Expand and Simplify: (3b+7a)(-5a+2b) Binomial Multiplication

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

Each term in the first binomial must multiply each term in the second binomial. With 2 terms × 2 terms, you get 4 separate products that you then combine.

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

Treat the sign as part of each term. So (3b + 7a)(−5a + 2b) has terms: +3b, +7a, −5a, +2b. Then apply sign rules: positive × negative = negative.

Q: What does 'like terms' mean when combining?

Like terms have exactly the same variables with the same exponents. For example: −15ab and +14ab are like terms, but 6b² and −35a² are not.

Q: Can I use FOIL for any binomial multiplication?

Yes! FOIL (First, Outer, Inner, Last) works for any two binomials. Just remember: First terms, Outer terms, Inner terms, Last terms.

Q: Why is my final answer different from the original expression?

That's the point! You're expanding the factored form into standard form. $ (3b+7a)(-5a+2b) $ and $ 6b^2-35a^2-ab $ are equivalent expressions.

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

00:33 Calculate the multiplications

01:03 Arrange the expression, group like terms

01:15 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:

$(3b+7a)(-5a+2b)=$

2

Step-by-step solution

Let's simplify the given expression by opening the parentheses using the extended distribution law:

$(\textcolor{red}{t}+\textcolor{blue}{k})(c+d)=\textcolor{red}{t}c+\textcolor{red}{t}d+\textcolor{blue}{k}c+\textcolor{blue}{k}d$

Note that in the formula template for the above distribution law, we take as a default that the operation between the terms inside the parentheses is addition. 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, which we'll open using the above formula, first as an expression where addition operation exists between all terms:

$(3b+7a)(-5a+2b) \\ (\textcolor{red}{3b}+\textcolor{blue}{7a})((-5a)+2b)\\$ Let's begin by opening the parentheses:

$(\textcolor{red}{3b}+\textcolor{blue}{7a})((-5a)+2b)\\ \textcolor{red}{3b}\cdot (-5a)+\textcolor{red}{3b}\cdot2b+\textcolor{blue}{7a}\cdot (-5a) +\textcolor{blue}{7a}\cdot2b\\ -15ab+6b^2-35a^2+14ab$

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

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

In the next step, we'll combine like terms, which we define as terms where the variable(s) (or each variable separately), in this case an and b, have identical exponents. (In the absence of one of the variables from the expression, we'll consider its exponent as zero power, given that raising any number to the zero power yields 1) We'll apply the commutative law of addition as well as arrange the expression from highest to lowest power from left to right (we'll treat the free number as zero power):
$\textcolor{purple}{-15ab}\textcolor{green}{+6b^2}-35a^2\textcolor{purple}{+14ab}\\ \textcolor{green}{6b^2}-35a^2\textcolor{purple}{-15ab}\textcolor{purple}{+14ab}\\ \textcolor{green}{6b^2}-35a^2\textcolor{purple}{-ab}$

In the combining of like terms performed above, 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,

The correct answer is answer B.

3

Final Answer

$6b^2-35a^2 -ab$

Key Points to Remember

Essential concepts to master this topic

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

Common Mistakes

Avoid these frequent errors

Forgetting to multiply all four term combinations
Don't just multiply first terms with first terms and second with second = missing half the answer! This gives incomplete results like 6b² − 35a². Always use FOIL or systematic distribution to get all four products: First×First, First×Second, Second×First, Second×Second.

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 four terms when multiplying two binomials?

+

Each term in the first binomial must multiply each term in the second binomial. With 2 terms × 2 terms, you get 4 separate products that you then combine.

How do I keep track of all the negative signs?

+

Treat the sign as part of each term. So (3b + 7a)(−5a + 2b) has terms: +3b, +7a, −5a, +2b. Then apply sign rules: positive × negative = negative.

What does 'like terms' mean when combining?

+

Like terms have exactly the same variables with the same exponents. For example: −15ab and +14ab are like terms, but 6b² and −35a² are not.

Can I use FOIL for any binomial multiplication?

+

Yes! FOIL (First, Outer, Inner, Last) works for any two binomials. Just remember: First terms, Outer terms, Inner terms, Last terms.

Why is my final answer different from the original expression?

+

That's the point! You're expanding the factored form into standard form. $(3b+7a)(-5a+2b)$ and $6b^2-35a^2-ab$ are equivalent expressions.

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Expand and Simplify: (3b+7a)(-5a+2b) Binomial Multiplication

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