Problem 29 - Precalculus : Building Concepts and Connections

Question

Determine together \(q(x)\) is a factor of \(p(x)\) Here, \(p(x)\) is the first polynomial and \(q(x)\) is the second polynomial. justify your answer. $$x^{3}-7 x+6 ; x-3$$

Step-by-Step Solution

Answer

Yes, \(q(x)\) (which is \(x-3\)) is a factor of \(p(x)\) (which is \(x^3 - 7x + 6\)) as the remainder of the synthetic division is zero.

Step 1: Arrange the Polynomials

First, reorder the polynomials. The polynomials given in the problem are \(p(x) = x^3 - 7x + 6\) and \(q(x) = x-3\). Here, the goal is to find out whether \(q(x)\) is a factor of \(p(x)\).

Step 2: Setting Polynomial q(x) equal to 0

Set \(q(x) = x-3 = 0\) to solve for x. This gives \(x = 3\). This is the root that will be used for synthetic division.

Step 3: Synthetic Division

Perform the synthetic division using the root found in step 2. Below is the synthetic division illustration: \n\n3 | 1 0 -7 6 \n 3 9 6 \n ------------- \n 1 3 2 0 \n\nThe coefficients in the last row are the coefficients of the quotient obtained by dividing by \(x-3\) and the last number is the remainder after division. The zero remainder indicates that \(x-3\) is indeed a factor of the given polynomial.

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing a polynomial by a binomial of the form \(x - c\). It is particularly useful because it requires fewer calculations than traditional long division. The process begins by writing down the coefficients of the polynomial you wish to divide. In our exercise, for \(p(x) = x^3 - 7x + 6\), the coefficients used are: 1, 0 (for the missing \(x^2\) term), -7, and 6.

Next, we use the root derived from setting \(q(x) = x - 3\) to zero, which gives \(x = 3\). This root is then applied as the divisor in synthetic division. Arrange the coefficients in a row and bring down the leading coefficient. Multiply this number by 3 and add it to the next coefficient. Continue this process across all coefficients.

• The remainder after the division indicates how well \(q(x)\) fits into \(p(x)\).
• If the remainder is zero, it signals that the division resulted in a whole number, showing that \(q(x)\) is a factor.
• In the exercise, the remainder is zero, confirming \(x - 3\) is a factor of the polynomial.

Polynomial Factors

Polynomial factors are expressions that can be multiplied to obtain the original polynomial. When a polynomial \(p(x)\) is divisible by another polynomial \(q(x)\) without leaving a remainder, \(q(x)\) is considered a factor of \(p(x)\). This concept is crucial for simplifying polynomials and finding roots.

In our problem, we needed to check if \(x-3\) is a factor of \(x^3 - 7x + 6\). Through synthetic division, it became evident that \(x-3\) divides evenly into the polynomial, yielding a zero remainder. Thus, \(x-3\) is confirmed as a factor of the polynomial.

• Having the factors helps in breaking down complex expressions.
• Once factors are identified, substitution or further simplification becomes manageable.

Remainder Theorem

The Remainder Theorem is a powerful tool for evaluating polynomial expressions, especially in determining factors. It states that for a polynomial \(p(x)\), divided by \(x-c\), the remainder of this division is equivalent to \(p(c)\). This theorem simplifies the checking for factors since if \(p(c) = 0\), \(x-c\) is indeed a factor of \(p(x)\).

Applying the theorem to our exercise, we set \(c = 3\), since \(q(x) = x - 3\). After performing the polynomial division using synthetic division, the remainder was 0, which matches the Remainder Theorem. This zero remainder indicates that substituting \(x = 3\) into the polynomial \(p(x)\) would result in zero, confirming \(x-3\) as a valid factor.

• This theorem provides a quick check for factors, bypassing the lengthy division process.
• It saves time and simplifies factorization tasks in polynomial algebra.