Problem 51 - Intermediate Algebra

Question

Let \(Q(x)=x^{4}-3 x^{3}+2 x^{2}+x-3 .\) Evaluate \(Q(x)\) by substituting the given value of \(x\) into the polynomial and simplifying. Then evaluate the polynomial by using the remainder theorem and synthetic division. See Example 4. $$ Q(2) $$

Step-by-Step Solution

Answer

Evaluate \(Q(2)\) directly or using synthetic division to find \(Q(2) = -1\).

Step 1: Substitute the Given Value into the Polynomial

Start by substituting the value of \(x = 2\) into the polynomial \(Q(x)\). Our polynomial is \(Q(x) = x^4 - 3x^3 + 2x^2 + x - 3\). Substitute \(2\) in place of \(x\): \[ Q(2) = (2)^4 - 3(2)^3 + 2(2)^2 + 2 - 3 \]

Step 2: Simplify the Polynomial Expression

Now calculate each term in the expression and simplify:1. \(2^4 = 16\)2. \(-3(2)^3 = -3(8) = -24\)3. \(2(2)^2 = 2(4) = 8\)4. \(2\)5. \(-3\)Combine these results:\[ Q(2) = 16 - 24 + 8 + 2 - 3 \] Simplify by performing the operations from left to right:\[ Q(2) = 16 - 24 = -8 \] \[ -8 + 8 = 0 \] \[ 0 + 2 = 2 \]\[ 2 - 3 = -1 \]

Step 3: Evaluate Using the Remainder Theorem

According to the remainder theorem, the value \(Q(a)\) is the remainder when \(Q(x)\) is divided by \(x-a\). Hence, to evaluate \(Q(2)\), use synthetic division to divide \(Q(x)\) by \(x-2\).

Step 4: Perform Synthetic Division

Set up the synthetic division:1. List the coefficients of \(Q(x)\) which are: 1 (for \(x^4\)), -3 (for \(x^3\)), 2 (for \(x^2\)), 1 (for \(x\)), and -3 (constant term).2. Use \(2\) from \(x-2\) as the divisor.3. Bring down the first coefficient (1) as is.4. Multiply \(2\) by the current value, and add this product to the next coefficient, continuing this process across all coefficients.Calculation steps:- Bring down 1.- \(2 \times 1 = 2\), add to -3 to get -1.- \(2 \times -1 = -2\), add to 2 to get 0.- \(2 \times 0 = 0\), add to 1 to get 1.- \(2 \times 1 = 2\), add to -3 to get -1 (this is the remainder).The remainder in synthetic division is the same as \(Q(2)\), which confirms that \(Q(2) = -1\).

Key Concepts

Remainder Theorem

The Remainder Theorem is a fantastic tool for evaluating polynomials. It states that if you divide a polynomial \( P(x) \) by the binomial \( x - a \), the remainder of this division is \( P(a) \). Simple, right? This means if you want to find \( P(a) \), you don't have to go through the whole process of plugging \( a \) into the polynomial and calculating it fully. Instead, you can get the remainder of this division, which will be \( P(a) \).
This theorem stands out because it offers a quick way to check the value of a polynomial at any given point. Understanding and applying the Remainder Theorem simplifies the process of polynomial evaluation. Also, when paired with synthetic division, it speeds up calculations considerably.

Synthetic Division

Synthetic division is like the swiss army knife for dividing polynomials. It's a simplified method designed exclusively for dividing a polynomial by a binomial of the form \( x - a \). Although it looks similar to long division, it’s faster and works wonders when paired with the Remainder Theorem.
Here's a brief rundown of how it works:

• First, list all the coefficients of the polynomial you are working with.
• Use the value of \( a \) from \( x-a \) for the divisor.
• Bring down the leading coefficient immediately to start the process.

From that initial drop, the process involves straightforward multiplication and addition to work through the rest of the coefficients. The beauty of synthetic division? You only need to deal with the numbers, making it super efficient. The final result gives you not only the quotient but also the remainder, which is what you're really after if you're verifying something with the Remainder Theorem.

Polynomial Simplification

Polynomial simplification is all about breaking down complex expressions into their simplest forms. This process clears the fog, making it easier to work with the polynomial whether you're differentiating, evaluating, or integrating it.
When evaluating a polynomial, you'll substitute a specific value for the variable, like in the problem where \( x = 2 \). Paying attention to details helps avoid common mistakes during simplification:

• Handle exponents first: calculate the powers of expressions before multiplying.
• Be mindful of signs: especially with subtraction and negative terms.
• Proceed from left to right and simplify step-by-step to avoid errors.

The goal of simplification is a neat and easy-to-read expression. Simplified polynomials make it easier to apply other mathematical operations or techniques, like using other theorems or syntheses for evaluations.