2x^3-6x+5¡Ô(x-2)(2x^2+ax+2)+b
I'm afraid I'm not very good at identities... what is a? I think b=9...
I cant read it.
its 2x cubed minus 6x plus 5 equals (x plus 2)(2x squared plus ax plus 2) plus b.
2x^3-6x+5=(x+2)(2x^2+ax+2)+b
multiply out the right.
= 2x^3+2ax^2+2x+2x^2+2ax+4+b
now compare coefficents of x^n
2x^3=2x^3 checks.
X^2: 0=2a + 2 or a=-1 check that
x: -6 = 2+2a a=-1
constants: 5=4+b
so b= ....
" 2x^3-6x+5≡(x-2)(2x^2+ax+2)+b "
What you would do is to expand the right hand side to be
(x-2)*(2*x^2+a*x+2)+b
=2*x^3+(a-4)*x^2+(2-2*a)*x+b-4
So the identity becomes:
2x^3-6x+5≡2*x^3+(a-4)*x^2+(2-2*a)*x+b-4
Then you compare the coefficients of each term with the left hand side,
For x³, you have 2 on the left, which equals to 2 on the right.
Proceed with x²
you will establish the equation
0 = (a-4) by comparing the coefficients on each side of the ≡ sign.
That gives us a=4.
Continue this way to find b and confirm all terms of the identity are consistent.
Note: If your computer is set to have encoding in a foreign language, be sure to revert to Western 8859-1 encoding before writing Math symbols so everyone can read. On most browsers, it would be under "view-encoding" and then choose the encoding (for the current page only).
To find the value of "a" in the expression 2x^3-6x+5¡Ô(x-2)(2x^2+ax+2)+b, we can use the distributive property of multiplication over addition to expand the parentheses. Let's start by multiplying (x-2) with (2x^2+ax+2):
(x-2)(2x^2+ax+2) = x(2x^2+ax+2) - 2(2x^2+ax+2)
= 2x^3 + ax^2 + 2x - 4x^2 - 2ax - 4
= 2x^3 - 4x^2 + ax^2 + 2x - 2ax - 4
Now, we can substitute this expanded expression back into the original equation:
2x^3 - 6x + 5 = (2x^3 - 4x^2 + ax^2 + 2x - 2ax - 4) + b
Rearranging the terms:
2x^3 - 6x + 5 = 2x^3 - 4x^2 + ax^2 + 2x - 2ax - 4 + b
We can now compare the coefficients of like terms on both sides of the equation to determine the value of "a".
Comparing the coefficients of x^2 on both sides:
-6x^2 = -4x^2 + ax^2
This tells us that ax^2 - 4x^2 = -6x^2. Simplifying further, we have:
-6x^2 = -4x^2 + ax^2
-6x^2 + 4x^2 = ax^2
-2x^2 = ax^2
From this equation, we can see that a must be equal to -2.
Therefore, the value of "a" is -2.