Solve (3b+7a)(-5a+2b): Step-by-Step Binomial Multiplication

Q: Do I have to use FOIL for this problem?

FOIL works great here! First terms: $ 3b \times (-5a) = -15ab $, Outer: $ 3b \times 2b = 6b^2 $, Inner: $ 7a \times (-5a) = -35a^2 $, Last: $ 7a \times 2b = 14ab $.

Q: Why do I get -ab instead of +ab in the final answer?

You get two 'ab' terms: $ -15ab $ and $ +14ab $. When you combine them: $ -15ab + 14ab = -1ab = -ab $. The negative coefficient wins!

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

Write each step clearly! Use parentheses: $ (+3b) \times (-5a) = -15ab $ and $ (+7a) \times (-5a) = -35a^2 $. Positive times negative always equals negative.

Q: Can I rearrange the terms in my final answer?

Yes! $ 6b^2 - ab - 35a^2 $ is the same as $ -35a^2 - ab + 6b^2 $. Most teachers prefer descending degree order or alphabetical variable order.

Q: What if I accidentally multiply the same terms twice?

This happens when you're not systematic! Use a multiplication grid or carefully check off each pair as you multiply. Each term should appear exactly once in your expansion.

Binomial Multiplication with Mixed Variable Terms

Solve the following equation:

$(3b+7a)\cdot(-5a+2b)=\text{?}$

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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

00:32 Calculate the multiplications

01:02 Positive times negative always equals negative

01:13 Collect terms

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 equation:

$(3b+7a)\cdot(-5a+2b)=\text{?}$

Step-by-step solution

We will use the expanded distribution law as seen below in order to simplify the given expression:

$(a+b)(c+d)=ac+ad+bc+bd$

First, we'll perform the multiplication between the pairs of parentheses using the distribution law mentioned, and then we will proceed to combine like terms if possible. We'll do this whilst observing the correct multiplication of signs:

$(3b+7a)(-5a+2b)= \\ -15ab+6b^2-35a^2+14ab=\\ \boxed{6b^2-ab-35a^2}$ Therefore, the correct answer is answer B.

Final Answer

_{$-ab+6b^2-35a^2$}

Key Points to Remember

Essential concepts to master this topic

Distribution Law: Each term in first binomial multiplies each term in second
FOIL Method: (3b)(−5a) = −15ab, then (3b)(2b) = 6b², continue systematically
Verification: Combine like terms −15ab + 14ab = −ab, final answer has three terms ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply every term with every other term
Don't just multiply the first terms and last terms = incomplete expansion! This misses crucial middle terms like 14ab. Always use FOIL or distribution law to multiply each term in the first binomial by each term in the second binomial.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Do I have to use FOIL for this problem?

FOIL works great here! First terms: $3b \times (-5a) = -15ab$ , Outer: $3b \times 2b = 6b^2$ , Inner: $7a \times (-5a) = -35a^2$ , Last: $7a \times 2b = 14ab$ .

Why do I get -ab instead of +ab in the final answer?

You get two 'ab' terms: $-15ab$ and $+14ab$ . When you combine them: $-15ab + 14ab = -1ab = -ab$ . The negative coefficient wins!

How do I keep track of all the signs correctly?

Write each step clearly! Use parentheses: $(+3b) \times (-5a) = -15ab$ and $(+7a) \times (-5a) = -35a^2$ . Positive times negative always equals negative.

Can I rearrange the terms in my final answer?

Yes! $6b^2 - ab - 35a^2$ is the same as $-35a^2 - ab + 6b^2$ . Most teachers prefer descending degree order or alphabetical variable order.

What if I accidentally multiply the same terms twice?

This happens when you're not systematic! Use a multiplication grid or carefully check off each pair as you multiply. Each term should appear exactly once in your expansion.

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