Factor each expression completely. show some sort of check to confirm your answer. (hint: think FOIL)
x^2+11+30
x^2-5x-14
9x-16
five and six is eleven
(x+5)(x+6)
seven - two
(x-7)(x+2)
I bet you mean
9x^2-16
which is difference of two squares
a^2-b^2 = (a+b)(a-b)
(3x+4)(3x-4)
Yes that's what I meant, Damon, sorry. I'm not getting the whole factoring thing with foil. I thought foil was using distributive with binomials. I understand that 5+6=11, but so does 10+1 or 7+4. Don't understand the second or third one either. Please explain how you got the answers and how to check it
for your second expression, you need to find 2 numbers that if you add or subtract will give you the middle term and if you multiply them, it will give you the last term. so going back to your example, the two number that if you multiply and they will give you (-14)is -7 and +2 --but why we chose these numbers? because if you add -7 and +2 you will get -5 which is your middle term. This is how you check if your foiling is correct.
To factor each expression completely, we need to break down each expression into its prime factors.
First Expression: x^2 + 11x + 30
To factor this expression, we will use the FOIL method. We need to find two numbers that multiply to give us 30 and add up to give us 11. The numbers that meet these criteria are 5 and 6. So, we can rewrite the expression as follows:
x^2 + 5x + 6x + 30
Next, we group the terms:
(x^2 + 5x) + (6x + 30)
Now, we find the common factors in each group:
x(x + 5) + 6(x + 5)
Finally, we factor out the common factor (x + 5):
(x + 5)(x + 6)
To check our answer, you can use the FOIL method which stands for First, Outer, Inner, Last. Multiply the first terms, then the outer terms, inner terms, and the last terms:
(x + 5)(x + 6)
= x^2 + 6x + 5x + 30
= x^2 + 11x + 30
This confirms that our factoring is correct.
Second Expression: x^2 - 5x - 14
To factor this expression, we need to find two numbers that multiply to give us -14 and add up to give us -5. The numbers that meet these criteria are -7 and +2. Rewriting the expression using these numbers, we get:
x^2 - 7x + 2x - 14
Next, we group the terms:
(x^2 - 7x) + (2x - 14)
Finding the common factors in each group:
x(x - 7) + 2(x - 7)
Factoring out the common factor (x - 7):
(x - 7)(x + 2)
To check our answer using the FOIL method:
(x - 7)(x + 2)
= x^2 + 2x - 7x - 14
= x^2 - 5x - 14
This confirms that our factoring is correct.
Third Expression: 9x - 16
This expression cannot be factored as it only has two terms. However, we can simplify it by finding any common factors. In this case, there are no common factors, so the expression remains as 9x - 16.
To check our answer, we can evaluate the expression for different values of x and compare the results with the original expression. For example, if we substitute x = 2:
Original expression: 9(2) - 16 = 18 - 16 = 2
Factored expression: 9(2) - 16 = 18 - 16 = 2
The values match, confirming our answer.