic equations. This technique involves using the coefficients of the quadratic equation, particularly focusing on the relationship between the terms.

To start, we need to identify the coefficients of the quadratic expression, which are typically expressed in the form of \(Ax^2 + Bx + C\). In the provided exercise, our equation is \(4x^2 + 28x + 40\), where we identify the coefficients as:

• \(A = 4\)
• \(B = 28\)
• \(C = 40\)

Then, we calculate the product of \(A\) and \(C\), which is called the AC product. So, \(4 \times 40 = 160\). This number is crucial because we need to find two numbers that multiply to \(160\) and add up to \(28\). This process simplifies the quadratic expression and sets the stage for further factoring steps.

coefficients

Coefficients are the numerical factors that multiply with variables in any algebraic expression or equation. In the expression \(4x^2 + 28x + 40\), the coefficients we focus on are \(4\), \(28\), and \(40\). These numbers serve different purposes in the equation:

• \(A = 4\), which is the coefficient of the \(x^2\) term
• \(B = 28\), which is the coefficient of the \(x\) term
• \(C = 40\), which is the constant term with no variable attached

Understanding these coefficients helps streamline the AC method as they enable the calculation of the AC product. Recognizing the role of coefficients assists in breaking down complex expressions, making it easier to manipulate and factor them. By managing coefficients effectively, you can separate parts of the expression to reteach it in more manageable bits.

factoring by grouping

Factoring by grouping is a strategic method used in breaking down complex expressions into simpler factors. This method comes in handy after you've rewritten the quadratic in a form that exposes common factors.

In this exercise, after the quadratic expression \(4x^2 + 28x + 40\) is rewritten using the numbers \(8\) and \(20\) as \(4x^2 + 8x + 20x + 40\), you then apply factoring by grouping:

• First, group terms to enable factorization: \((4x^2 + 8x)\) and \((20x + 40)\)
• Factor out the Greatest Common Factor (GCF) from each grouped pair: from the first group, \(4x(x + 2)\), and from the second group, \(20(x + 2)\)

Notice that \((x + 2)\) is a common factor in both groups. This insight allows us to factor it out, simplifying our expression considerably to \((4x + 20)(x + 2)\). Thus, grouping simplifies complex expressions and helps reveal hidden factors.

greatest common factor

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. It plays a crucial role in the process of factoring expressions, particularly in factoring by grouping.

During the grouping of terms, identifying the GCF allows us to simplify groups of terms. In the expression \((4x^2 + 8x)\) and \((20x + 40)\):

• The first group, \(4x^2 + 8x\), has a GCF of \(4x\). This means \(4x\times (x + 2)\) is a simplified form, where \(4x\) is factored out.
• For the second group, \(20x + 40\), the GCF is \(20\). This simplifies to \(20 \times (x + 2)\).

Once these GCFs are found and factored out, other common factors, like \(x + 2\), in this case, become apparent and simplify further factoring. The identification and extraction of the GCF not only simplify expressions but also ensure accurate and complete factoring.