Problem 51 - Elementary and Intermediate Algebra

Question

Subtract the polynomials. $$ \left(-4 h^{3}+5 h^{2}+15\right)-\left(h^{3}-15\right) $$

Step-by-Step Solution

Answer

\(-5h^3 + 5h^2 + 30\)

Step 1: Understand the problem

We need to subtract the second polynomial \((h^3 - 15)\) from the first polynomial \((-4h^3 + 5h^2 + 15)\). To do this, we'll distribute the negative sign across the second polynomial.

Step 2: Distribute the negative sign

Distribute the negative sign across the terms of the second polynomial: \((h^3 - 15)\) becomes \(-h^3 + 15\). This is equivalent to changing the sign of each term in the polynomial.

Step 3: Combine like terms

Write the expression as \(-4h^3 + 5h^2 + 15 - h^3 + 15\). Now, combine the like terms from both polynomials:- Combine the \(h^3\) terms: \(-4h^3 - h^3 = -5h^3\).- There are no other \(h^2\) terms, so \(5h^2\) stays as is.- Combine the constant terms: \(15 + 15 = 30\).

Step 4: Simplify the expression

Put it all together to form the simplified polynomial: \(-5h^3 + 5h^2 + 30\).

Key Concepts

Distributing Negative Sign

When subtracting one polynomial from another, the first crucial step is to distribute the negative sign across the terms of the polynomial being subtracted. Imagine the negative sign as a small multiplier of \(-1\) that affects every term inside the parenthesis. If you have a polynomial, say \((h^3 - 15)\), and it's being subtracted, you need to change the signs of all its terms as follows:

• \(h^3\) becomes \(-h^3\)
• \(-15\) becomes \(+15\)

This step is crucial as it correctly flips the sign of each term, ensuring that the subtraction is executed correctly. Keep in mind that failure to distribute the negative sign will lead to incorrect results in your polynomial subtraction.

Combining Like Terms

Once you've distributed the negative sign, the next step in subtracting polynomials is to combine like terms. Like terms are terms that have the same variable raised to the same power. In the expression\(-4h^3 + 5h^2 + 15 - h^3 + 15\), you will tackle this task by identifying and then combining them:

• For \(h^3\) terms: \(-4h^3\) and \(-h^3\) combine to make \(-5h^3\).
• The \(h^2\) term \(5h^2\) has no other terms to combine with, so it remains the same.
• Constant terms like \(15 + 15 = 30\), added together, form a single term.

By combining like terms, you're grouping similar variables, which simplifies the polynomial expression and makes it easier to understand integer sums or differences.

Simplifying Polynomial Expressions

To finalize the polynomial subtraction process, the expression needs to be simplified by putting all combined terms together. Simplifying helps in obtaining the most straightforward form of the polynomial expression after all operations have been carefully applied. For the polynomial \(-5h^3 + 5h^2 + 30\), the process involved:

• Gathering all like terms into one compact expression.
• Ensuring that operations such as addition and subtraction have been accurately performed.
• Looking for a chance to factor further, though in this case, the expression is already in simple form.

Each component, such as \(-5h^3\), \(5h^2\), and the constant \(30\), are ordered starting from the highest degree to the lowest, ensuring clarity and a standard form. Always verify that you've combined terms correctly to avoid mistakes in your solutions.