Problem 38 - College Algebra

Question

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$ x-2,4 x^{3}-3 x^{2}-8 x+4 $$

Step-by-Step Solution

Answer

No, \( x - 2 \) is not a factor of the polynomial.

Step 1: Set up synthetic division

First, write down the divisor, which is 2 (since the divisor is in the form \( x - c \), we use 2 for \( c \)). Beside it, list the coefficients of the dividend polynomial \( 4x^3 - 3x^2 - 8x + 4 \), which are 4, -3, -8, and 4.

Step 2: Perform the synthetic division

Bring down the leading coefficient, which is 4. Multiply it by 2 and write the result (8) under the next coefficient (-3). Add -3 and 8 to get 5. Then multiply 5 by 2 to get 10, place it under -8, and add to get 2. Finally, multiply 2 by 2 to get 4, place it under the last coefficient (4), and add to get 8.

Step 3: Interpret the result

The numbers at the bottom are the coefficients of the quotient polynomial, and the last number is the remainder. In this case, the quotient is \( 4x^2 + 5x + 2 \) and the remainder is 8.

Step 4: Check for factorization

Since the remainder is not zero, \( x - 2 \) is not a factor of \( 4x^3 - 3x^2 - 8x + 4 \).

Key Concepts

Factorization

Factorization is a fundamental concept in algebra and is essential for simplifying polynomials. It involves breaking down a complex expression into simpler components—often into products of its factors. When dealing with polynomials, factorization helps simplify equations and find roots more efficiently.

• Powers sum: Often begins with the identification of the polynomial's highest power.
• Common factor extraction: Identify common factors in the polynomial terms to reduce the complexity.
• Recognizing patterns: Be vigilant for standard forms like the difference of squares.

In the given exercise, the focus is to determine if the expression \( x-2 \) can be a factor of the polynomial. If successful, the polynomial should have no remainder when divided. Remember, the factorization in polynomials works by expressing the original polynomial as a product of its factors if the remainder is zero.

Polynomial Division

Polynomial division is a method similar to long division but applied to algebraic polynomials. It's a systematic approach for dividing a polynomial by another, providing quotient and remainder.

• Synthetic Division: This simplified form of polynomial division is quick and less prone to error, especially when dividing by linear expressions like \( x-c \).
• Leading coefficient: In synthetic division, this is the first entry in the row and often brought down immediately.
• Structure: Follows steps of multiplication and addition repeatedly throughout the operation.

In the example provided, synthetic division aids in systematically dividing \( 4x^3 - 3x^2 - 8x + 4 \) by \( x-2 \), swiftly revealing whether \( x-2 \) is a factor of the polynomial.

Remainder Theorem

The Remainder Theorem is a powerful tool in algebra that allows you to find the remainder of a polynomial division quickly and confirm factor presence.

• Evaluation: States that the remainder of a polynomial \( f(x) \) divided by \( x-c \) is simply \( f(c) \).
• Factor Confirmation: If the remainder is zero, \( x-c \) is a factor of \( f(x) \).
• Application: Adds efficiency to division problems where factors need to be checked.

In our exercise, the remainder from synthetic division is 8. According to the Remainder Theorem, since it is not zero, \( x-2 \) is confirmed not to be a factor of the polynomial \( 4x^3 - 3x^2 - 8x + 4 \). This illustrates the direct applicability and usefulness of the theorem in factorization tasks.