Problem 1 - College Algebra

Question

If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?

Step-by-Step Solution

Answer

The binomial is a factor of the polynomial.

Step 1: Understand the Polynomial Division

In polynomial division, we often divide a polynomial, say \( P(x) \), by a binomial, say \( (x - a) \). The result of this division can be represented in the form \( P(x) = (x - a)Q(x) + R(x) \), where \( Q(x) \) is the quotient and \( R(x) \) is the remainder.

Step 2: Set Condition for Remainder

In the given problem, the remainder after division is zero. This implies \( R(x) = 0 \). Therefore, the equation becomes \( P(x) = (x - a)Q(x) \).

Step 3: Deduce Meaning of Zero Remainder

Since \( R(x) = 0 \), it indicates that \( (x - a) \) perfectly divides \( P(x) \). Hence, \( (x - a) \) is a factor of \( P(x) \). In the context of roots, \( a \) is a root of the polynomial \( P(x) \).

Step 4: Conclude the Factor Theorem

Based on the above deduction, we can conclude that if the remainder is zero, it confirms the Factor Theorem, which states that a polynomial \( P(x) \) has a factor \( (x-a) \) if and only if \( a \) is a root of \( P(x) \), meaning \( P(a) = 0 \).

Key Concepts

Polynomial Division

Polynomial division is analogous to long division with numbers, but instead it uses polynomials. When you divide a polynomial, let's call it \( P(x) \), by a binomial of the form \( (x - a) \), you can express the division in this format: \( P(x) = (x - a)Q(x) + R(x) \).
This equation tells us that the original polynomial \( P(x) \) is the product of the divisor \( (x - a) \) and the quotient \( Q(x) \), plus the remainder \( R(x) \).

• \( Q(x) \) is the quotient obtained after the division.
• \( R(x) \) is the remainder left after the division.

When dividing, our aim is often to see if the division yields a zero remainder, which is crucial in determining certain properties about \( P(x) \) and \( (x - a) \). This leads us to the relevance of roots in polynomial division.

Roots of Polynomials

Roots often play a central role in understanding and manipulating polynomials. A root of a polynomial is a value of \( x \) that satisfies the polynomial equation, meaning the entire polynomial equals zero.
If \( a \) is a root of the polynomial \( P(x) \), then \( P(a) = 0 \). This is because when you substitute \( a \) into \( P(x) \), the result yields zero.
When you divide a polynomial by a binomial \((x - a)\) and get a zero remainder, it confirms that \( a \) is indeed a root of \( P(x) \).
Here's why:

• Since \( P(a) = 0 \), the remainder \( R(x) \) must be zero when \( P(x) \) is divided by \( (x - a) \).
• This relationship is utilized in the Factor Theorem, linking factors and roots closely together.

Recognizing roots helps in evaluating, simplifying, and factoring polynomials concisely.

Zero Remainder

When a polynomial \( P(x) \) is divided by \( (x - a) \) and results in a zero remainder, it signifies that \( (x - a) \) is a factor of the polynomial \( P(x) \). This means \( P(x) \) can be completely divided by \( (x - a) \) without any leftover terms.
The zero remainder confirms an important mathematical principle known as the Factor Theorem, which relates to finding roots and factors of polynomials.
This theorem states:

• A polynomial \( P(x) \) has \( (x - a) \) as a factor if \( a \) is a root of \( P(x) \).
• Conversely, if a polynomial \( P(x) \) has a zero remainder when divided by \( (x - a) \), it proves that \( a \) is a root of the polynomial.

Thus, determining whether the remainder is zero clarifies both the factorization and the root identification of polynomials.