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### Course: Get ready for Algebra 2>Unit 1

Lesson 7: Factoring quadratics by grouping

# Factoring quadratics: common factor + grouping

Sal factors 35k^2+100k-15 as 5(k+3)(7k-1). Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• Are polynomials the same as quadracts?
• Quadratics are a special kind of polynomial. Here are some examples of various kinds of polynomials:

(1) x^2 + 3x + 9
(2) x^3 + x^2 - 9x
(3) x^5 - 5x^3 - 2x^2 + x - 20
(4) x^10 + x - 1

While each of the above is a polynomial, only (1) is called a quadratic -- this is because its largest exponent is a 2. Another way of saying this is that (1) is a "second-degree polynomial". (2) is a "third-degree polynomial" because its largest exponent is 3 (these kinds of polynomials are often called "cubics"). (3) is a "fifth-degree polynomial", (4) would be a "tenth-degree polynomial", and so on.

So basically, the term "quadratic" is just another word for any second-degree polynomial -- or any polynomial where the largest exponent is 2.

(Also, I used x as my variable in all the above examples, but you could use any variable you like... it doesn't need to be x)
• Okay, so we found out that this factors into 5(7k-1)(k+3), but what does k equal?
How do you find out what k is when there's a 5 outside the factored terms?
• Scotty,
The original problem was
Factor 35k² + 100k - 15
This is an expression, but it is not an equation. So you cannot solve the equation for k. All you can do is factor the expression.

If instead the original problem was
35k² + 100k - 15 = 0 you could slove for k
You would first factor the left side, just like in the video and you would get
5(7k-1)(k+3) = 0
The expression on the right would be equal to zero when either (7k-1) = 0 or if (k+3) = 0 because anythng times zero is zero.
And the solutions for k would be k=-3 and k= 1/7

But that was not the solution to the expression in the video because it was not an equaltiy, all we could do is factor the expression.

Factoring an expression may not seem like it has a purpose until you can then use it to solve something, but learning how to factor expressions is the first step to solving quadratic equations.

I hope that helps you make some sense of things.
• Hi at - , Sal mentions finding a number whose product is 7 * -3. Why isn't he simply referring to the number -3 as his product?

• The product of two numbers is the number that results from multiplying two numbers together. For example, -3×7 equals -21. In this instance, -21 is the product.

I hope this helps!
• I understand the concept, but I have a problem that doesn't make sense. If the highest exponent variable doesn't have a constant before it, how would I find the common factor?
The problem is:
``y^4 + 10y^2 + 24=___``

• For now, ignore the first term and focus on the last two. Ask yourself what adds to 10 and multiplies to 24? The correct answer is 6 and 4. So the factored form of this expression would be (y^2+6x)(y^2+4x)
• is this the same as factorising using common prime factors?
• Mmm not really - I don't see what the connection is between prime factoring a number and this.
• Would using brackets be acceptable?
• Are there by any chance any videos explaining how to factor problems like:

2x^3 - 16x^2 + 24x

If so, I would appreciate if someone could provide me a link to where I can learn how to solve these types of problems, or teach me themselves.
Thanks a bunches!
(1 vote)
• 2x³ - 16x² + 24x
Step 1: Factor out the GCF, in this case it is 2x
2x[x² - 8x + 12]
Step 2; Factor [x² - 8x + 12]
(x-6)(x-2)
Step 3: put it all together
(2x)(x-6)(x-2)
If you need to know how to factor cubic equations that don't have a GCF to factor out that makes it easy, let me know and I can walk you through that.
• Consider the polynomial function
Px=x4-3x3+ax2-6x+14,
where a is an unknown real number. If x-2is a factor of this polynomial, what is the value of a?
• Interesting question.
Well since x - 2 is a factor of P(x), we know that 2 is a root of P(x). So we must have P(2) = 0. This means:
2⁴ - 3(2³) + 4a - 12 + 14 = 0
Now we can just solve for "a":
16 - 24 + 4a + 2 = 0
8 - 12 + 2a + 1 = 0
-3 + 2a = 0
2a = 3
a = 2/3
Comment if you have any questions.
• why do the questions have upvote and downvote buttons like Reddit?
(1 vote)
• So people can upvote the good questions so that other people with the same question can just read the response. The answers also have upvotes and downvotes so people can upvote the best answer.
• Does it matter, in this example, if the 21k went first or the -k would. When I practice, I have no idea which one I should put first and sometimes get it wrong because of that. I'm really confused, please help!
• It can matter. It's not wrong to put the -1k first, but sometimes you'll get stuck trying to factor like that.

You want to think about the idea of "grouping" - which term groups better with it's neighbor. So 21k would be better to group with 7 k^2 because 21 is divisible by 7.

In this example, 21 is also divisible by 3, so you can group the other way:
5(7k^2 -1k + 21k - 3)
5[k(7k-1) + 3(7k-1)]
5(7k-1)(k+3)
Here you get the same answer, but it might not always factor as neatly both ways.

[Hmm, looking at the examples from the videos again, it seems like it would factor either way. Does anyone have an example where it doesn't? Or know the rule for it?]