The Correct Answer and Explanation is:
Of course. Based on the provided image, here is the solution and a detailed explanation for one of the central problems on the worksheet, which involves multiplying binomials.
Problem: Multiply each of the following polynomials: (x – 3)(x + 7)
Correct Answer: x² + 4x – 21
Explanation
To solve the problem (x – 3)(x + 7), we need to multiply two binomials. A common and effective method for this is the FOIL technique. FOIL is an acronym that helps you remember the steps for multiplying the terms in the correct order. It stands for First, Outer, Inner, and Last. This process is an application of the distributive property, ensuring every term in the first binomial is multiplied by every term in the second one.
Let’s break down the process step by step using the FOIL method.
- First: Multiply the first term of each binomial together. In this case, we multiply x from the first binomial by x from the second binomial.
x * x = x² - Outer: Multiply the two outermost terms. This means we take the first term of the first binomial and the last term of the second.
x * 7 = 7x - Inner: Multiply the two innermost terms. This involves the last term of the first binomial and the first term of the second. Be sure to include the negative sign with the 3.
-3 * x = -3x - Last: Multiply the last term of each binomial together. Again, remember the negative sign.
-3 * 7 = -21
After completing these four multiplication steps, we have four separate terms: x², 7x, -3x, and -21. The next step is to combine these terms into a single expression and simplify it by combining any like terms. Like terms are terms that have the same variable raised to the same power.
Our expression is: x² + 7x – 3x – 21
In this expression, 7x and -3x are like terms because they both contain the variable x to the first power. We can combine them:
7x – 3x = 4x
The term x² and the constant -21 do not have any other like terms, so they remain as they are. After combining the like terms, we write the final simplified polynomial in standard form, from the highest power to the lowest.
This gives us the final answer: x² + 4x – 21
