x³+4x²+5x+2 factor theorm
Answers
Answer:
Let f(x)=x
3
−4x
2
+5x−2
Sum of the coefficients =1−4+5−2=0
∴(x−1) is a factor.
Remainder =0
f(x)=x
3
−4x
2
+5x−2
=(x−1)(x
2
−3x+2)
=(x−1)(x−1)(x−2)
=(x−1)
2
(x−2)
-1, -1, -2
x^3 + 4*x^2 + 5*x + 2
[write x+1 terms and then figure out the coefficients of x+1, explained below]
x^2(x+1) + 3x(x+1) + 2(x+1)
(x+1) (x^2+3x+2)
(x+1) (x^2 + x +2x + 2)
(x+1) ( x (x+1) + 1 (x+2) )
(x+1) ( (x+1) (x+2) )
(x+1) (x+1) (x+2)
(x+1)^2 (x+2)
-1, -1, -2
Step-by-step explanation:
Write the steps on a paper as you read statements and follow step by step rather than reading everything at once and then solving
Use a small trick to start with:
1. Add the coefficients of alternate powers, i.e. odd powers and even power, constant number, here 2 has the power 0 hence, x^0.
so here,
we have the sum of odd powers as 1+5=6( x^3 has coefficient 1 and x has coefficient as 5) &
The sum of even powers as 2+4=6( x^2 has coefficient2 and x^0 has coefficient as 4)
Now, if the are sum of both odd and even is the same then -1 is a solution of the equation,i.e. "x+1" is a solution since the value of the equation is 0 at -1.
Additionally, if odd and even power sums are conjugate to each other, say 6 and -6, then 1 is a solution since the value of the equation is 0 at 1
to understand this put -1 in the given equation so we get,
(-1)^3 + 4*(-1)^2 + 5(-1) + 2
= -1 + 4*(1) + 5(-1) + 2
= -1 + 4 -5 + 2
= 0
We know that, the value of the equation at its solution is zero, i.e. the graph of the equation cuts the x-axis at that particular point, here -1.
Similarly, you can understand the trick of 1 as a solution by changing the 4x^2 term to -4x^2 and 2 to -2 - the odd sum would be 6 and the ven sum would be -6. So basically the value of equation comes out to be 6 + (-6) which is 0. hence, that is the solution.
2. After that. this is a cubic equation so here the highest power is 3, then write three terms of (x+1).
3. Now, let's focus on the first "x+1", so you need to figure by multiplying what number to "x" would you get x^3, so that is x^2, so write x^2 there and follow the next step
3. Then, to get the next term -> multiply x^2 that you wrote above with 1 from the first x+1, so you get x^2 itself, remember this value.
4. Now let's move to the second "x+1".. so you had x^2 and your goal is to 4*x^2(given in question) so subtract x^2 from 4x^2...4x^2-x^2= 3x^2, so you got 3x^2 now focus on the second "x+1" term..you need to get 3x^2 and you already have x with you so what you need is 3x, hence you write 3x there.
5. Similar to the third step, multiply 3x with 1 of your second (x+1), you get 3x and remember this.
6. Let's move to the third (x+1), you had 3x previously and here you need to get to 5x so what you do you basically add 2x to 3x to get to 5x. hence we need to add 2x somehow. for that we already x with us from "x+1" term.. so just write 2 there and you'll have 2x similar to the previous steps and if you look carefully you also have got 2..2 in multiplication with 1 from x+1 term..so there you go.
7. Take x+1 common, and write the next step.
8. Split the middle term of quadratic eq. so that the sum of the new terms is 3x and the product is 2*x^2, just a way we solve quadratic eqns.
9. Again, you get x+1 in the quadratic part, so write that and continue.