factor:
9w - w^3 = 3w(3-w^2)
m^4 - n^4 = (m-n)^2(m-n)^2
are these correct?
the next few I wasn't sure how to work...
factor:
2h^2 - h - 3 = 0
x^2 - 36 = 0
x^3 = 4x
Your answers aren't right. You can check factorizations pretty easily by just multiplying.
(3w)(3 - w^2) = (3w)(3) - (3w)(w^2)
=9w - 3w^2
((m-n)^2)((m-n)^2) = (m-n)^4 =/= m^4 - n^4.
Some hints:
9w - w^3 : Factor out the w first. This leaves a difference of two squares. Remember that (x^2 - y^2) = (x+y)(x-y).
m^4 - n^4 : Difference of two squares. One of the factors is also a difference of two squares.
2h^2 - h - 3 = 0 : Quadratic, so factor the left side into (2h +/- a)(h +/- b).
x^2 - 36 = 0 : Left side is a difference of two squares.
x^3 = 4x : Move everything to the left side. Factor out the x. You'll be left with a difference of two squares.
9w - w^3
Most of your questions can be reduced to the difference of two squares.
a^2 - b^2 = (a+b)(a-b)
for example:
9w-w^3
first factor out the common variable w
w (9-w^2)
NOW we have w times (square of 3 - square of w
w (3+w)(3-w)
Bow look at the second one
m^4 - n^4
that is (m^2)^2 - (n^2)^2
which we know is
(m^2+n^2)(m^2-n^2)
now that second factor is also the difference of two squares so
(m^2+n^2)(m+n)(m-n)
2h^2 - h - 3 = 0
Now try to factor that
(2 h ? 3)(h ? 1)
or
(2 h ? 1)(h ? 3)
well
(2 h -3)(h+1) = 0 works
the left is zero if either factor is zero so two solutions
h = 3/2
or
h = -1
x^2 - 36 = 0
back to difference of two squares
(x+6)(x-6) = 0
then do like the last one
x^3 = 4x
well that is
x (x^2-4) = 0
or
x(x+2)(x-2) = 0
then proceed as before x = 0, x = 2, x = -2
To verify whether the given factorizations are correct, you need to expand the expressions on the right-hand side and check if they simplify to the original expressions on the left-hand side.
1. Let's confirm the first factorization:
9w - w^3 = 3w(3 - w^2)
Expanding the right side: 3w(3) - 3w(w^2) = 9w - 3w^3
This matches the original expression, so the factorization is correct.
2. Checking the second factorization:
m^4 - n^4 = (m - n)^2(m + n)^2
Expanding the right side: (m - n)^2(m + n)^2
= (m - n)(m - n)(m + n)(m + n)
= (m - n)(m - n)(m + n)^2
This matches the original expression, so the factorization is correct.
Moving on to the next expressions, let's work on factoring them:
3. 2h^2 - h - 3 = 0
Since this is a quadratic equation, we can try factoring or using the quadratic formula.
To factor the quadratic, we look for two numbers that multiply to give -6 and add up to -1.
The factors that satisfy this condition are -3 and 2 since (-3) * (2) = -6 and (-3) + (2) = -1.
Therefore, we can factor the equation as:
(2h + 3)(h - 1) = 0
4. x^2 - 36 = 0
This is a difference of squares, which can be factored as follows:
(x - 6)(x + 6) = 0
5. x^3 = 4x
We can rearrange the equation to have all terms on one side:
x^3 - 4x = 0
Factor out the common factor, which is x:
x(x^2 - 4) = 0
The expression x^2 - 4 is a difference of squares, so we can further factor it:
x(x - 2)(x + 2) = 0
So, the factorizations are:
1. 9w - w^3 = 3w(3 - w^2)
2. m^4 - n^4 = (m - n)^2(m + n)^2
3. 2h^2 - h - 3 = (2h + 3)(h - 1)
4. x^2 - 36 = (x - 6)(x + 6)
5. x^3 = 4x becomes x(x - 2)(x + 2) = 0