Expand the Expression: (a-5b)(7a+3b) Using Binomial Multiplication

Q: What does FOIL stand for and why do I need it?

FOIL stands for First, Outer, Inner, Last. It helps you systematically multiply two binomials without missing any terms. For $ (a-5b)(7a+3b) $, you get 4 products that you then combine.

Q: Why is my middle term different from the answer choices?

You probably made a sign error! Remember that $ a \cdot 3b = +3ab $ but $ -5b \cdot 7a = -35ab $. When you combine these: 3ab - 35ab = -32ab, not +32ab.

Q: How do I keep track of all the negative signs?

Write out each multiplication step carefully: $ a(7a) + a(3b) + (-5b)(7a) + (-5b)(3b) $. Notice that -5b times anything gives a negative result!

Q: Can I use a different method instead of FOIL?

Yes! You can use the distributive property by distributing each term in the first binomial to the entire second binomial. Both methods give the same result when done correctly.

Q: What if I get a different answer than the choices?

Double-check your like term combination. The most common error is in combining the middle terms: $ 3ab - 35ab $. Make sure you're subtracting, not adding!

Binomial Multiplication with Negative Terms

Solve the following problem:

$(a-5b)(7a+3b)=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solution

00:05 Open parentheses properly, multiply each factor by each factor

00:30 Calculate the multiplications

01:00 Collect the factors

01:17 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following problem:

$(a-5b)(7a+3b)=$

Step-by-step solution

In order to solve the given problem, we'll follow these steps:

Step 1: Use the distributive property to expand the expression.
Step 2: Combine like terms to simplify the expression.

Let's proceed to work through each step:

Step 1: Begin by applying the distributive property to expand the expression:

$(a-5b)(7a+3b) = a(7a) + a(3b) - 5b(7a) - 5b(3b)$

Calculate each term:

$a \cdot 7a = 7a^2$
$a \cdot 3b = 3ab$
$-5b \cdot 7a = -35ab$
$-5b \cdot 3b = -15b^2$

Merge together, as follows:

$7a^2 + 3ab - 35ab - 15b^2$

Step 2: Combine like terms:

The terms $3ab$ and $-35ab$ are like terms, hence we combine them:

$7a^2 + (3ab - 35ab) - 15b^2 = 7a^2 - 32ab - 15b^2$

Therefore, the solution to the problem is $7a^2 - 32ab - 15b^2$ .

Final Answer

$7a^2-32ab-15b^2$

Key Points to Remember

Essential concepts to master this topic

FOIL Method: First, Outer, Inner, Last terms multiply systematically
Technique: $a \cdot 7a = 7a^2$ and $-5b \cdot 3b = -15b^2$
Check: Combine like terms $3ab - 35ab = -32ab$ for final answer ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute negative signs to all terms
Don't multiply -5b by only the first term (7a) = incomplete expansion! This creates missing terms and wrong coefficients. Always distribute each term in the first binomial to every term in the second binomial, keeping track of negative signs.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

What does FOIL stand for and why do I need it?

FOIL stands for First, Outer, Inner, Last. It helps you systematically multiply two binomials without missing any terms. For $(a-5b)(7a+3b)$ , you get 4 products that you then combine.

Why is my middle term different from the answer choices?

You probably made a sign error! Remember that $a \cdot 3b = +3ab$ but $-5b \cdot 7a = -35ab$ . When you combine these: 3ab - 35ab = -32ab, not +32ab.

How do I keep track of all the negative signs?

Write out each multiplication step carefully: $a(7a) + a(3b) + (-5b)(7a) + (-5b)(3b)$ . Notice that -5b times anything gives a negative result!

Can I use a different method instead of FOIL?

Yes! You can use the distributive property by distributing each term in the first binomial to the entire second binomial. Both methods give the same result when done correctly.

What if I get a different answer than the choices?

Double-check your like term combination. The most common error is in combining the middle terms: $3ab - 35ab$ . Make sure you're subtracting, not adding!

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