The document presents solutions to two mathematical problems involving inequalities and polynomial factors. It details the process of solving the inequality |x-4|≤3|x+2| and finding values for constants z and r in the polynomial 4x^3 + zx^2 + rx - 2, given specific factors. The final values determined are z=11 and r=5, with the remainder when p(x) is divided by (x^2+1) being x-13.

1. ALGEBRA
By: Frilla Marita and Shakira Wildan

2. Modulus Function
QUESTION
Solve this inequality
|X-4|≤3|X+2|

3. ANSWER
(X-4)+3(X+2) (X-4-3(X+2)≤0
(X-4+3X+6) (X-4-3X-6)≤0
(4X+2) (-2X-10)≤0
X= -1/2 X= -5
X≤-5 or X≥-1/2
- + -

4. Polynomial
QUESTION
The polynomial 4x^3 + zx^2 +rx-2, where z and r are
constants, is donated by p(x). It is given that (x+1)
and (x+2) are factors of p(x).
(i) Find the values of of z and r
(ii) When z and r have these values, find the
remainder when p(x) is divided by (x^2+1)

5. I) ANSWER
EQUATION 1
P(-1)= 4(-1)^3 + z(-1)^2 + r(-1)-2=0
-4+z-r-2=0
z-r-6=0

6. EQUATION 2
p(-2)= 4(-2)^3+z(-2)^2+r(-2)-2
= -32+4z-2r-2
= 4z-2r-34

7. 4z-2r-34=0
z-r-6=0
4z-2r-34=0
2z-2r-12=0
x1
x2
-
2z-22= 0
2z=22
z=22/2
z=11
ELIMINATION

8. 11-r-6=0
-r=-11+6
-r=-5
r=5
SUBSTITUTION
(i) Therefore,
z=11 and r=5

9. II) ANSWER
NVJHV
(i) Therefore, the
remainder is x-13

10. THANK YOU