Question

Find the product. $$ (2.5 z-6.1)(z+4.3) $$

Short answer

\(2.5z^2 + 4.65z - 26.23 \)

Step-by-step solution

Apply the Distributive Property for the First Terms

Multiply the first terms in each binomial. So, \(2.5z \times z = 2.5z^2 \)

Apply the Distributive Property for the Outer Terms

Multiply the outer terms in the two binomials. so, \(4.3 \times 2.5z = 10.75z \)

Apply the Distributive Property for the Inner Terms

Multiply the inner terms in both binomials. That is, \(-6.1 \times z = -6.1z \)

Apply the Distributive Property for the Last Terms

Multiply the last terms in both binomials. In this case, \(-6.1 \times 4.3 = -26.23 \)

Combine Like Terms

Now, combine the like terms from Steps 2 and 3. \(10.75z + -6.1z = 4.65z \)

Write the Final Answer

So, integrating the results from Step 1, Step 5, and Step 4, we get \(2.5z^2 + 4.65z - 26.23 \)

Key concepts

Distributive Property

Understanding the distributive property is vital when it comes to multiplying binomials. It's a rule that allows us to multiply a single term by each term within a parenthesis systematically. For example, when we have \(ax(b + c)\), the distributive property tells us to multiply \(a\) by both \(b\) and \(c\) separately. The result would be \(ab + ac\).

In the given exercise, the distributive property is used to expand the product of two binomials: \(2.5z-6.1\) and \(z+4.3\). Essentially, each term of the first binomial is distributed across the terms of the second binomial.

For instance, take the first term of the first binomial (2.5z) and multiply it by each term of the second binomial (z and 4.3), yielding \(2.5z \times z\) and \(2.5z \times 4.3\) respectively. Applying the same approach to each consecutive term ensures all combinations are covered, which is crucial for correct polynomial multiplication.

Combining Like Terms

After using the distributive property, we often end up with terms that can be combined because they have the same variables raised to the same powers. These are known as 'like terms'. Combining like terms simplifies the expression and is essential to finding a final, concise answer.

In our exercise, after the binomials are multiplied, we notice that some terms have the same variable with the same exponent. Specifically, \(10.75z\) and \( -6.1z\) are like terms because both contain the variable \(z\) raised to the first power. By combining the coefficients (the numerical parts) of these like terms, we condense the expression into a more simplified form. This process yields a single term: \(4.65z\).

Combining like terms requires careful attention to signs and coefficients. It's an opportunity to reduce complexity in polynomial expressions and a step that should not be overlooked for achieving the correct answer.

Polynomial Multiplication

Multiplying polynomials involves a few steps including the distributive property and combining like terms, which we've discussed earlier. When working with polynomials, especially with binomials, we're often dealing with a method called 'FOIL'. This stands for First, Outer, Inner, Last and refers to the terms of each binomial multiplied in a specific order.

Here's how FOIL applies to our exercise: First, we multiplied 'First' terms of each binomial (\(2.5z \times z\)). Next, we moved on to 'Outer' terms (\(2.5z \times 4.3\)), then 'Inner' terms (\(-6.1 \times z\)), and finally the 'Last' terms (\(-6.1 \times 4.3\)). After these multiplications, we combined like terms to arrive at the final, simplified polynomial.

Through polynomial multiplication, we obtain an expression that is a combination of the original binomials' features, capturing their interaction in the expanded form. As seen in the exercise, the final product \(2.5z^2 + 4.65z - 26.23\) is that expanded form, expressing the result of this systematic multiplication process.