Problem 36 - College Algebra

Question

For the following exercises, use synthetic division to find the quotient. $$ \left(4 x^{4}-2 x^{3}-4 x+2\right) \div(2 x-1) $$

Step-by-Step Solution

Answer

The quotient is \(4x^3 - 4\) with a remainder of 0.

Step 1: Set Up the Synthetic Division

To perform synthetic division, identify the root of the divisor \((2x-1)\), which is \(x = \frac{1}{2}\). Write down the coefficients of the dividend \(4x^4 - 2x^3 + 0x^2 - 4x + 2\). Note that we include a 0 for the missing \(x^2\) term. The coefficients are \([4, -2, 0, -4, 2]\).

Step 2: Bring Down the Leading Coefficient

Start the synthetic division by bringing down the leading coefficient, which is 4, to the bottom row.

Step 3: Multiply and Add

Multiply the root \(\frac{1}{2}\) by the leading coefficient 4 and place it under the next coefficient (-2). Add -2 and 2 to get 0. Continue the process by multiplying \(\frac{1}{2}\) by the result (0) and add to the next coefficient (0). Repeat this for all coefficients. You should have:- Multiply \(\frac{1}{2}\) by 4, add to -2, to get 0.- Multiply \(\frac{1}{2}\) by 0, add to 0, to get 0.- Multiply \(\frac{1}{2}\) by 0, add to -4, to get -4.- Multiply \(\frac{1}{2}\) by -4, add to 2, to get 0.

Step 4: Interpret the Result

The numbers in the bottom row represent the coefficients of the quotient polynomial: \(4x^3 + 0x^2 + 0x - 4\). The last number (0) is the remainder.

Key Concepts

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients, structured in a way that they include terms of the form \( ax^n \).These terms are combined using addition, subtraction, and multiplication operators.Here are some key points to understand polynomials:

• The variable \( x \) can have non-negative integer exponents.
• A polynomial is usually expressed in a standard form, ordered from the highest power of \( x \) to the lowest.
• Each term in a polynomial contains a coefficient, which is a real number.
• The highest exponent of the variable in the polynomial indicates its degree. For example, a degree four polynomial will have the highest term as \( x^4 \).

In the given exercise, the expression \( 4x^4 - 2x^3 - 4x + 2 \) is a polynomial.It has four terms, and its degree is 4 because the highest power of \( x \) is 4.

Roots of Polynomials

The roots of a polynomial are the solutions where the polynomial equals zero.Finding roots is essential in simplifying and solving polynomial equations.Here are some important aspects about polynomial roots:

• For a polynomial equation \( P(x) = 0 \), the roots are the values of \( x \) that make the equation true.
• The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) will have exactly \( n \) roots, which may be real or complex, some of which might be repeated.
• In synthetic division, verifying roots is essential as the method is used to divide polynomials by linear factors \((x - r)\), where \( r \) is a root of the polynomial.

In the exercise, the divisor is \( 2x-1 \), extracted from which we find a root \( x = \frac{1}{2} \).This root is then used in the synthetic division process.

Synthetic Division Steps

Synthetic division is a handy method for dividing polynomials when the divisor is in the linear form \((x - r)\).The approach simplifies long division by focusing on coefficients.Here's how to perform synthetic division:

• First, determine the root \( r \) from the divisor, such as \((2x-1)\), which leads to \( r = \frac{1}{2} \).
• Next, list the coefficients of the polynomial in descending order of power. Don’t forget to include 0 for any missing terms.
• Place the root \( \frac{1}{2} \) to the left of the setup. Bring down the first coefficient directly to the bottom row.
• Multiply this coefficient by the root \( r \) and carry the result to the line of coefficients, then add it to the next coefficient.
• Continue the process: multiply the new number in the bottom row by \( r \) and add it to the next coefficient above, repeating till the end.
• The resulting numbers at the bottom row are the coefficients of the quotient and any remainder.

In the solution given, these steps yield the result \( 4x^3 + 0x^2 + 0x - 4 \) for the quotient, and a remainder of 0.