Need help with factoring and solving polynomials PLEASE?

Question

I need to get answears to these math problems and i'm really stuck. So any help would be appreciated.

1) x4 + 10x2 + 16

2) 8x3 - 12x2 - 2x + 3

3) x4 + 10x2 + 16

4) x3 - 3x2 = 0

5) x3 + 27 = 0

6) x3 + 8x2 = -16x

7)125x3 - 8

8) x4 + 7x3 - 8x - 56 = 0

Answer 1

I'll show you the evens, and leave it to you to work out the rest. Most of these look harder than they really are; they're set up to be straightforward to factor. You just separate terms out much like you would quadratic factoring. Here we go:

2) 8x^3-12x^2-2x+3
= 4x^2 (2x-3)-(2x-3)
= (4x^2-1)(2x-3)
= (2x-1)(2x+1)(2x-3)

4) x^3-3x^2 = 0;
x^2(x-3) = 0;
x=3; 0 (x=0 has two roots)

6) x^3+8x^2=-16x;
x^3+8x^2+16x = 0
x(x^2+8x+16)=0
x(x+4)(x+4) = 0
x=0,-4 (x=-4 has two roots)

8) x^4+7x^3-8x-56=0
x^3(x+7) -8(x+7) = 0
(x^3-8)(x+7) = 0
Each parenthetic term is zero, so
(x^3-8)=0; x^3=8; x=2 (x=2 has three roots)
x+7=0; x=-7

Good luck, work hard, and stay away from drugs.

Answer 2

ok.
1. first you would set y = x^2 so you can make it a quadratic equation in y.
y^2 + 10y + 16.
now to factor this, you have to find 2 numbers that add to 10 and multiply to 16 (8 and 2), so the factored form is
(y+8)(y+2)

2. Well, this one's longer but still doable I think. First you have to use the factor theorem (i forgot the name of the rule) but you have to try a value of x that has the numerator of one of the factors of 3 and denominator of one of the factors of 8.

so it could be any of : 1/1, 1/2, 1/4, 1/8, 3/1, 3/2, 3/4, or 3/8.

if you try them in the equation, you see that 3/2 works. (just put x= these values and see if it equals 0)
(3/2 works because 8x^3 = 27, -12x^2 = -27, -2x = -3 and iff you add 3 to all these, you get 0)

so that means that one of the factors is x=3/2, or (2x-3)

so now you have to long divide the equation by 2x-3.
...........................4x^2 - 1---------(this is the result of the division)
(2x-3) | (8x^3 - 12x^2 - 2x + 3)
.............8x^3 - 12x^2
.............------------------------------
.....................................-2x + 3
......................................-2x + 3
......................................----------
................................................0

now you have to factor 4x^2 - 1, using the (a^2 - b^2) = (a+b)(a-b) rule, you get (2x-1)(2x+1)

so the factored form is (2x-3)(2x-1)(2x+1).

3. let y = x^2
y^2 + 10y + 16 is the same as the first question.

4. first factor x^2 out of that
x^2 (x - 3), which is the factored form. The roots are x = 0 (double root) and x = 3

5. use the (a^3 + b^3) = (a+b)(a^2 - ab + b^2) rule with a = x and b = 3.
so you get (x+3)(x^2 - 3x + 9) is the factored form.
one of the roots is x=-3 and to find the other 2 roots, use the quadratic equation. I think they are complex roots though so don't worry about that if you've never heard of it.

6. bring everything to the left side first
x^3 + 8x^2 + 16x = 0
factor an x out of everything
x (x^2 + 8x + 16) and now find 2 numbers that add to 8 and multiply to 16 (4 and 4)
x (x+4)^2 is the factored form and the roots are x=0 and x=-4 (double root)

7. use the (a^3 - b^3) = (a-b)(a^2 + ab + b^2) thing with a = 5x and b = 2.
(5x - 2)(25x^2 + 10x + 4) is the factored form

8. ummm, I guess you would have to find the factors using the factor theorem (might be called remainder theorem too, but not sure). one solution is one of the factors of 56.
the possible solutions are x = 1, 2, 4, 8, 14, 28, 56.
try each of these values in the solution to see if any of them gives you 0.
if you do that, you see that x=2 works. so a factor is (x-2)

now long divide the equation by x-2.
.....................x^3 + 9x^2 + 18x + 28
(x-2) | (x4 + 7x3 + 0x^2 - 8x - 56)
...........x^4 - 2x^3
..........--------------------
................9x^3 + 0x^2
................9x^3 - 18x^2
................-------------------
...........................18x^2 - 8x
...........................18x^2 - 36x
........................................28x - 56
........................................28x - 56
..................................................0

so now factor x^3 + 9x^2 + 18x + 28 using the same method. this time, x = 1 works. so divide by (x-1)
....................x^2 + 10x + 28
(x-1) | x^3 + 9x^2 + 18x + 28
...........x^3 - x^2
..................10x^2 + 18x
..................10x^2 - 10x
.................................28x + 28
.................................28x + 28
.............................................0

the factored form is (x^2 + 10x + 28)(x-1)(x-2).

Hope that helps.

Answer 3

1) x4 + 10x2 + 16 = 2(5x^2 + 2x+8)
2) 8x^3 - 12x^2 - 2x + 3 = (2x-3)(4x^2-1) = (2x-3)(2x+1)(2x-1)
3) See 1
4) x3 - 3x2 = 0 ; (x-3)x^2 = 0; x = 0, x = 3
5) x^3 + 27 = 0; x = -3
6) x3 + 8x2 = -16x ; x^3+x^2+16x = 0; x(x+4)^2 = 0, x = 0, -4
7) 125x3 - 8 = (5x-2)^3
8) x4 + 7x3 - 8x - 56 = 0; (x-7)(x-2)^3 = 0; x = 7, 2

Answer 4

1. Substitute w for the x^2 and w^2 for the w^4 term and factor as usual. Be sure to substitute back to find x.

2. Factor by grouping.

3. Same as #1.

4. Factor out x^2.

5. Sum of 2 cubes.

6. Make one side = 0, factor out an x first and factor what remains.

7. Difference of 2 cubes.

8. Factor by grouping.

Good luck!

Answer 5

1. (x^2+8)(x^2+2)
2. (4x^2+1)(2x-3)
3. same as #1
4. x=0, x=3
5. x= -3, x=3+5i/2, x=3-5i/2

6. x=0, x= -4
7. x=2/5, x= -5+5i squareroot3 x= -5 -5i squareroot3
8. x=2, x= -7, x=2+i squareroot3, x=2-i squareroot3

Answer 6

You definitely need some review. Calculus weeds out hopeful science and engineering majors at colleges every semester. Your algebra (I and II, not just part of it), trigonometry, and (to a lesser extent) geometry skills all need to be top notch. If you haven't mastered these things, don't take the class. Mastery of such basic math is assumed at the calculus level and beyond. Introductory statistics classes are very easy and require literal algebra.

Answer 7

Dang straight. I hate it when people use this as a homework solutions center. If you really are having trouble, talk to a teacher or fellow student. I think I can speak on behalf of most answerers in that we would be happy to answer one or two, but eight? How about no? Pay attention in class, seriously.

Answer 8

1: Learn how write it properly. 8 X squared is 8x^2.

2. Pick ONE. Why don't you put SOME effort in LEARNING how to do it on your OWN.