Problem 11 - Algebra for College Students

Question

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(3 x^{3}+8 x^{2}-8\right) \div(x+2) $$

Step-by-Step Solution

Answer

The quotient is \( 3x^2 + 2x - 4 \) with a remainder of 0.

Step 1: Setup the Synthetic Division

To use synthetic division, we first set up the process by writing the coefficients of the polynomial \(3x^3 + 8x^2 + 0x - 8\). Note the missing \(x\) term, which we represent with a 0 in the coefficients: \([3, 8, 0, -8]\). The divisor \((x + 2)\) gives a root of \(-2\), which we use in synthetic division.

Step 2: Start the Division Process

Write the number \(-2\) to represent the root on the left side. Draw a horizontal line, and bring down the first coefficient (3). This number starts the bottom row.

Step 3: Perform Multiplications and Additions

Multiply \(-2\) by the number you just brought down (3), which equals \(-6\). Write \(-6\) under the second coefficient, 8. Add these two numbers: \(8 + (-6) = 2\). Write 2 below the line.

Step 4: Continue the Multiplication and Addition

Repeat the same process: Multiply \(-2\) by the result 2 to get \(-4\). Add \(0 + (-4) = -4\), and write \(-4\) below the line.

Step 5: Final Multiplication and Addition

Multiply \(-2\) by \(-4\) to yield 8. Add \(-8 + 8 = 0\). Write 0 below the line. This final number is the remainder.

Step 6: Write the Result

The bottom row (from left to right) gives us the coefficients of the quotient: \([3, 2, -4]\), representing \(3x^2 + 2x - 4\). The remainder is 0.

Key Concepts

Polynomial Division

Polynomial division is akin to long division but is applied to algebraic expressions. Unlike numerical division, this requires careful tracking of coefficients and variables. For any polynomial of degree "n" divided by another polynomial of lesser or equal degree "m", polynomial division helps determine the quotient and remainder.

In our exercise, we have a cubic polynomial, which is a polynomial of degree three, divided by a linear polynomial, which is a polynomial of degree one. Synthetic division simplifies this process significantly by focusing purely on the coefficients of the polynomial rather than the variables themselves. By stripping the polynomial to its numerical essence, the process becomes both faster and less prone to error.

Root of Polynomial

The root of a polynomial function is a solution for which the polynomial evaluates to zero. It represents the x-value where the graph of the polynomial crosses the x-axis.

In synthetic division, the term "root" refers to the number that makes the divisor zero. For example, for a divisor \( (x + 2) \), the root is "\(-2\)". This is because when \( -2 \) is substituted for \( x \) in the divisor, the entirety becomes zero.

• The divisor \( x + 2 \) reflects the transformation into a root by using \(-2\).
• Using this root in synthetic division provides the framework to successfully divide the given polynomial.

Quotient and Remainder

During polynomial division, the outcome includes both a quotient and a remainder. The quotient is what we get after performing the division without the remainder. When you divide, just like dividing numbers, sometimes you have leftover bits - these are called the remainder in polynomial division.

For the given problem, upon division using synthetic division, our quotient results in \(3x^2 + 2x - 4\) and our remainder is zero. This means our divisor evenly divides the polynomial.

• The absence of a remainder suggests the divisor is a factor of the polynomial.
• Writing quotients and remainders helps break down otherwise complex expressions into simpler ones.

Algebraic Expressions

Algebraic expressions involve variables and numbers applied through various operations like addition, subtraction, multiplication, or division. At the heart of this structure lies polynomial expressions—combinations of variables raised to powers and multiplied by coefficients.

Understanding algebraic expressions aids in deciphering how these terms interplay during polynomial division.

• Each term in the polynomial \(3x^3 + 8x^2 - 8\) contributes to the complexity of the division operation.
• Using synthetic division to reveal the quotient \(3x^2 + 2x - 4\) highlights how algebraic behaviors translate numerically.

By effectively managing these terms, polynomial division can be approached with confidence and clarity.