Problem 84 - Algebra 2

Question

Divide. Tell whether each divisor is a factor of the dividend. $$ \left(y^{3}-64\right) \div(y+4) $$

Step-by-Step Solution

Answer

Yes, \((y + 4)\) is a factor of \((y^3 - 64)\), the simplified result of the division is \((y-4)\)

Step 1: Understanding the difference of two cubes

The difference of cubes formula is \(A^3 - B^3 = (A-B)(A^2 + AB + B^2)\). This formula is helpful in this case because it simplifies the division process. For the current problem, \(A = y, B = 4\)

Step 2: Apply the difference of cubes formula

Let's rewrite \((y^3 - 64)\) using the difference of cubes formula: \((y^3 - 4^3) = (y-4)(y^2 + 4y + 4^2)\)

Step 3: Carry out the division

Now, divide \((y^3 - 64)\) by \((y + 4)\). \((y-4)(y^2 + 4y +16) \div (y + 4)\). After dividing, you are left with \((y-4)\)

Step 4: Determine if the divisor is a factor of the dividend

Because the division was successful without any remainder, it tells us that \((y + 4)\) is a factor of \((y^3 - 64)\)

Key Concepts

Difference of cubes

When you encounter expressions like \(y^3 - 64\), it helps to recognize that they might represent a difference of cubes. The formula for the difference of two cubes is \(A^3 - B^3 = (A-B)(A^2 + AB + B^2)\). Here, in the expression \(y^3 - 64\), we can see that \(y\) is \(A\) and \(4\) is \(B\) since \(64 = 4^3\).

Applying this formula, \(y^3 - 64\) can be expressed as \((y - 4)(y^2 + 4y + 16)\). This simplifies the polynomial division process by breaking down the expression into manageable parts. Recognizing these patterns is a key skill in algebra, making complex expressions easier to handle.

• Rewrite the expression using difference of cubes: \((y - 4)(y^2 + 4y + 16)\).
• It simplifies the division process greatly.

Factor theorem

The factor theorem is a fundamental concept in polynomial algebra. It states that \(x-c\) is a factor of a polynomial \(P(x)\) if and only if \(P(c) = 0\). This theorem is useful in checking if a divisor is indeed a factor of the original polynomial.

In the exercise, we divided \(y^3 - 64\) by \(y + 4\) and ended up with \((y-4)\) without a remainder. According to the factor theorem, since there was no remainder, \(y + 4\) is a factor of \(y^3 - 64\).

• If dividing results in zero remainder, the divisor is a factor.
• Check factor validity by substituting the root in the polynomial.
• For \(y + 4\), substitute \(-4\) into the polynomial to see if we get zero.

Synthetic division

Synthetic division is a simplified method of dividing polynomials, particularly when dealing with linear divisors like \(x - c\). It's much faster than long division and useful for quickly identifying factors.

In this specific exercise, once the expression was rewritten using the difference of cubes, long division could be simplified. However, for linear divisors, synthetic division can provide quicker confirmation.

To use synthetic division, align the coefficients of the polynomial and apply the division using the root of the divisor. For \(y+4\), you'd use \(-4\) in synthetic division, ensuring fast confirmation of the factor without performing full polynomial division.

• Simplifies division with linear divisors.
• Always use the root \(x-c\) for division.
• Provides quick checks for factor validity.