Problem 50 - 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. $$ Q(1) $$

Step-by-Step Solution

Answer

The value of \(Q(1)\) is -2.

Step 1: Substitute and Simplify

To evaluate \(Q(1)\), substitute \(x = 1\) into the polynomial. This gives:\[ Q(1) = 1^{4} - 3 \cdot 1^{3} + 2 \cdot 1^{2} + 1 - 3. \]Simplify each term:- \(1^{4} = 1\)- \(-3 \cdot 1^{3} = -3\)- \(+ 2 \cdot 1^{2} = 2\)- \(+ 1 = 1\)- \(- 3 = -3\)Combine these results:\[ Q(1) = 1 - 3 + 2 + 1 - 3 = -2. \]

Step 2: Set Up Synthetic Division

Prepare to use synthetic division to verify the result using the Remainder Theorem. Set up synthetic division using \(x = 1\) as the divisor. The coefficients of the polynomial \(Q(x)\) are: 1, -3, 2, 1, -3.

Step 3: Perform Synthetic Division

Write down the coefficients: 1, -3, 2, 1, -3. Begin with the leading coefficient (1) and place it under the division line.1. Bring down the first coefficient (1) as is.2. Multiply this by 1 (the divisor) and add to the next coefficient: - 1 \(\cdot\) 1 = 1, then -3 + 1 = -2.3. Repeat the multiplication and addition: - -2 \(\cdot\) 1 = -2, then 2 + (-2) = 0.4. Repeat the step: - 0 \(\cdot\) 1 = 0, then 1 + 0 = 1.5. Final step: - 1 \(\cdot\) 1 = 1, then -3 + 1 = -2.The remainder is -2.

Step 4: Verify the Remainder Theorem

The Remainder Theorem states that the remainder of the division of a polynomial \(Q(x)\) by \(x - c\) is \(Q(c)\). Here, the remainder for \(x - 1\) division is -2, which matches the evaluation from Step 1. Therefore, \(Q(1) = -2\) is verified.

Key Concepts

Synthetic Division

Synthetic division is a streamlined method for dividing polynomials, removing much of the complexity associated with traditional long division. This technique is particularly useful when you're checking how a polynomial behaves when evaluated at a given point, often referred to as finding whether a number is a root or obtaining the remainder. To perform synthetic division for the polynomial \(Q(x) = x^4 - 3x^3 + 2x^2 + x - 3\) when \(x = 1\), you would use the coefficients of the polynomial, which are: 1, -3, 2, 1, and -3.

• Write these coefficients horizontally.
• Place \(1\), the value for substitution, outside the division framework.

Start by bringing the first coefficient (1) directly down. Multiply this by 1, the number outside, and add to the next coefficient (-3), resulting in -2. Continue this operation: multiply and add sequentially through the polynomial. The final number is the remainder, \(-2\) in this instance. The beauty of synthetic division is its ability to handle these operations quickly, without the mess of long polynomial division.

Remainder Theorem

The Remainder Theorem provides a powerful link between division and evaluating polynomials. This theorem states that when you divide a polynomial \(Q(x)\) by \(x - c\), the remainder of this division is precisely the value of the polynomial when \(c\) is substituted for \(x\). In simpler terms, \(Q(c)\) gives the remainder you get when dividing \(Q(x)\) by \(x - c\).In the exercise, you determined \(Q(1)\) by evaluation, finding \(-2\) through direct substitution. By also using synthetic division, you confirmed the remainder was \(-2\). These matching values bolster the reliability of both approaches. In practice:

• When you find the remainder \(-2\) using synthetic division, it reinforces that \(Q(1) = -2\).
• Whenever working with polynomials, this theorem can simplify checking specific values.

This connection allows for quick evaluations of polynomial expressions at specific points.

Algebraic Substitution

Algebraic substitution is a fundamental tool for evaluating polynomials. It involves replacing the variable \(x\) in the polynomial expression \(Q(x)\) with a specific value, in this case, \(x = 1\). By substituting, you change a complex algebraic expression into simple arithmetic.Start with \[ Q(x) = x^4 - 3x^3 + 2x^2 + x - 3 \] and plug in 1 for every instance of \(x\):

• \(1^4 \to 1\)
• \(-3 \times 1^3 \to -3\)
• \(+ 2 \times 1^2 \to 2\)
• \(+ 1 \to 1\)
• \(- 3 \to -3\)

When you calculate these values, sum them up: 1 - 3 + 2 + 1 - 3 = -2.Substitution is not only helpful for evaluation; it also allows deeper understanding of how a polynomial behaves when specific values are applied. It simplifies comparison with other methods, like the synthetic division, ensuring comprehension.