Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Find the zero of the finction \(f\). Do not use a calculator. \(f(x)=-4(2 x-3)+8(2 x+1)\)

Step-by-Step Solution

Answer

The zero of the function is \(x = -\frac{5}{2}\).

Step 1: Simplify the Expression

Start by expanding the function expression. Apply the distributive property to expand both terms: \(-4(2x-3) = -8x + 12\) and \(+8(2x+1) = 16x + 8\). Combine these to get:\[f(x) = -8x + 12 + 16x + 8\].

Step 2: Combine Like Terms

Next, simplify the expression by combining like terms. Group the \(x\) terms together and the constant terms together:\(-8x + 16x = 8x\) and \(12 + 8 = 20\). So the simplified expression for \(f(x)\) is:\[f(x) = 8x + 20\].

Step 3: Set the Function to Zero

Find the zero of the function by setting \(f(x) = 0\). This means:\[0 = 8x + 20\].

Step 4: Solve for x

Solve the equation \(8x + 20 = 0\) by isolating \(x\). First, subtract 20 from both sides to get:\[8x = -20\]. Then, divide both sides by 8 to solve for \(x\):\[x = -\frac{20}{8}= -\frac{5}{2}\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions and solve equations more easily. It involves distributing a multiplier over the terms inside a parenthesis. For example, in the expression \(a(b + c)\), you can apply the distributive property to rewrite it as \(ab + ac\). This means you multiply the term outside the parenthesis by each term inside the parenthesis separately.
This property is vital when working with mathematical functions like \(f(x) = -4(2x-3) + 8(2x+1)\), where you need to distribute \(-4\) across \(2x-3\) and \(+8\) across \(2x+1\).

When you apply the distributive property correctly:

• Distribute \-4\ across \(2x-3\), which gives you \(-8x + 12\).
• Distribute \+8\ across \(2x+1\), which results in \(+16x + 8\).

This way, you effectively eliminate the parenthesis and prepare the expression for further simplification.

Combining Like Terms

Combining like terms is another essential algebraic process that involves simplifying expressions by grouping terms with the same variable parts. Like terms are terms that have identical variable components raised to the same power, even if their coefficients differ. For instance, \(3x\) and \(5x\) are like terms because they both have the variable \(x\).
You simplify expressions by adding or subtracting the coefficients of like terms.

For the expression derived from applying the distributive property, \(f(x) = -8x + 12 + 16x + 8\), you identify like terms:

• The \(x\) terms \(-8x\) and \(+16x\) are combined to give you \(+8x\).
• The constant terms, \(12\) and \(8\), combine to equal \(20\).

So, the simplified expression becomes \(f(x) = 8x + 20\), which is much easier to work with for solving equations.

Solving Linear Equations

Solving linear equations requires finding the value of the variable that makes the equation true. In the equation \(8x + 20 = 0\), you need to isolate \(x\) to solve it.
You'll perform several steps to achieve this:

• First, eliminate the constant term on the side with the variable by subtracting \(20\) from both sides, giving you \(8x = -20\).
• Next, isolate \(x\) by dividing every term by \(8\), resulting in \(x = -\frac{20}{8}\).
• Simplify the fraction: \(-\frac{20}{8} = -\frac{5}{2}\).

The solution \(x = -\frac{5}{2}\) is the value that makes the function equal zero, and thus, is the zero or root of the function.