identify what kind of factoring polynomials is applicable to each algebraic expressions to solve for their factors
CHOICES
(common monomial factor), (factoring difference of two squares), (factoring difference of two cube), (factoring perfect square trinomial), ( factoring general/quadratic trinomial)
1. 9m2-54m+81
2. 27x9+8
3. X2+17x+60
4. F3-g
5. 25x4-y2
6. 9x10-81
7. 36a2-12ab+b2
8. 2y+3xy+2xy2
9. 14x2-41x+15
10. 2x3y+6x2y2-4x4y
Answers
Step-by-step explanation:
Is x2 + 10x + 25 a perfect square trinomial? If so, write the trinomial as the square of a binomial.
Well, the first term, x2, is the square of x. The third term, 25, is the square of 5. Multiplying these two, I get 5x.
Multiplying this expression by 2, I get 10x. This is what I'm needing to match, in order for the quadratic to fit the pattern of a perfect-square trinomial. Looking at the original quadratic they gave me, I see that the middle term is 10x, which is what I needed. So this is indeed a perfect-square trinomial:
(x)2 + 2(x)(5) + (5)2
But what was the original binomial that they'd squared?
Affiliate
I know that the first term in the original binomial will be the first square root I found, which was x. The second term will be the second square root I found, which was 5. Looking back at the original quadratic, I see that the sign on the middle term was a "plus". This means that I'll have a "plus" sign between the x and the 5. Then this quadratic is:
a perfect square, with
x2 + 10x + 25 = (x + 5)2
Write 16x2 – 48x + 36 as a squared binomial.
The first term, 16x2, is the square of 4x, and the last term, 36, is the square of 6.
(4x)2 – 48x + 62
Actually, since the middle term has a "minus" sign, the 36 will need to be the square of –6 if the pattern is going to work. Just to be sure, I'll make sure that the middle term matches the pattern:
(4x)(–6)(2) = –48x
It's a match to the original quadratic they gave me, so that quadratic fits the pattern of being a perfect square:
(4x)2 + (2)(4x)(–6) + (–6)2
I'll plug the 4x and the –6 into the pattern to get the original squared-binomial form:
16x2 – 48x + 36 = (4x – 6)2
Is 4x2 – 25x + 36 a perfect square trinomial?
The first term, 4x2, is the square of 2x, and the last term, 36, is the square of 6 (or, in this case, –6, if this is a perfect square).
According to the pattern for perfect-square trinomials, the middle term must be:
(2x)(–6)(2) = –24x
However, looking back at the original quadratic, it had a middle term of –25x, and this does not match what the pattern requires. So:
this is not a perfect square trinomial.
Content Continues Below
Factor x4 – 2x2 + 1 fully.
If I use the regular methods for factoring quadratic-type polynomials, I can factor this just fine. But what if this is in the homework for the section in my textbook on perfect-square binomials? Naturally, I'm going to be thinking that the author is expecting me to notice a perfect square. So:
The first term is x4, whose square root is x2. The third term is 1, whose square root is just 1. Does the middle term, 2x2, fit the pattern for perfect-square binomials? I'll check:
2(x2)(1) = 2x2
It's a match to the original polynomial, so this is a perfect-square trinomial. With the "minus" on the middle term of what they gave me, the original squared-binomial form looks like:
(x2 – 1)2
Hmm... The instructions say to "factor fully". That's often a clue that there may be some more factoring that I could, after the usual bit is completed. Can I factor any more here?
Yes, I can. Looking inside the parentheses, I notice that I have a difference of squares, which I can factor:
x2 – 1 = (x – 1)(x + 1)
Putting the square on everything, I end up with a fully-factoring answer of:
x4 – 2x2 + 1 = (x2 – 1)2
= ((x – 1)(x + 1))2
= (x – 1)2(x + 1)2
That's really all there is to perfect squares.
You can use the Mathway widget below to practice checking if a trinomial is a perfect square. Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)
(Clicking on "Tap to view steps" on the widget's answer screen will take you to the Mathway site for a paid upgrade.)
Share This Page
Terms of Use
Privacy
Contact
About Purplemath
About the Author
Tutoring from PM
Advertising
Linking to PM
Site licencing
Visit Our Profiles
© 2021 Purplemath, Inc. All right reserved. Web Design by
Answer:
1. Factoring perfect square trinomial
2. Factoring adding of two cubes
3. Factoring general/quadratic trinomial.
4. Factoring difference of two cubes
5. Factoring difference of two squares.
6. Factoring difference of two squares.
7. Factoring perfect square trinomial
8. Common monomial factor.
9. Factoring general/quadratic trinomial
10. Common monomial factor.
Step-by-step explanation:
1. 9m² - 54m + 81
= 9(m² - 6m + 9)
Therefore, as it is a perfect square,
It will be solved using factoring perfect square trinomial
2. 27x⁹ + 8
= (3x³)³ + 2³
Therefore, it will be solved using factoring adding of two cubes
3. x² + 17x + 60
It will be solved using factoring general/quadratic trinomial.
4. F³ - g
= (F)³ - (∛g)³
Therefore, it will be solved using factoring difference of two cubes
5. 25x⁴ - y²
= (5y²)² - y²
Therefore, it will be solved using factoring difference of two squares.
6. 9x¹⁰ - 81
= 9[(x⁵)² - (3)²]
Therefore, it will be solved using factoring difference of two squares.
7. 36a² - 12ab + b²
= (6a)² - 2*6a*b + b²
Therefore, it will be solved using factoring perfect square trinomial
8. 2y + 3xy + 2xy²
= y(2 + 3x + 2xy)
Therefore, it will be solved using common monomial factor.
9. 14x² - 41x + 15
We will solve this by mid term splitting
Therefore, it will be solved using factoring general/quadratic trinomial
10. 2x³y + 6x²y² - 4x⁴y
= x²y (2x + 6y - 4x²)
Therefore, it will be solved using common monomial factor.