Problem 31 - Algebra A Combined Function

Question

Subtract using a vertical format. $$ \begin{array}{r} 4 z^{2}-8 z+3 \\ -\left(6 z^{2}+8 z-3\right) \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The difference is \(-2z^2 - 16z + 6\).

Step 1: Set Up the Problem

Write the subtraction problem in vertical format, lining up like terms:\[\begin{array}{r}4z^2 - 8z + 3 \-(6z^2 + 8z - 3)\\hline\end{array}\]Make sure the subtraction sign is in front of the second polynomial.

Step 2: Subtract the Constant Terms

Subtract the constant term of the second polynomial (-3) from the constant term of the first polynomial (3):\[3 - (-3) = 3 + 3 = 6\]Therefore, the constant term in the difference is 6.

Step 3: Subtract the Linear Terms

Subtract the coefficients of the linear terms (-8z) in the first polynomial from the linear term in the second ((+8z)) polynomial:\[-8z - 8z = -16z\]Therefore, the linear term in the difference is -16z.

Step 4: Subtract the Quadratic Terms

Subtract the quadratic term in the second polynomial (6z^2) from the quadratic term in the first (4z^2):\[4z^2 - 6z^2 = -2z^2\]Therefore, the quadratic term in the difference is -2z^2.

Step 5: Write the Final Difference

Combine all the terms obtained from previous steps:\[-2z^2 - 16z + 6\]This is the result of the given subtraction.

Key Concepts

Vertical Format

Polynomial subtraction can be greatly simplified using a vertical format. This format involves writing polynomials in columns that align similar types of terms, making subtraction more straightforward. To set up for subtraction:

• Align the polynomials one above the other.
• Ensure that like terms, such as those with the same degree of polynomial, are in the same vertical line.
• Place the subtraction sign in front of the polynomial being subtracted.

This organization helps in clearly visualizing the subtraction process and keeps the operations organized.

Like Terms

When dealing with polynomial subtraction, identifying and aligning like terms is crucial. Like terms are terms that have the same variable raised to the same power, and only these terms can be combined algebraically.

• For example, in the polynomials we are working with, terms like \( 4z^2 \) and \( 6z^2 \) are like terms because they both contain \( z^2 \).
• Similarly, \( -8z \) and \( 8z \) are like terms because they both consist of the linear term \( z \).

By grouping and processing like terms, we simplify the subtraction process, ensuring each term type is subtracted correctly.

Step-by-Step Solution

A step-by-step solution is a methodical approach that involves breaking down the polynomial subtraction process into manageable parts. Here's an overview of the necessary steps:1. **Setup**: Arrange polynomials in vertical format, aligning the like terms to keep each operation organized.
2. **Subtract Constants**: Starting with constant terms, compute \( 3 - (-3) \) which simplifies to \( 6 \). The double negative effectively makes it addition.
3. **Subtract Linear Terms**: Next, handle the linear terms, for instance, \(-8z - 8z = -16z \).
4. **Subtract Quadratic Terms**: Finally, subtract the quadratic terms \( 4z^2 - 6z^2 = -2z^2 \).
5. **Combine**: Put the results of each step together to get the final polynomial: \(-2z^2 - 16z + 6 \).
Taking it one step at a time prevents mistakes and clarifies the process, fostering better comprehension of polynomial operations.