Problem 88 - Algebra A Combined Function

Question

See the Concept Check in this section. Without multiplying, decide which of the following factored forms is not a factored form of \(4 x^{2}+19 x+12\) a. \((2 x+4)(2 x+3)\) b. \((4 x+4)(x+3)\) c. \((4 x+3)(x+4)\) d. \((2 x+2)(2 x+6)\)

Step-by-Step Solution

Answer

Options A, B, and D are not factored forms.

Step 1: Understand the Problem

We need to determine which of the given expressions is not a factored form of the polynomial \(4x^2 + 19x + 12\). We will do this by checking which option, when expanded, does not equal the given polynomial.

Step 2: Expand Option A

Expand \((2x+4)(2x+3)\).First, distribute \(2x\):\[2x \cdot 2x + 2x \cdot 3 = 4x^2 + 6x\]Then, distribute \(4\):\[4 \cdot 2x + 4 \cdot 3 = 8x + 12\]Combine all terms:\[4x^2 + (6x + 8x) + 12 = 4x^2 + 14x + 12\]This is not equal to \(4x^2 + 19x + 12\).

Step 3: Expand Option B

Expand \((4x+4)(x+3)\).First, distribute \(4x\):\[4x \cdot x + 4x \cdot 3 = 4x^2 + 12x\]Then, distribute \(4\):\[4 \cdot x + 4 \cdot 3 = 4x + 12\]Combine all terms:\[4x^2 + (12x + 4x) + 12 = 4x^2 + 16x + 12\]This does not match \(4x^2 + 19x + 12\).

Step 4: Expand Option C

Expand \((4x+3)(x+4)\).First, distribute \(4x\):\[4x \cdot x + 4x \cdot 4 = 4x^2 + 16x\]Then, distribute \(3\):\[3 \cdot x + 3 \cdot 4 = 3x + 12\]Combine all terms:\[4x^2 + (16x + 3x) + 12 = 4x^2 + 19x + 12\]This matches the original polynomial.

Step 5: Expand Option D

Expand \((2x+2)(2x+6)\).First, distribute \(2x\):\[2x \cdot 2x + 2x \cdot 6 = 4x^2 + 12x\]Then, distribute \(2\):\[2 \cdot 2x + 2 \cdot 6 = 4x + 12\]Combine all terms:\[4x^2 + (12x + 4x) + 12 = 4x^2 + 16x + 12\]This is not equal to \(4x^2 + 19x + 12\).

Step 6: Determine the Incorrect Option

From the expansions, only Option C results in the expression \(4x^2 + 19x + 12\). Thus, all other options are not valid factored forms of the polynomial.

Key Concepts

Expanding Expressions

To fully understand factoring polynomials, it's crucial to comprehend expanding expressions. This involves distributing each term of one polynomial to every term of another. By doing this, you can combine like terms to present the expression in its expanded form.

For instance, let's consider expanding the expression \((2x+4)(2x+3)\). You start by distributing the first term from the first binomial, \(2x\), to both terms of the second binomial:

• \(2x \cdot 2x = 4x^2\)
• \(2x \cdot 3 = 6x\)

Then do the same with the second term, \(4\), from the first binomial:

• \(4 \cdot 2x = 8x\)
• \(4 \cdot 3 = 12\)

After distributing, combine all the terms: \(4x^2 + 6x + 8x + 12\). Combining like terms gives you \(4x^2 + 14x + 12\). This process of expansion allows you to solve problems and check if certain factored forms work.

Factored Form Verification

Factored form verification helps determine if a polynomial expression has been correctly factored. By expanding the factored form and comparing it with the original polynomial, you can verify its correctness.

For example, in our exercise, you had to verify possible factored forms of \(4x^2 + 19x + 12\) by expansion. Expanding the options gives results which you compare to see if they equal the original polynomial. The only correct factored form here was option C: \((4x+3)(x+4)\), which successfully expands back to \(4x^2 + 19x + 12\).

The verification process mainly ensures your factorization is accurate and whether a given factored form truly represents the polynomial in its expanded form. This skill is essential for solving polynomial equations correctly.

Quadratic Equations

Quadratic equations take the form \(ax^2 + bx + c = 0\). Solving these involves various techniques, and one of them is factoring. A quadratic is factorable when you can break it into binomials that multiply to give the original quadratic. This process makes solving quadratic equations easier.

For the polynomial \(4x^2 + 19x + 12\), the correct factorization \((4x+3)(x+4)\) allows you to set each binomial equal to zero, revealing the solutions to the equation.

• Set each factor to zero: \((4x+3) = 0\) or \((x+4) = 0\)
• Solve each equation: \(x = -\frac{3}{4}\) and \(x = -4\)

These solutions are where the quadratic equation intercepts the x-axis. Understanding this principle aids in graphing functions and solving various polynomial equations.