Expand (a+15)(5+a): Solving Binomial Multiplication Step-by-Step

Q: What does FOIL stand for and why is it important?

FOIL stands for First, Outside, Inside, Last. It ensures you multiply every term with every other term: First terms (a×5), Outside terms (a×a), Inside terms (15×5), Last terms (15×a). This prevents missing any combinations!

Q: How do I know which terms are 'like terms' to combine?

Like terms have the exact same variable parts. In this problem: $ 5a $ and $ 15a $ are like terms (both have just 'a'), but $ a^2 $ is different because it has 'a squared'.

Q: Why do I get a² in my answer?

The $ a^2 $ term comes from multiplying a × a when you use FOIL. This happens when you multiply the 'a' from the first binomial with the 'a' from the second binomial. Remember: $ a \cdot a = a^2 $!

Q: Can I check my answer by substituting a number for 'a'?

Absolutely! Try a = 2: Original expression $ (2+15)(5+2) = 17 \times 7 = 119 $. Your answer: $ 2^2 + 20(2) + 75 = 4 + 40 + 75 = 119 $ ✓ Perfect match!

Q: What if the terms are in different order like (5+a)(a+15)?

The order doesn't matter! Multiplication is commutative, so $ (a+15)(5+a) $ equals $ (5+a)(a+15) $. You'll still get $ a^2 + 20a + 75 $ after combining like terms.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 Open brackets properly, multiply each factor by each factor

00:26 Calculate the multiplications

00:46 Arrange the expression, collect terms

00:55 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:

$(a+15)(5+a)=$

2

Step-by-step solution

Let's simplify the given expression, using the extended distribution law to open the parentheses :

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

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. Remember that the sign preceding the term is an inseparable part of it. Apply the rules of sign multiplication so that we can present any expression in parentheses. We'll open the parentheses using the above formula, as an expression where addition operation exists between all terms. In this expression it's clear, all terms have a plus sign prefix.

Therefore we'll proceed directly to opening the parentheses as shown below:

:

$(\textcolor{red}{a}+\textcolor{blue}{15})(5+a)\\ \textcolor{red}{a}\cdot 5+\textcolor{red}{a}\cdot a+\textcolor{blue}{15}\cdot 5 +\textcolor{blue}{15}\cdot a\\ 5a+a^2+75+15a$

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 (or variables each separately), in this case a, have identical exponents. (In the absence of one of the variables from the expression, we'll consider its exponent as zero power, this is due to the fact that any number raised to the power of zero equals 1) Apply the commutative law of addition and arrange the expression from highest to lowest power from left to right (we'll treat the free number as power of zero):
$\textcolor{purple}{5a}\textcolor{green}{+a^2}+75\textcolor{purple}{+15a}\\ \textcolor{green}{a^2}\textcolor{purple}{+5a+15a}+75\\ \textcolor{green}{a^2}\textcolor{purple}{+20a}+75\\$ 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,

We therefore got that the correct answer is answer B.

3

Final Answer

$a^2+20a+75$

Key Points to Remember

Essential concepts to master this topic

FOIL Method: Multiply First, Outside, Inside, Last terms systematically
Distribution: Each term multiplies every term: $a \cdot 5 = 5a$ , $15 \cdot a = 15a$
Combine Like Terms: Group same variables: $5a + 15a = 20a$ ✓

Common Mistakes

Avoid these frequent errors

Only multiplying some terms instead of all four combinations
Don't just multiply a×5 and 15×a = 5a + 15a! This misses the squared term and constant. You get 20a instead of the correct a² + 20a + 75. Always multiply each term in the first binomial by each term in the second binomial using FOIL.

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 is it important?

+

FOIL stands for First, Outside, Inside, Last. It ensures you multiply every term with every other term: First terms (a×5), Outside terms (a×a), Inside terms (15×5), Last terms (15×a). This prevents missing any combinations!

How do I know which terms are 'like terms' to combine?

+

Like terms have the exact same variable parts. In this problem: $5a$ and $15a$ are like terms (both have just 'a'), but $a^2$ is different because it has 'a squared'.

Why do I get a² in my answer?

+

The $a^2$ term comes from multiplying a × a when you use FOIL. This happens when you multiply the 'a' from the first binomial with the 'a' from the second binomial. Remember: $a \cdot a = a^2$ !

Can I check my answer by substituting a number for 'a'?

+

Absolutely! Try a = 2: Original expression $(2+15)(5+2) = 17 \times 7 = 119$ . Your answer: $2^2 + 20(2) + 75 = 4 + 40 + 75 = 119$ ✓ Perfect match!

What if the terms are in different order like (5+a)(a+15)?

+

The order doesn't matter! Multiplication is commutative, so $(a+15)(5+a)$ equals $(5+a)(a+15)$ . You'll still get $a^2 + 20a + 75$ after combining like terms.

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Expand (a+15)(5+a): Solving Binomial Multiplication Step-by-Step

Binomial Multiplication with Distributive Property

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