Problem 51 - Elementary Algebra

Question

\(\left(9 z^{2}+12 z+4\right) \div(3 z+2)\)

Step-by-Step Solution

Answer

3z + 2

Step 1: Set up the long division

Write down the polynomial division as a long division problem. Divide \[9z^2 + 12z + 4\] by \[3z + 2\].

Step 2: Divide the first terms

Divide the first term of the numerator (9z²) by the first term of the denominator (3z). \[\frac{9z^2}{3z} = 3z\] Write 3z above the division line.

Step 3: Multiply and subtract

Multiply \[3z\] by \[3z + 2\] to get \[9z^2 + 6z\]. Subtract \[9z^2 + 6z\] from \[9z^2 + 12z + 4\] to get the new polynomial \[6z + 4\].

Step 4: Repeat the division

Divide the first term of the new polynomial (6z) by the first term of the denominator (3z). \[\frac{6z}{3z} = 2\] Write 2 above the division line next to 3z.

Step 5: Multiply and subtract again

Multiply \[2\] by \[3z + 2\] to get \[6z + 4\]. Subtract \[6z + 4\] from \[6z + 4\] to get a remainder of 0.

Step 6: Write the final answer

Since the remainder is 0, the quotient of the division is the final answer, \[3z + 2\].

Key Concepts

Polynomial Division

Polynomial division involves dividing one polynomial by another. This process is similar to long division with numbers, but here we work with variables and coefficients. The goal is to determine how many times the divisor polynomial can go into the dividend polynomial. Here's how it works:

• Set up the division problem, arranging the polynomials in decreasing order of their degrees.
• Divide the first term of the dividend by the first term of the divisor.
• Multiply the entire divisor by the result and subtract it from the dividend.
• Repeat the process with the new polynomial formed until there are no more terms to bring down.

In our exercise, we divided \(9z^2 + 12z + 4\) by \(3z + 2\). The answer we got after completing the polynomial division process is \(3z + 2\).

Long Division

Long division is a method used to divide larger numbers or polynomials. For polynomials, the process is quite similar but involves variables. Here’s a step-by-step breakdown:

• Setting Up: Arrange the terms of both polynomials. Write the dividend (the polynomial we are dividing) inside the long division symbol and the divisor (the polynomial doing the dividing) outside it.
• Divide Terms: Divide the first term of the dividend by the first term of the divisor, then write the result on top.
• Multiply and Subtract: Multiply the entire divisor by the term obtained and subtract it from the dividend. Bring down the next term.
• Repeat: Continue the process until every term of the polynomial has been brought down and processed.

In our example, we divided each term step-by-step: first \(9z^2\) by \(3z\), then \(6z\) by \(3z\), giving us the quotient \(3z + 2\).

Remainder

In polynomial division, the remainder is what's left after completing the division process. If the remainder is 0, the divisor divides the dividend exactly. In our solution:

• After performing the division steps, if any terms are left that cannot be divided by the divisor, this is the remainder.
• Sometimes the remainder can be a polynomial of lower degree than the divisor.
• The remainder is added as a term in the final answer, often expressed as \( \text{quotient} + \frac{\text{remainder}}{\text{divisor}} \).

In the given exercise, after processing all terms, we found there was no remainder, indicated by the fact that the number left after the last subtraction was 0. Therefore, \(3z + 2\) is the complete quotient with no remainder.