Problem 33 - 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. $$x^{5}-3 x^{3}+2 x-8 ; x-4$$

Step-by-Step Solution

Answer

No, \(q(x)=x-4\) is not a factor of \(p(x)=x^{5}-3 x^{3}+2 x-8\) since the remainder of the division of \(p(x)\) by \(q(x)\) is not zero.

Step 1: Identify the polynomials

The first polynomial \(p(x)\) is \(x^{5}-3 x^{3}+2 x-8\) and the second polynomial \(q(x)\) is \(x-4\). The goal is to understand if \(q(x)\) is a factor of \(p(x)\).

Step 2: Perform Polynomial Division

By dividing \(p(x)=x^{5}-3x^{3}+2x-8\) by \(q(x)=x-4\), generates the equation \((x^{5}-3x^{3}+2x-8)÷(x-4)\). We carry out the division (you can use synthetic division or long division of polynomials). The result of this division is \(x^{4}+4x^{3}+16x^{2}+64x+256\).

Step 3: Check the Remainder

The polynomial division performed in Step 2 gives us a remainder of 1028. Since the remainder is not zero, that means, \(q(x)=x-4\) is not a factor of \(p(x)=x^{5}-3x^{3}+2x-8\).

Key Concepts

Factor Theorem

The Factor Theorem is a fundamental concept in algebra that connects factors and roots of a polynomial. If you have a polynomial such as \( p(x) \), and you want to determine if \( x - a \) is a factor, the Factor Theorem tells us that:\[ p(a) = 0 \] implies \( x - a \) is a factor of \( p(x) \).
The idea is straightforward: if substituting \( x \) with a specific value (here, \( a \)) makes the polynomial equal to zero, it means \( x - a \) divides the polynomial without leaving a remainder.

In our exercise, we check if \( x - 4 \) is a factor of \( p(x) = x^5 - 3x^3 + 2x - 8 \) by evaluating \( p(4) \). Upon performing synthetic division (or polynomial division), we find that \( p(4) \) is not zero as the remainder is 1028. This implies that \( x - 4 \) is not a factor through the application of the Factor Theorem.

Remainder Theorem

The Remainder Theorem is closely linked to the Factor Theorem and offers a quick way to determine not only if a polynomial is divisible by another linear polynomial but also what the remainder is. When you divide a polynomial \( p(x) \) by a linear divisor \( x - a \), the theorem states that the remainder is simply \( p(a) \).

This makes checking for factors efficient, especially when combined with synthetic division. If you get a zero remainder, \( x - a \) is a factor. But if the remainder is not zero, like in our exercise where we obtained 1028, it confirms \( x - a \) is not a factor.

• The theorem helps identify the remainder without complete division.
• A zero remainder indicates \( x - a \) is a factor.
• The remainder is the polynomial evaluated at \( a \).

This simple result can save significant time and effort when dealing with polynomial division.

Synthetic Division

Synthetic Division is a simplified form of polynomial division, especially suited to dividing polynomials by linear factors of the form \( x - a \). Unlike long division, synthetic division does not involve writing down variables and powers, making it faster and easier to execute for many students.

To use synthetic division, you follow these steps

• Write down the coefficients of the polynomial.
• Use the opposite of the coefficient \( a \) from \( x - a \).
• Bring down the first coefficient, multiply it by \( a \), add to the next coefficient, and continue until complete.
• The last number is the remainder, while the rest are coefficients of the quotient.

In our example, synthetic division shows that dividing \( x^5 - 3x^3 + 2x - 8 \) by \( x - 4 \) yields a remainder of 1028, confirming \( x - 4 \) is not a factor. The ease and speed of synthetic division make it a valuable tool in polynomial mathematics.