Problem 53 - Algebra for College Students

Question

For problems \(53-64\), use synthetic division to determine the quotient and remainder. $$ \left(x^{2}-8 x+12\right) \div(x-2) $$

Step-by-Step Solution

Answer

The quotient is \(x - 6\) and the remainder is 0.

Step 1: Set up the Synthetic Division

Write the coefficients of the dividend polynomial \(x^2 - 8x + 12\). These are \(1, -8, 12\). The divisor \(x-2\) has a root of \(2\). Place this to the left.

Step 2: Bring Down the Leading Coefficient

Bring the leading coefficient (1) down. This will be the starting point of the quotient.

Step 3: Multiply and Add

Multiply the root \(2\) by the number just brought down (1). The product is \(2\), write this under the next coefficient \(-8\). Add \(-8\) and \(2\) to get \(-6\).

Step 4: Repeat Multiply and Add

Multiply \(2\) by \(-6\) to get \(-12\). Write this under the last coefficient (12). Add \(12\) and \(-12\) to get a remainder of \(0\).

Step 5: Write the Quotient

The numbers at the bottom (1 and -6) represent the coefficients of the quotient polynomial \(x - 6\). Therefore, the quotient is \(x - 6\) and the remainder is \(0\).

Key Concepts

Polynomial Division

Polynomial division is a method for dividing one polynomial by another. It's like long division for numbers, but applied to expressions consisting of variables and coefficients. This technique is pivotal when dealing with polynomial functions, as it helps to simplify expressions and solve polynomial equations.

• There are two major methods: long division and synthetic division. The latter is preferable when dividing by a linear divisor, such as in our example \(x - 2\).
• To perform synthetic division, only the coefficients of the polynomial are required, making the computation more straightforward and less prone to errors than long division.
• The final result of a polynomial division is expressed in the form of a quotient and a remainder.

Understanding polynomial division is vital in algebra and calculus, as it underlies many more advanced concepts.

Quotient and Remainder Theorem

The Quotient and Remainder Theorem is a fundamental concept in algebra that states when you divide a polynomial by a linear divisor, the result is a quotient and a remainder. This theorem is crucial as it tells us exactly how a polynomial decomposes.

• For any polynomial \(f(x)\), when divided by a non-zero linear polynomial \(x - r\), you will get a quotient polynomial \(q(x)\) and a remainder \(r\).
• Mathematically, this is written as: \ f(x) = (x - r)q(x) + r \.
• The remainder will always be a constant, and in the case where the divisor is a factor of the polynomial, this remainder \(r\) will be zero.

In our exercise, the remainder is zero, showing that \(x - 2\) is a factor of \(x^2-8x+12\). Understanding this theorem helps solve equations and find factorizations.

Roots of Polynomials

Finding the roots of a polynomial is akin to solving for the values of the variable that make the polynomial zero. For a quadratic polynomial like \(x^2 - 8x + 12\), these roots hold particular significance as they reveal the polynomial's factors.

• The roots represent the values for which the polynomial evaluates to zero.
• Using synthetic division, verifying the roots becomes efficient as the remainder gives a direct indication if our divisor matches a root.
• In our solution, the polynomial being divided by \(x - 2\) leads to a quotient of \(x - 6\) and remainder of zero, signifying \(x = 2\) is indeed a root.

This ability to identify roots plays a crucial role in many mathematical applications, including graph sketching and solving polynomial equations.

Polynomial Coefficients

Coefficients in polynomials are the numbers that multiply the variable terms. Understanding their role is key in performing operations like synthetic division and analyzing polynomial behavior.

• The coefficients directly influence the solution obtained when dividing polynomials, as we see in the example \(x^2 - 8x + 12\) where the coefficients are \(1, -8, 12\).
• Synthetic division simplifies computations by focusing solely on these coefficients. It reduces potential calculation errors associated with variables.
• Changes in coefficients will alter the shape and position of the polynomial's graph, affecting both roots and overall function characteristics.

By focusing on the coefficients, especially in synthetic division, we gain valuable insights into the function without unnecessary complexity.