The document discusses sign charts of factorable polynomials and rational expressions. It defines a factorable polynomial as one that can be written as the product of real linear factors. An example polynomial is fully factored. Roots of the polynomial are defined as the values making each linear factor equal to zero. The order of a root is defined as the power of the corresponding factor. The Even/Odd-Order Sign Rule is stated: for a factorable polynomial, signs are the same on both sides of an even-ordered root and different on both sides of an odd-ordered root. An example sign chart is constructed applying this rule.

1. Sign Charts of Factorable Formulas

2. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas

3. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3

4. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16)

5. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2

6. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2

7. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable

8. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
Hence r1, r2,.. ,rk are the roots of P(x).
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable

9. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
Hence r1, r2,.. ,rk are the roots of P(x).
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable with roots x = 0, –2, and 2.

10. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
Hence r1, r2,.. ,rk are the roots of P(x).
The order of a root is the corresponding power
raised in the factored form, i.e.
the order of the r1 is N1, order of r2 is N2 ,etc..
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable with roots x = 0, –2, and 2.

11. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
Hence r1, r2,.. ,rk are the roots of P(x).
The order of a root is the corresponding power
raised in the factored form, i.e.
the order of the r1 is N1, order of r2 is N2 ,etc..
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable with roots x = 0, –2, and 2.
x = 0 has order 3,

12. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
Hence r1, r2,.. ,rk are the roots of P(x).
The order of a root is the corresponding power
raised in the factored form, i.e.
the order of the r1 is N1, order of r2 is N2 ,etc..
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable with roots x = 0, –2, and 2.
x = 0 has order 3, x = –2 and x = 2 have order 2.

13. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
Hence r1, r2,.. ,rk are the roots of P(x).
The order of a root is the corresponding power
raised in the factored form, i.e.
the order of the r1 is N1, order of r2 is N2 ,etc..
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable with roots x = 0, –2, and 2.
x = 0 has order 3, x = –2 and x = 2 have order 2.

14. Sign Charts of Factorable Formulas
An important property of a root is whether it is an
even-ordered root or an odd-ordered root.

15. Sign Charts of Factorable Formulas
An important property of a root is whether it is an
even-ordered root or an odd-ordered root.
For example, for P(x) = 2x3(x + 2)2(x – 2)2,

16. Sign Charts of Factorable Formulas
An important property of a root is whether it is an
even-ordered root or an odd-ordered root.
For example, for P(x) = 2x3(x + 2)2(x – 2)2,
a. x = 0 is an odd-ordered root (its order is 3)

17. Sign Charts of Factorable Formulas
An important property of a root is whether it is an
even-ordered root or an odd-ordered root.
For example, for P(x) = 2x3(x + 2)2(x – 2)2,
a. x = 0 is an odd-ordered root (its order is 3)
b. x = 2 or –2 are even-ordered roots
(each has order 2).

18. Sign Charts of Factorable Formulas
An important property of a root is whether it is an
even-ordered root or an odd-ordered root.
For example, for P(x) = 2x3(x + 2)2(x – 2)2,
a. x = 0 is an odd-ordered root (its order is 3)
b. x = 2 or –2 are even-ordered roots
(each has order 2).
Theorem (The Even/Odd–Order Sign Rule)

19. Sign Charts of Factorable Formulas
An important property of a root is whether it is an
even-ordered root or an odd-ordered root.
For example, for P(x) = 2x3(x + 2)2(x – 2)2,
a. x = 0 is an odd-ordered root (its order is 3)
b. x = 2 or –2 are even-ordered roots
(each has order 2).
Theorem (The Even/Odd–Order Sign Rule)
For the sign-chart of a factorable polynomial
1. the signs are the same on both sides of
an even-ordered root,
2. the signs are different on two sides of
an odd-ordered root.

20. Sign Charts of Factorable Formulas
An important property of a root is whether it is an
even-ordered root or an odd-ordered root.
For example, for P(x) = 2x3(x + 2)2(x – 2)2,
a. x = 0 is an odd-ordered root (its order is 3)
b. x = 2 or –2 are even-ordered roots
(each has order 2).
Theorem (The Even/Odd–Order Sign Rule)
For the sign-chart of a factorable polynomial
1. the signs are the same on both sides of
an even-ordered root,
2. the signs are different on two sides of
an odd-ordered root.
This theorem simplifies the construction of sign-charts
and graphs (later) of factorable polynomials.

21. Sign Charts of Factorable Formulas
Example B. Make the sign-chart of x3(x + 3)2 (x – 3)

22. 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.

23. 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.

24. 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)

25. 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.

26. 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.
+
x=4

27. 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.
+
x=4

28. 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.
change
sign +
x=4

29. 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 "+".
change
sign
change
sign+ +
x=4

30. 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
change
sign
change
sign+ +
x=4

31. 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
change
sign
change
sign+
sign
unchanged+ +
x=4

32. 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

33. 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.

34. 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

35. 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:

36. 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

37. 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)

38. 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).

39. 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).

40. 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 " – ".
x=0 (even) x=2 (odd)x= 1 (even) x=3

41. 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) x=3

42. 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)
+ change
x=3

43. 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)
+ changeunchanged
+
x=3

44. 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
++
x=3

45. 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

46. 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

47. 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√ √
√ √

48. (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

49. 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

50. 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

51. 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

52. 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

53. 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