-a^3+10a^2-31a+30=0
To solve the equation -a^3 + 10a^2 - 31a + 30 = 0, we can use a factoring method.
Step 1: Group the terms into pairs:
(-a^3 + 10a^2) + (-31a + 30) = 0
Step 2: Factor out the greatest common factor from each pair:
a^2 (-a + 10) - 1(31a - 30) = 0
Step 3: Simplify the equation:
a^2 (-a + 10) - (31a - 30) = 0
Step 4: Expand the equation by distributing:
-a^3 + 10a^2 - 31a + 30 = 0
Step 5: Set each group equal to zero and solve for 'a':
a^2 = 0
This gives us a possible solution of a = 0.
-a + 10 = 0
Solving for 'a', we have a = 10.
31a - 30 = 0
Solving for 'a', we have a = 30/31.
Therefore, the solutions to the equation -a^3 + 10a^2 - 31a + 30 = 0 are a = 0, a = 10, and a = 30/31.
To solve the equation -a^3 + 10a^2 - 31a + 30 = 0, you can use the method of factoring or the rational roots theorem. Let's use the factoring method.
Step 1: Group the terms in pairs, if possible. Looking at the equation, we notice that the first two terms and the last two terms can be grouped.
-a^3 + 10a^2 - 31a + 30 = 0
(a^2 - 10a) + (-31a + 30) = 0
Step 2: Factor out the common terms from each pair. From the first pair, we can factor out "a," and from the second pair, we can factor out "-1."
a(a - 10) - 1(31a - 30) = 0
Step 3: Simplify the expression.
a(a - 10) - (31a - 30) = 0
a(a - 10) + (30 - 31a) = 0
Step 4: Rearrange the terms.
a(a - 10) - (31a - 30) = 0
a(a - 10) - (30 - 31a) = 0
Step 5: Factor by grouping.
a(a - 10) - (30 - 31a) = 0
a(a - 10) + (31a - 30) = 0
(a - 1)(a - 10) + (31a - 30) = 0
(a - 1)(a - 10) + (31a - 30) = 0
Step 6: Use the zero-product property to set each factor equal to zero and solve for "a."
(a - 1) = 0 or (a - 10) = 0
a = 1 or a = 10
Therefore, the solutions to the equation -a^3 + 10a^2 - 31a + 30 = 0 are a = 1 and a = 10.
always try a = 1, no
a = -1 gives -1+10+31+30 no
a = 2 gives -8+40-62+30 YES!
so we know a = 2 is a solution
so now factor. You know one factor is (a-2) so divide.