Problem 11 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use long division to divide and use the result to factor the dividend completely. $$\left(x^{3}+5 x^{2}-12 x-36\right) \div(x+2)$$

Step-by-Step Solution

Answer

The factorization of the dividend \(x^{3}+5x^{2}-12x-36\) is \((x+2)(x-3)(x+6)\).

Step 1: Polynomial Division

Divide the given polynomial \(x^{3}+5x^{2}-12x-36\) by \(x+2\) using long division. Start by dividing the first term of the dividend \(x^{3}\) by the first term of the divisor \(x\), to get the first term of the quotient \(x^{2}\). Subtract the product of the divisor and \(x^{2}\) from the dividend to get the new dividend. Repeat this process until the degree of the remainder is less than the divisor.

Step 2: Get Quotient

After performing the polynomial division, the obtained quotient is \(x^2+3x-18\).

Step 3: Factorization

Next, factorize the quotient \(x^2+3x-18\). The expression can be written as \(x^2+3x-18=(x-3)(x+6)\).

Step 4: Factorization of the Dividend

The resulting factorization of the original dividend \(x^{3}+5x^{2}-12x-36\) is thus \((x+2)(x-3)(x+6)\). This is obtained by multiplying the divisor \(x+2\) by the factors of the quotient.

Key Concepts

Long Division

Polynomial long division is quite similar to numerical long division. When dividing two polynomials, our goal is to break down a complex polynomial (the dividend) by a simpler polynomial (the divisor). The result is a quotient and sometimes a remainder, just like in regular division.

To start with long division, consider each term of the polynomials. Divide the first term in the dividend by the first term in the divisor. This gives us the first term of the quotient.

• Divide: In our example, divide the term \(x^3\) by \(x\), resulting in \(x^2\).
• Multiply: Multiply the entire divisor by this term of the quotient (\(x^2\) \(\times\) \((x+2) = x^3 + 2x^2\)).
• Subtract: Subtract this product from the original polynomial.

Repeat this process until what remains of the dividend is a polynomial whose degree is less than that of the divisor.

This technique helps arrange the terms logically and build the quotient correctly from higher to lower degrees.

Factorization

Factorization is about expressing a polynomial as a product of simpler polynomials. After getting the quotient from polynomial division, the next step is breaking it down into its simplest parts.

For example, if we have the quotient \(x^2 + 3x - 18\), our task is to look for two numbers whose product is the constant term (\(-18\)) and whose sum is the coefficient of the middle term (\(3\)).

• Factor: For \(x^2 + 3x - 18\), the numbers \(6\) and \(-3\) fit these requirements, enabling us to factor it as \((x - 3)(x + 6)\).

Factorization is a crucial step because it allows us to see the individual root factors of the polynomial. Each factor represents a "solution" or "root" of the polynomial when set equal to zero.

Once the quotient is fully factorized, we can multiply these factors by the original divisor to get a complete factorization of the initial dividend. This uncovers the structure of the polynomial, leading us towards solving polynomial equations effectively.

Quotient

In the context of polynomial division, the quotient is what you get when you divide the dividend by the divisor without considering any remainder. It's essential to build up the quotient correctly, as it represents the simplified version of the original polynomial divided by the divisor.

Through the long division process, we sequentially determine each term of the quotient. It's important to match the degrees and terms accurately. In our exercise, after dividing \(x^3 + 5x^2 - 12x - 36\) by \(x + 2\), the quotient we get is \(x^2 + 3x - 18\).

• The quotient provides insight into how the original polynomial is structured in relation to the divisor.
• It simplifies our ability to factorize the dividend further.

Understanding the quotient forms the foundational step before moving to further factorization. It simplifies complex expressions, making polynomial equations more approachable.

Remainder

In polynomial division, sometimes a remainder appears when the dividend cannot be neatly divided by the divisor, especially when the degrees don't match perfectly. The remainder has a lower degree than the divisor and represents the "leftover" part of the dividend.

When performing long division, the remainder is what's left after subtracting out all the multiples of the divisor from the dividend. To reach a remainder, you must stop the division process when the degree of the new dividend formed during subtraction is less than that of the divisor.

• Our exercise concluded without a remainder, aiming for a clean factorization.
• Absence of a remainder implies that the divisor is a factor of the dividend.

The remainder is important because if it is not zero, it indicates that the divisor is not a perfect factor. Knowing the remainder helps us verify the completion and correctness of the division process. Concluding the division with no remainder—as in our problem—assists in validating that the factorization was accurate and complete.