The document discusses sign charts for factorable formulas. It provides examples of drawing sign charts, including marking the roots of factors and determining the sign of the expression between roots based on whether roots are odd- or even-ordered. Exercises have students draw sign charts for additional formulas and use them to solve inequalities or determine domains where expressions are defined.

1. Sign Charts of Factorable Formulas
Example B. Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x = 0, –3 and 3. x = 0 and 3 are odd-
ordered roots and root x = –3 is an even-ordered root.
Draw a line, mark off these roots and their types.
x=0 (odd) x=3 (odd)x=-3 (even)
Sample a point, say x = 4 and we get P(4) positive.
Mark the segment as such. By the theorem, across
the root x = 3 the sign changes to "–" because x = 3
is odd-ordered. Similarly, across the root x = 0, the
sign changes again to "+". But across x = -3 the
sign stays as "+" because it is even-ordered
and the chart is completed.
change
sign
change
sign+
sign
unchanged+ +
x=4

2. Sign Charts of Factorable Formulas
The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example C. Solve the inequality 2x2 – x3
x2 – 2x + 1
< 0
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered). Draw and
test x = 3, we get " – ". Complete the sign-chart.
x=0 (even) x=2 (odd)x= 1 (even)
+ changeunchangedunchanged
++
Hence the solution is 2 < x.
x=3

3. 1. x2(x – 2) 2. x2(x – 2)2 3. x(x – 2)2
4. x3(x – 2) 5. x(x + 2)2(5 – x) 6. x2(x + 2)2(5 – x)
8. 9x2 – x47. x2(x + 2)2(x – 5)3 9. x4 – 4x3 + 4x2
11. 3x2(2 – x)7(x – 1)410. 3(2x– 5) 2(x + 2)7(x – 1)4
12. (5 – x )2(3 – x)7(2x – 1)5
B. Draw the sign–chart of each formula below.
1.
7.
2. 3.
4. 5. 6.
x2 – 4
x2 – 4x + 4
x – 4
x2
(x + 3)2
x + 4
x + 2
x2 – 3x + 2
x2(x – 2)3
(x + 4)2(x + 2) 8.
x(x – 4)3
(x2 + 4)(x + 4)2
Exercise A. Draw the sign–chart of each formula below.
Sign Charts of Factorable Formulas
x + 4
x2 x2(x – 4)3x2(2 + x)2

4. 5. 6. (x – 2)2
x2(x – 6)
x + 6
x(x + 5)2
7.
(1 – x2)3
x2
C. Solve the following inequalities using sign–charts.
1. x2(x – 2) > 0 2. 0 > (x + 2)2(x – 5)3
4. 0 ≤ 2x4 – 8x2
3. x2(x – 4)3 ≤ 0
> 0
≥ 0
Sign Charts of Factorable Formulas
≤ 0
8.
x4 – x2
(x + 3)2(6 – x)
≤ 0
D. In order for the formula √R to be defined,
the radicand R must not be negative, i.e. R must be ≥ 0.
Draw the domain of the following formulas.
5. 6.
(x – 2)2
x2(x – 6)
x + 6
x(x + 5)2
7. x2
1. √x2(x + 2) 2. √(x + 2)2(x – 5)3
4. √2x4 – 8x2
3. √ x2(x – 4)3
8. (x + 3)2(6 – x)
x4 – x2(1 – x2)3√ √
√ √

7. (Answers to odd problems) Exercise A.
1. 𝑥2
(𝑥 – 2)
x=0 x=2
+
3. 𝑥(𝑥 − 2)2
x=0 x=2
++
5. 𝑥(𝑥 + 2)2(5 – 𝑥)
x= – 2 x=5x= 0
+
Sign Charts of Factorable Formulas

8. 11. 3𝑥2 2 − 𝑥 7(𝑥 − 1)4
x= 0 x=2x= 1
+
7. (𝑥 + 2)2(𝑥 − 5)3
x= – 2 x=5x= 0
+
9. 𝑥4 − 4𝑥3 + 4𝑥2
x= – 3 x=3x= 0
++
++
Sign Charts of Factorable Formulas

9. Exercise B.
1.
𝑥−4
𝑥+2
x= – 2 x=4
++
3.
𝑥2−4
𝑥+4
x= – 4 x=2x= -2
++
UDF f=0
UDF f=0 f=0
5.
𝑥2(2+𝑥)2
𝑥2−3𝑥+2
+UDF
x= –2 x=1x= 0
+
x=2
UDFf=0 f=0
+ +
Sign Charts of Factorable Formulas

10. 7.
𝑥2(2+𝑥)2
𝑥2−3𝑥+2
+ UDF
x= 2x= -2 x=0x=-4
UDF f=0f=0
++
Exercise C.
3. 𝑥2(𝑥 − 4)3
≤ 0
1. 𝑥2 𝑥 – 2 > 0
x=0 x=2
+
The solution is (2, ∞)
x=0 x=4
+
The solution is (−∞, 4]
Sign Charts of Factorable Formulas

11. 5.
𝑥2(𝑥−6)
𝑥+6
> 0
The solution is (−∞, −6) ∪ (6, ∞)
x= – 6 x=6x= 0
++ UDF f=0 f=0
7.
𝑥2
(1−𝑥2)3 ≥ 0
x= – 1 x=1x= 0
+
UDF f=0 UDF
+
The solution is (−1, −1)
Sign Charts of Factorable Formulas

12. Exercise D.
1. 𝑥2(𝑥 + 2)
3. 𝑥2(𝑥 − 4)3
x=-2
x=4
x=-6 x=6
x=-1 x=1
5.
𝑥2(𝑥−6)
𝑥+6
7.
𝑥2
(1−𝑥2)3
Sign Charts of Factorable Formulas