Problem 55 - 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. $$ Q(-3) $$

Step-by-Step Solution

Answer

The value of \( Q(-3) \) is 174.

Step 1: Direct Substitution into the Polynomial

To evaluate \( Q(-3) \), substitute \( -3 \) for \( x \) in the polynomial. Start by replacing each \( x \) in the expression \[ Q(x) = x^{4} - 3x^{3} + 2x^{2} + x - 3 \]with \( -3 \):\[ Q(-3) = (-3)^{4} - 3(-3)^{3} + 2(-3)^{2} + (-3) - 3 \]Calculate each term:- \((-3)^{4} = 81\)- \(-3 (-3)^{3} = -3 imes -27 = 81\)- \(2(-3)^{2} = 2 imes 9 = 18\)- \(-3 = -3\)- Final \(-3 = -3\)Thus, the expression simplifies to\[ Q(-3) = 81 + 81 + 18 - 3 - 3 \]Calculate the sum:\[ Q(-3) = 174 \]

Step 2: Evaluate using the Remainder Theorem

According to the Remainder Theorem, the remainder of the division of a polynomial \( f(x) \) by \( x-c \) is \( f(c) \). Let's evaluate \( Q(-3) \) using synthetic division.Step into synthetic division with \(-3\) as the divisor and the coefficients of the polynomial \(1, -3, 2, 1, -3\).1. Write the coefficients for the polynomial: - Initial row: \([1, -3, 2, 1, -3]\)2. Perform synthetic division: - Start by writing down \(1\), the leading coefficient. - Multiply \(1\) by \(-3\) and write it under the next coefficient: - New row: \([1, -3, 2, 1, -3]\) - Calculations: - \(1 \times (-3) = -3\) - place under \(-3\) - Add \(-3 + (-3) = -6\) - Multiply \(-6\) by \(-3\) and write the result under next coefficient: - \(-6 \times (-3) = 18\) - Place under \(2\). Then: \(2 + 18 = 20\) - Multiply \(20\) by \(-3\) and place under next coefficient: - \(20 \times (-3) = -60\) - Place under \(1\). Then: \(1 - 60 = -59\) - Multiply \(-59\) by \(-3\) and place under next coefficient: - \(-59 \times (-3) = 177\) - Place under \(-3\). Then: \(-3 + 177 = 174\).The remainder is \(174\), confirming \( Q(-3) = 174 \).

Step 3: Conclusion

Both methods, direct substitution and synthetic division, validated the result. The value of the polynomial \( Q(x) \) at \( x = -3 \) is \( 174 \).

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear divisor of the form \(x-c\). It simplifies the process compared to long division and is especially helpful when evaluating polynomials at specific points.
To perform synthetic division:

• Write down the coefficients of the polynomial.
• Use the value of \(c\) (the opposite of the divisor) for calculations.
• Start by writing the leading coefficient in a new row.
• Multiply it by \(c\) and add to the next coefficient.
• Repeat the process for all coefficients until you obtain the remainder.

For the polynomial \(Q(x) = x^4 - 3x^3 + 2x^2 + x - 3\) and \(x = -3\), synthetic division helped us verify that the remainder, and thus \(Q(-3)\), equals 174. This method not only confirms polynomial evaluations but also ensures accuracy and quicker calculations.

Remainder Theorem

The Remainder Theorem is a useful tool when evaluating polynomials. It states that, for a polynomial \( f(x) \), if you divide it by \( x-c \), the remainder of this division will be equal to \( f(c) \).
This theorem provides a powerful link between division and evaluation:

• It allows you to find the value of a polynomial at any given \(x\) value without fully computing the polynomial.
• You can quickly check if a number is a root of the polynomial (a zero) by checking if the remainder is zero.

Applying the Remainder Theorem to \(Q(x)\) for \(x=-3\) confirmed that the remainder was indeed the value of the polynomial, 174. This double-checks the results obtained through direct substitution or synthetic division.

Direct Substitution

Direct substitution is the most straightforward way to evaluate a polynomial at a specific point. It involves replacing each \(x\) in the polynomial with the given value and simplifying the expression.
For \(Q(x) = x^4 - 3x^3 + 2x^2 + x - 3\) and \(x = -3\):

• Replace each \(x\) with \(-3\).
• Calculate each power and product.
• Add or subtract the terms as needed to simplify.

Direct substitution for \(Q(-3)\) yielded the value of \(174\). While it is directly effective, it can be time-consuming for complex polynomials. However, it is a straightforward method that clarifies the process of polynomial evaluation.