Expand (2x-y)(4-3x): Solving Binomial Multiplication Step-by-Step

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

Each term in the first binomial must multiply with each term in the second binomial. With 2 terms × 2 terms, you get 4 products total before combining like terms.

Q: How do I keep track of positive and negative signs?

Treat the sign as part of each term. So $ (2x-y) $ becomes $ (2x + (-y)) $. Then use sign multiplication rules: positive × negative = negative.

Q: What if I don't have any like terms to combine?

That's completely normal! In this problem, all four terms ($ -6x^2, 3xy, 8x, -4y $) are different, so the expanded form is your final answer.

Q: Should I arrange terms in a specific order?

Yes! Arrange from highest degree to lowest degree: $ x^2 $ terms first, then $ xy $ terms, then $ x $ terms, then constants.

Q: How do I know which terms are like terms?

Like terms have identical variable parts with the same exponents. For example, $ 3x^2 $ and $ -5x^2 $ are like terms, but $ x^2 $ and $ xy $ are not.

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

00:27 Calculate the multiplications

00:55 Positive times negative always equals negative

01:10 Arrange the expression

01:17 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

$(2x-y)(4-3x)=$

2

Step-by-step solution

Let's simplify the given expression by factoring the parentheses using the expanded distributive law:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$ Note that that the sign before the term is an inseparable part of it.

We will also apply the laws of sign multiplication and thus we can present any term in parentheses to make things simpler.

$(2x-y)(4-3x)\\ (\textcolor{red}{2x}+\textcolor{blue}{(-y)})(4+(-3x))\\$ Let's start then by opening the parentheses:

$(\textcolor{red}{2x}+\textcolor{blue}{(-y)})(4+(-3x))\\ \textcolor{red}{2x}\cdot 4+\textcolor{red}{2x}\cdot(-3x)+\textcolor{blue}{(-y)}\cdot 4+\textcolor{blue}{(-y)} \cdot(-3x)\\ 8x-6x^2-4y+3xy$ In the operations above we used the sign multiplication laws, and the exponent law for multiplying terms with identical bases:

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

In the next step we will combine similar terms. We will define similar terms as terms in which the variables, in this case, x and y, have identical powers (in the absence of one of the unknowns from the expression, we will relate to its power as zero power, since raising any number to the power of zero will yield the result 1).

We will arrange the expression from the highest power to the lowest from left to right (we will relate to the free term as the power of zero),

Note that in the expression we received in the last step there are four different terms, since there is not even one pair of terms in which the unknowns (the variables) have the same power, so the expression we already received, is the final and most simplified expression.

We will settle for arranging it again from the highest power to the lowest from left to right:
$\textcolor{purple}{ 8x}\textcolor{green}{-6x^2}-4y\textcolor{orange}{+3xy}\\ \textcolor{green}{-6x^2}\textcolor{orange}{+3xy}\textcolor{purple}{ +8x}-4y\\$ We highlighted the different terms using colors, and as already emphasized before, we made sure that the sign before the term is correct.

We thus received that the correct answer is answer D.

3

Final Answer

$-6x^2+3xy +8x-4y$

Key Points to Remember

Essential concepts to master this topic

FOIL Method: Multiply First, Outer, Inner, Last terms systematically
Technique: $2x \cdot (-3x) = -6x^2$ using sign rules
Check: Count terms: should have 4 before combining like terms ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply all four term combinations
Don't skip the cross-multiplication of inner and outer terms = missing $3xy$ term! Students often only multiply matching variables and miss mixed terms. Always use FOIL to multiply every term in the first binomial by every 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 four terms when multiplying two binomials?

+

Each term in the first binomial must multiply with each term in the second binomial. With 2 terms × 2 terms, you get 4 products total before combining like terms.

How do I keep track of positive and negative signs?

+

Treat the sign as part of each term. So $(2x-y)$ becomes $(2x + (-y))$ . Then use sign multiplication rules: positive × negative = negative.

What if I don't have any like terms to combine?

+

That's completely normal! In this problem, all four terms ( $-6x^2, 3xy, 8x, -4y$ ) are different, so the expanded form is your final answer.

Should I arrange terms in a specific order?

+

Yes! Arrange from highest degree to lowest degree: $x^2$ terms first, then $xy$ terms, then $x$ terms, then constants.

How do I know which terms are like terms?

+

Like terms have identical variable parts with the same exponents. For example, $3x^2$ and $-5x^2$ are like terms, but $x^2$ and $xy$ are not.

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Expand (2x-y)(4-3x): Solving Binomial Multiplication Step-by-Step

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