# Algebra : Simplifying Expressions using the Remaider Theorem and Solving Equations

1. Divide, then write the result as a division statement.
a) (x3-3x2+5x-4)/(x-2)
X not equal to 2
___________
x-2)x3-3x2+5x-4
x3-2x2
-x2+5x
-x2+2x
3x-4
3x

b) (x3-3x-2)/(x+3)

2. Use the Remainder Theorem to determine the remainders ( do not divide).

a) (x3-7x2+2x-4)/(x+2)
P(x)=x3-7x2+2x-4
x-b=x+2
therefore x=-2
Evaluating P(-2)
P(-2)=(-2)3-7(-2)2+2(-2)-4
=-8-28-4-4
=-44

b) (4x3-2x2+7x-1)/(x-2)
P(x)=4x3-2x2+7x-1
x-b=x-2
therefore x=2
Evaluating P(2)
P(2)=4(2)3-2(2)2+7(2)-1
=32-8+14-1
=37

3. Use the Factor Theorem to show that (x-1) is a factor of (2x4-11x3+12x2+x-4).

If x-1 is a factor, then P(1)=0
P(1)=2(1)4-11(1)3+12(1)2+(1)-4
=2-11+12+1-4
=0
Therefore (x-1) is a factor because P(1)=0.

4. Find k so that km3-10m2+2m+3 has (m-3) as a factor.
If m-3 is a factor, then P(3)=0
k(3)3-10(3)2+2(3)+3=0
27k-90+6+3=0
27k=81
k=3

5. Solve the following equation. Express roots to the nearest hundredth.
2m2-m-7=0

6. Solve the following equation by finding its zeros.
x3+9x2+26x+24=0

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