Question39. Let . When is divided by , the remainder is 30 . When is divided by , the quotient is .
(a) Find the remainder when is divided by .
Explain (b) How many real roots does the equation have? Explain your answer. Example 14
Studdy Solution
STEP 1
1. We are given the polynomial .
2. When is divided by , the remainder is 30.
3. When is divided by , the quotient is .
4. We need to find the remainder when is divided by .
5. We need to determine how many real roots the equation has.
STEP 2
1. Use the Remainder Theorem to find conditions on .
2. Use the division information to set up an equation for .
3. Solve for the remainder when is divided by .
4. Analyze the polynomial to determine the number of real roots.
STEP 3
Use the Remainder Theorem for :
Simplify:
STEP 4
Given that the quotient when dividing by is , we can express as:
where is the remainder, a linear polynomial since the divisor is quadratic:
STEP 5
Substitute and into using the division form to find and .
For :
For :
These give us two equations:
1.
2.
STEP 6
Use to find and :
Solve the equations for and .
STEP 7
Substitute the known values into the equations:
Solve these simultaneous equations to find and .
STEP 8
Analyze the polynomial for real roots.
1. The degree of is 3, so it can have at most 3 real roots.
2. Use the information from the remainder and division to check for possible roots.
The remainder when is divided by is:
The number of real roots of depends on the discriminant and the nature of the roots found.
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