Solve (x-8)(x+y): Binomial Expression Multiplication

Q: Why do I get 4 terms when multiplying two binomials?

Each binomial has 2 terms, so when you multiply them, you get $ 2 \times 2 = 4 $ products! This is why $ (x-8)(x+y) $ gives you four separate terms.

Q: How do I keep track of the signs correctly?

Remember that the sign is part of the term! So $ (x-8) $ really means $ (x + (-8)) $. When you multiply $ (-8) \times x = -8x $ and $ (-8) \times y = -8y $.

Q: Can I combine any of these terms?

No! The terms $ x^2 $, $ xy $, $ -8x $, and $ -8y $ all have different variables or powers, so they cannot be combined.

Q: How can I check if my answer is right?

Try substituting simple values! If $ x=1 $ and $ y=1 $, then $ (1-8)(1+1) = (-7)(2) = -14 $. Your expanded form should also equal $ 1+1-8-8 = -14 $ ✓

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Open parentheses properly, multiply each factor by each factor

00:21 Calculate the multiplications

00:43 Positive times negative always equals negative

00:47 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Solve the following problem:

$(x-8)(x+y)=$

2

Step-by-step solution

Let's simplify the given expression, using the expanded distribution law in order to open the parentheses :

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Note that in the formula template for the above distribution law, we take by default that the operation between the terms inside of the parentheses is addition. We must remember that the sign preceding the term is an inseparable part of it. We'll also apply the rules of sign multiplication and thus we can present any expression inside of the parentheses. We'll open the parentheses using the above formula, first as an expression where addition operation exists between all terms:

$(x-8)(x+y)\\ (\textcolor{red}{x}+\textcolor{blue}{(-8)})(x+y)\\$ Proceed to open the parentheses:

$(\textcolor{red}{x}+\textcolor{blue}{(-8)})(x+y)\\ \textcolor{red}{x}\cdot x+\textcolor{red}{x}\cdot y+\textcolor{blue}{(-8)}\cdot x +\textcolor{blue}{(-8)}\cdot y\\ x^2+xy-8x -8y$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Note that in the expression that we obtained in the last stage there are four different terms, this is due to the fact that there isn't even one pair of terms where the variables (different ones) have the same exponent. Additionally the expression is already organized therefore the expression that we obtain is the final and most simplified form:
$\textcolor{purple}{ x^2}\textcolor{green}{+xy}-8x \textcolor{orange}{-8y}\\$ We highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term is an inseparable part of it,

Therefore the correct answer is answer A.

3

Final Answer

$x^2+xy-8x-8y$

Key Points to Remember

Essential concepts to master this topic

Rule: Each term multiplies each term: (a+b)(c+d) = ac+ad+bc+bd
Technique: For (x-8)(x+y), calculate x·x + x·y + (-8)·x + (-8)·y
Check: Count terms in answer: x² + xy - 8x - 8y has 4 terms ✓

Common Mistakes

Avoid these frequent errors

Not multiplying all terms together
Don't just multiply x·x and -8·y = incomplete expansion! This misses the middle terms xy and -8x, giving the wrong answer. Always ensure each term in the first binomial multiplies each term in the second binomial.

Practice Quiz

Test your knowledge with interactive questions

$(3+20)\times(12+4)=$

FAQ

Everything you need to know about this question

Why do I get 4 terms when multiplying two binomials?

+

Each binomial has 2 terms, so when you multiply them, you get $2 \times 2 = 4$ products! This is why $(x-8)(x+y)$ gives you four separate terms.

How do I keep track of the signs correctly?

+

Remember that the sign is part of the term! So $(x-8)$ really means $(x + (-8))$ . When you multiply $(-8) \times x = -8x$ and $(-8) \times y = -8y$ .

Can I combine any of these terms?

+

No! The terms $x^2$ , $xy$ , $-8x$ , and $-8y$ all have different variables or powers, so they cannot be combined.

What's the difference between this and FOIL?

+

FOIL is just a memory trick for this same process! First: x·x, Outer: x·y, Inner: (-8)·x, Last: (-8)·y. Both methods give the same answer.

How can I check if my answer is right?

+

Try substituting simple values! If $x=1$ and $y=1$ , then $(1-8)(1+1) = (-7)(2) = -14$ . Your expanded form should also equal $1+1-8-8 = -14$ ✓

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Solve (x-8)(x+y): Binomial Expression Multiplication

Distributive Property with Binomial Multiplication

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