If f(x)=x3+lx+m leaves the same reminder when divided by (x-1) or by (x+1). Find values of l and m?
To find the values of \( l \) and \( m \) that satisfies the condition, we can use the Remainder Theorem.
According to the Remainder Theorem, if the polynomial \( f(x) \) yields the same remainder when divided by \( x-1 \) and \( x+1 \), this means that when we substitute \( x = 1 \) and \( x = -1 \) into \( f(x) \), we should get the same remainder.
Let's evaluate \( f(x) \) for \( x = 1 \) and \( x = -1 \):
For \( x = 1 \):
\( f(1) = 1^3 + l(1) + m = 1 + l + m \)
For \( x = -1 \):
\( f(-1) = (-1)^3 + l(-1) + m = -1 + l + m \)
Since the remainders are the same, we have:
\( 1 + l + m = -1 + l + m \)
By canceling out the common terms, we get:
\( 1 = -1 \)
However, this equation is contradictory and cannot be satisfied. This means there are no values of \( l \) and \( m \) that would yield the same remainder when divided by \( x-1 \) and \( x+1 \).
Therefore, there are no solutions for the values of \( l \) and \( m \) that satisfy the given condition.
To find the values of l and m, we need to use the Remainder Theorem. According to the Remainder Theorem, if f(x) leaves the same remainder when divided by (x-1) or by (x+1), then plugging in x=1 and x=-1 into f(x) should give the same result.
Let's start by evaluating f(x) at x=1:
f(1) = (1)^3 + l(1) + m
= 1 + l + m
Now, let's evaluate f(x) at x=-1:
f(-1) = (-1)^3 + l(-1) + m
= -1 - l + m
Since f(x) leaves the same remainder when divided by (x-1) or by (x+1), we can set these two expressions equal to each other and solve for l and m:
1 + l + m = -1 - l + m
By simplifying the equation, we get:
2l = -2
Dividing both sides by 2, we find:
l = -1
Now, substitute this value of l back into one of the equations to solve for m. Let's use the equation f(1) = 1 + l + m:
f(1) = 1 + (-1) + m
= 0 + m
= m
Therefore, m = 0.
So, the values of l and m are l = -1 and m = 0.
do a little synthetic division.
R(x-1) = m+(l+1)
R(x+1) = m-(l+1)
m-(l+1) = m+(l+1)
2(l+1) = 0
l = -1
and m can be anything. So,
f(x) = x^3-x+m
the remainder is always m.