Problem 67 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{3 y^{4}+9 y^{3}-2 y^{2}-6 y+4}{y+3} $$

Step-by-Step Solution

Answer

Question: Perform the division of the polynomial \(3y^4 + 9y^3 - 2y^2 - 6y + 4\) by the polynomial \(y + 3\). Answer: \(\frac{3 y^{4}+9 y^{3}-2 y^{2}-6 y+4}{y+3} = 3y^3 - 2y + \frac{4}{y+3}\)

Step 1: Write down the division problem in long division format

The dividend is the polynomial \(3y^4 + 9y^3 - 2y^2 - 6y + 4\) and the divisor is the polynomial \(y + 3\). Set up the long division by placing the dividend inside the division symbol and the divisor on the outside.

Step 2: Divide the lead term of the dividend by the lead term of the divisor

We'll divide the lead term of the dividend (\(3y^4\)) by the lead term of the divisor (\(y\)). This gives us \(3y^3\). Write \(3y^3\) above the dividend as the first term of the quotient.

Step 3: Multiply the divisor by the result obtained in step 2

Now multiply the divisor, \(y + 3\), by \(3y^3\). This gives us \((3y^3(y + 3))\), which simplifies to \(3y^4 + 9y^3\). Write this result under the dividend.

Step 4: Subtract the result of step 3 from the dividend

Subtract the result from step 3 (\(3y^4 + 9y^3\)) from the dividend (\(3y^4 + 9y^3 - 2y^2 - 6y + 4\)). This gives us the new dividend, \(-2y^2 - 6y + 4\).

Step 5: Repeat steps 2-4 for the new dividend

Now we have a new dividend, \(-2y^2 - 6y + 4\). Repeat steps 2-4 with this new dividend: 1. Divide the lead term of the new dividend (\(-2y^2\)) by the lead term of the divisor (\(y\)) -> \(-2y\) 2. Multiply the divisor (\(y+3\)) by the result obtained above (\(-2y\)) -> \((-2y(y+3))\) -> \(-2y^2 - 6y\) 3. Subtract the result from the new dividend: \((-2y^2 - 6y + 4) - (-2y^2 - 6y)\) -> \(4\)

Step 6: Write down the final quotient and remainder

The quotient we obtained through the division process is \(3y^3 - 2y\). The remainder is \(4\). Therefore, the final answer is: $$ \frac{3 y^{4}+9 y^{3}-2 y^{2}-6 y+4}{y+3} = 3y^3 - 2y + \frac{4}{y+3} $$

Key Concepts

Dividing Polynomials

Dividing polynomials is an essential skill in algebra that involves breaking down a complex polynomial into simpler parts. The division of polynomials can be performed using various methods, with the most common being long division and synthetic division.

Think of it as a process similar to long division with numbers, where a dividend is divided by a divisor to find a quotient and sometimes a remainder. In the case of the given exercise, the dividend is the polynomial 3y^4 + 9y^3 - 2y^2 - 6y + 4, and the divisor is y + 3. To tackle this, you systematically divide each term, starting with the highest degree, and work your way down to the constant.

As with numerical division, there can be a remainder in polynomial division. The remainder is what's left over when the polynomial is not fully divisible by the divisor. In our exercise, the remainder was 4. This means that the divisor does not perfectly divide into the dividend, and so we express the answer as a quotient with an additional fraction that represents the remainder over the original divisor.

Synthetic Division

Synthetic division is a shorthand, or simplified, method of dividing polynomials which is particularly useful when the divisor is a first-degree binomial of the form x - c. It's a quick way to divide without writing the variables, their exponents, and the polynomial's terms every time which can be cumbersome, especially with larger polynomials.

Here's how synthetic division would work in the context of our exercise. Rather than using the long division format, we would use the coefficients of the dividend - 3, 9, -2, -6, and 4 - and the zero of the divisor, which is -3 in the case of y + 3. We would then perform a series of steps that involve multiplication and addition with these numbers to find the quotient and remainder.

How Synthetic Division Works:

• Write down the coefficients of the dividend.
• Write the zero of the divisor outside the synthetic division symbol.
• Bring down the leading coefficient.
• Multiply the zero of the divisor by the number you just brought down, and write the result in the next column.
• Add that result to the next coefficient of the dividend, writing the sum underneath.
• Repeat the multiplication and addition steps until all coefficients have been used.
• The numbers at the bottom are the coefficients of the quotient, and the last number represents the remainder.

Although the exercise provided uses long division, understanding synthetic division can make your work much quicker and efficient.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation - addition, subtraction, multiplication, or division. In our given exercise, 3y^4 + 9y^3 - 2y^2 - 6y + 4 is an algebraic expression that represents a polynomial with a degree of four.

Each part of an algebraic expression has a name:

• Terms: The separate elements split by the plus or minus sign, such as 3y^4 or -6y.
• Coefficients: The numerical factor of each term, like the 3 in 3y^4.
• Variables: The letters representing numbers, in this case, y.
• Exponents: The power to which the variable is raised, like the 4 in y^4.
• Constant: A term without a variable, such as 4.

Understanding how to work with algebraic expressions is crucial, as it serves as a foundation for many topics in algebra, including simplifying expressions, solving equations, and factoring polynomials. Learning to identify the parts and manipulate them following algebraic rules paves the way for a solid comprehension of polynomials and their behavior.