How can the polynomial 6d^4+9d^3-12d^2 be factored?
the answers for unit 4, lesson 2, in connections academy are
b
c
a
d
d
the answers are actually
d
c
a
d
d
yw :)
Zoie is right 100%
Well, factoring this polynomial is no joke, but I'll give it a shot! Let's see if we can find some common terms or factors.
First, let's look at the coefficients 6, 9, and -12. What do they have in common? They are all divisible by 3. So, let's factor out a 3 from each term, and we get:
3(2d^4 + 3d^3 - 4d^2)
Now, let's see if we can find any common terms or factors within the parentheses. We can see that each term has a factor of d^2, so let's factor that out, too:
3d^2(2d^2 + 3d - 4)
And there you have it! The polynomial 6d^4 + 9d^3 - 12d^2 can be factored as 3d^2(2d^2 + 3d - 4). Keep spreading laughter in your Math class!
To factor the polynomial 6d^4 + 9d^3 - 12d^2, we can follow these steps:
Step 1: Look for the common factor
First, we check if there is a common factor among the terms in the polynomial. In this case, we can factor out 3d^2, as it is a common factor for all terms:
Common factor: 3d^2
Factored polynomial: 3d^2(2d^2 + 3d - 4)
Step 2: Factor the remaining quadratic expression
Next, we focus on factoring the remaining quadratic expression 2d^2 + 3d - 4. This can be done using factoring techniques or the quadratic formula. However, in this case, the quadratic expression does not factor nicely. Therefore, we proceed to use the quadratic formula to find the roots.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic expression 2d^2 + 3d - 4, we have:
a = 2, b = 3, c = -4
Using the quadratic formula, we calculate the roots:
d = (-3 ± √(3^2 - 4(2)(-4))) / (2(2))
d = (-3 ± √(9 + 32)) / 4
d = (-3 ± √41) / 4
Since the roots do not simplify, we leave them as (-3 ± √41) / 4.
Step 3: Write the factored form
By combining the common factor and the quadratic roots, we can express the factored form of the original polynomial:
Factored polynomial: 3d^2(2d - (-3 + √41)/4)(2d - (-3 - √41)/4)
Please note that the quadratic expression may not always factor in simple terms, and resorting to the quadratic formula or other methods may be necessary.
6d^4+9d^3-12d^2
first, factor out the d^2
d^2(6d^2+9d-12)
Now, how about a 3?
3d^2(2d^2+3d-6)
Now, note that the discriminant is 57, so the roots will contain √57. No rational roots exist. Can't factor it any further using integers.