Problem 13 - Algebra 2

Question

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=x^{4}-6 x-8 $$

Step-by-Step Solution

Answer

\(g(3) = 55\) and \(g(-4) = 272\).

Step 1: Define Synthetic Substitution Setup for g(3)

First, we're going to use synthetic substitution to find \(g(3)\). We need to set up the numbers without the variable \(x\) using the coefficients of the polynomial \(g(x)=x^4 - 0x^3 - 0x^2 - 6x - 8\). Thus, the coefficients are \([1, 0, 0, -6, -8]\). We use \(3\) as the value we'll substitute in.

Step 2: Perform Synthetic Division for g(3)

1. Write down the coefficients: \([1, 0, 0, -6, -8]\).2. Bring down the first coefficient: \(1\).3. Multiply it by \(3\) and add to the next coefficient: \(1 \cdot 3 = 3\), \(0 + 3 = 3\).4. Repeat for each position: - Multiply \(3\) by \(3\) and add to the next: \(3 \cdot 3 = 9\), \(0 + 9 = 9\). - Multiply \(9\) by \(3\) and add to \(-6\): \(9 \cdot 3 = 27\), \(-6 + 27 = 21\). - Multiply \(21\) by \(3\) and add to \(-8\): \(21 \cdot 3 = 63\), \(-8 + 63 = 55\). 5. The remainder is \(55\), which is \(g(3)\).

Step 3: Define Synthetic Substitution Setup for g(-4)

Now, let's find \(g(-4)\). Again, set up with the coefficients \([1, 0, 0, -6, -8]\) and using \(-4\) as the substitution value.

Step 4: Perform Synthetic Division for g(-4)

1. Write down those coefficients: \([1, 0, 0, -6, -8]\).2. Bring down the first coefficient: \(1\).3. Multiply by \(-4\) and add to the next coefficient: \(1 \cdot -4 = -4\), \(0 - 4 = -4\).4. Continue this process: - Multiply \(-4\) by \(-4\) and add: \(-4 \cdot -4 = 16\), \(0 + 16 = 16\). - Multiply \(16\) by \(-4\) and add: \(16 \cdot -4 = -64\), \(-6 - 64 = -70\). - Multiply \(-70\) by \(-4\) and add: \(-70 \cdot -4 = 280\), \(-8 + 280 = 272\).5. The remainder is \(272\), which is \(g(-4)\).

Key Concepts

Polynomial Functions

A polynomial function is like a math expression made up of variables and coefficients. It's usually written with powers, or exponents, attached to variables like "x." For example, in the function \(g(x) = x^4 - 6x - 8\), we see a polynomial of degree 4.

Polynomials give us a way to generalize arithmetic, and they appear everywhere in algebra, calculus, and beyond. Here are some key features:

• The highest power of the variable is called the "degree" of the polynomial.
• The coefficients are the numbers in front of the variables. In \(g(x)\), these are 1, 0, 0, -6, and -8.
• Polynomials are called "functions" because each \(x\) input gives exactly one output \(g(x)\).

Understanding polynomials is crucial because it helps with graphing them, finding their roots, and much more.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear factor, making it quick and tidy. It simplifies calculations, especially without needing to write out every step like in long division. Here's the general idea:

• Write the coefficients of the polynomial. For \(g(x) = x^4 - 6x - 8\), they're \([1, 0, 0, -6, -8]\).
• Use the value you want to substitute (like 3 or -4 in the example) at the side.
• Bring down the first coefficient, then multiply and add across each step.

This technique helps find remainders easily, which actually represent the value of the polynomial at specific points—this is a powerful and efficient tool in algebra.

Evaluating Polynomials

Evaluating polynomials might sound tricky, but it's all about finding the output value by plugging in numbers into the polynomial. This is where synthetic substitution shines.

In practice:

• Substitute the chosen number into the polynomial using synthetic division for speed.
• Calculate the results at each step to finally reach the remainder, which is the polynomial evaluated at that number.

For example, when computing \(g(3)\) and \(g(-4)\) from our polynomial \(g(x) = x^4 - 6x - 8\), we used synthetic division to find the remainders. These remainders, \(55\) and \(272\) respectively, give us the evaluated value of the polynomial at those points. Evaluating polynomials helps reveal the function's behavior, like its growth, decay, and other important characteristics.