Expand (2x-3)×(5x-7): Complete Binomial Multiplication

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

FOIL stands for First, Outer, Inner, Last. It helps you remember to multiply all four combinations: (2x)(5x), (2x)(-7), (-3)(5x), (-3)(-7). Without FOIL, you might forget some multiplications!

Q: Why do I get a trinomial instead of a binomial?

When you multiply two binomials, you get four terms initially. After combining like terms (the middle x terms), you end up with three terms: $ ax^2 + bx + c $, which is called a trinomial.

Q: How do I handle the negative signs correctly?

Be extra careful with signs! Remember: positive × negative = negative and negative × negative = positive. In our problem, (-3)(-7) = +21, not -21.

Q: Can I use a different method besides FOIL?

Yes! You can use the distributive property twice, or draw a multiplication grid. FOIL is just the most popular method because it's easy to remember for binomial multiplication.

Q: How do I know if I expanded correctly?

Your final answer should be a trinomial in standard form (highest degree first). Also, try substituting a simple number like x=1 into both the original expression and your answer - they should give the same result!

Binomial Multiplication with FOIL Method

$(2x-3)\times(5x-7)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solution

00:05 Let's open parentheses properly, multiply each factor by each factor

00:30 Let's calculate the products

00:53 Let's collect the terms

01:01 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

$(2x-3)\times(5x-7)$

Step-by-step solution

To answer this exercise, we need to understand how the extended distributive property works:

For example:

(a+1)∗(b+2)

To solve this type of exercises, the following steps must be taken:

Step 1: multiply the first factor of the first parentheses by each of the factors of the second parentheses.

Step 2: multiply the second factor of the first parentheses by each of the factors of the second parentheses.

Step 3: group like terms together.

ab∗2a∗b∗2

We start from the first number of the exercise: 2x

2x*5x+2x*-7

10x²-14x

We will continue with the second factor: -3

-3*5x+-3*-7

-15x+21

We add all the data together:

10x²-14x-15x+21

10x²-29x+21

Final Answer

$10x^2-29x+21$

Key Points to Remember

Essential concepts to master this topic

FOIL Rule: First, Outer, Inner, Last terms multiply systematically
Technique: (2x)(5x) = 10x², (2x)(-7) + (-3)(5x) = -29x
Check: Count terms: quadratic + linear + constant = 10x² - 29x + 21 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply all four combinations
Don't just multiply first terms and last terms = 10x² + 21! This skips the middle term completely and gives a wrong binomial instead of a trinomial. 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 do I need it?

FOIL stands for First, Outer, Inner, Last. It helps you remember to multiply all four combinations: (2x)(5x), (2x)(-7), (-3)(5x), (-3)(-7). Without FOIL, you might forget some multiplications!

Why do I get a trinomial instead of a binomial?

When you multiply two binomials, you get four terms initially. After combining like terms (the middle x terms), you end up with three terms: $ax^2 + bx + c$ , which is called a trinomial.

How do I handle the negative signs correctly?

Be extra careful with signs! Remember: positive × negative = negative and negative × negative = positive. In our problem, (-3)(-7) = +21, not -21.

What if my middle terms don't combine nicely?

Sometimes they don't! Just write them separately if they can't be combined. But in most textbook problems like this one, the middle terms will combine to give you a clean trinomial.

Can I use a different method besides FOIL?

Yes! You can use the distributive property twice, or draw a multiplication grid. FOIL is just the most popular method because it's easy to remember for binomial multiplication.

How do I know if I expanded correctly?

Your final answer should be a trinomial in standard form (highest degree first). Also, try substituting a simple number like x=1 into both the original expression and your answer - they should give the same result!

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