Problem 54 - 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. $$ Q(0) $$

Step-by-Step Solution

Answer

The value of \( Q(0) \) is \(-3\).

Step 1: Substitute x with 0 in Q(x)

To evaluate the polynomial \( Q(x) = x^4 - 3x^3 + 2x^2 + x - 3 \), substitute \( x = 0 \).Compute:\[Q(0) = (0)^4 - 3(0)^3 + 2(0)^2 + 0 - 3\]Simplifying gives:\[Q(0) = 0 - 0 + 0 + 0 - 3 = -3\]So, \( Q(0) = -3 \).

Step 2: Understand the Remainder Theorem

According to the remainder theorem, for a polynomial \( P(x) \), the remainder of the division of \( P(x) \) by \( x - a \) is \( P(a) \). This means substituting \( x = a \) into the polynomial gives the remainder directly. Since \( a = 0 \), we already found the remainder to be \( -3 \) in Step 1.

Step 3: Perform Synthetic Division

Perform synthetic division of \( Q(x) \) by \( x - 0 \) (which is just \( x \)) to find the remainder:1. Write down the coefficients of \( Q(x) \): \( 1, -3, 2, 1, -3 \).2. Bring down the leading coefficient 1.3. Multiply by 0 (since we're dividing by \( x \)) and add to the next coefficient.4. Repeat this process for each coefficient, leaving the constant at the right as the remainder.Coefficients:\[\begin{array}{c|ccccc}0 & 1 & -3 & 2 & 1 & -3 \ & & 0 & 0 & 0 & 0 \\hline & 1 & -3 & 2 & 1 & -3 \\end{array}\]The remainder is \(-3\), consistent with Step 1.

Key Concepts

Remainder Theorem

The Remainder Theorem is a simple but powerful concept in algebra that helps us understand the relationship between division of polynomials and the evaluation of polynomial functions. When we talk about the remainder theorem, we mean that if you divide a polynomial \( P(x) \) by \( x-a \), the remainder is the same as \( P(a) \). This makes evaluating polynomials much easier; instead of doing long division, you can just substitute \( a \) into the polynomial.
For example, for the polynomial \( Q(x) = x^4 - 3x^3 + 2x^2 + x - 3 \), the remainder when divided by \( x-0 \) is the value of \( Q(0) \). This gives us a quick way to find what the value of the polynomial is at certain points without extensive calculations.
The application of the remainder theorem provides not just a shortcut, but also insight into the behavior of polynomial functions and their roots. If the remainder is zero, that means \( a \) is a root of the polynomial.

Synthetic Division

Synthetic division is a quick method for dividing a polynomial by a linear divisor of the form \( x - a \). This technique is much simpler than long division and involves working only with the coefficients of the polynomial.
Let's break down the process using the polynomial \( Q(x) = x^4 - 3x^3 + 2x^2 + x - 3 \). If we want to find the value of the polynomial at \( x = 0 \), which would also be the remainder when the polynomial is divided by \( x \), synthetic division provides an efficient way to do this:

• List the coefficients: \( [1, -3, 2, 1, -3] \).
• Drop the first coefficient down (1) and multiply by 0, add this to each subsequent coefficient.
• The final number in the last column of your synthetic division setup will be your remainder, which matches the value of \( Q(0) \).

This simplified approach makes evaluating polynomials straightforward, particularly when checking the polynomial's value or verifying the theorem.

Substitution Method

The substitution method for evaluating polynomials is exactly what it sounds like - you substitute a particular value of \( x \) into the polynomial and simplify to find the result. This method gives you the exact value of the polynomial for that specific \( x \).
For the polynomial \( Q(x) = x^4 - 3x^3 + 2x^2 + x - 3 \), evaluating it at \( x = 0 \) using substitution involves plugging 0 into the equation and simplifying the result:

• Substitute: \( Q(0) = 0^4 - 3(0)^3 + 2(0)^2 + 0 - 3 \).
• Calculate each term: \( 0 - 0 + 0 + 0 - 3 \).
• Simplify to get \( Q(0) = -3 \).

This method is direct and often used when verifying the results obtained from the Remainder Theorem or synthetic division.