1.5 sign charts and inequalities ii
TRANSCRIPT
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Sign Charts of Factorable Formulas
http://www.lahc.edu/math/precalculus/math_260a.html
![Page 2: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/2.jpg)
In this section we give a theorem about sign-charts of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas
![Page 3: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/3.jpg)
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
![Page 4: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/4.jpg)
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)
![Page 5: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/5.jpg)
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
![Page 6: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/6.jpg)
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
![Page 7: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/7.jpg)
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
![Page 8: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/8.jpg)
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
![Page 9: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/9.jpg)
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.
![Page 10: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/10.jpg)
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.
![Page 11: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/11.jpg)
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,
![Page 12: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/12.jpg)
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.
![Page 13: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/13.jpg)
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.
![Page 14: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/14.jpg)
Sign Charts of Factorable FormulasAn important property of a root is whether it is an even-ordered root or an odd-ordered root.
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Sign Charts of Factorable FormulasAn 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,
![Page 16: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/16.jpg)
Sign Charts of Factorable FormulasAn 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)
![Page 17: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/17.jpg)
Sign Charts of Factorable FormulasAn 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).
![Page 18: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/18.jpg)
Sign Charts of Factorable FormulasAn 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)
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Sign Charts of Factorable FormulasAn 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 polynomial1. 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.
![Page 20: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/20.jpg)
Sign Charts of Factorable FormulasAn 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 polynomial1. 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.
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Sign Charts of Factorable FormulasExample B. Make the sign-chart of x3(x + 3)2 (x – 3)
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Sign Charts of Factorable FormulasExample B. Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x = 0, –3 and 3.
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Sign Charts of Factorable FormulasExample 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.
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Sign Charts of Factorable FormulasExample 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)
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Sign Charts of Factorable FormulasExample 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.
![Page 26: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/26.jpg)
Sign Charts of Factorable FormulasExample 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
![Page 27: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/27.jpg)
Sign Charts of Factorable FormulasExample 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
![Page 28: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/28.jpg)
Sign Charts of Factorable FormulasExample 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.
changesign +
x=4
![Page 29: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/29.jpg)
Sign Charts of Factorable FormulasExample 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 "+".
changesign
changesign+ +
x=4
![Page 30: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/30.jpg)
Sign Charts of Factorable FormulasExample 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
changesign
changesign+ +
x=4
![Page 31: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/31.jpg)
Sign Charts of Factorable FormulasExample 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
changesign
changesign+
sign unchanged+ +
x=4
![Page 32: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/32.jpg)
Sign Charts of Factorable FormulasExample 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.
changesign
changesign+
sign unchanged+ +
x=4
![Page 33: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/33.jpg)
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat are factorable, that is, both the numerator and the denominator are factorable.
![Page 34: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/34.jpg)
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat are factorable, that is, both the numerator and the denominator are factorable.
Example C. Solve the inequality 2x2 – x3 x2 – 2x + 1
< 0
![Page 35: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/35.jpg)
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat 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:
![Page 36: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/36.jpg)
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat 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
![Page 37: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/37.jpg)
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat 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)
![Page 38: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/38.jpg)
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat 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).
![Page 39: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/39.jpg)
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat 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).
![Page 40: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/40.jpg)
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat 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
![Page 41: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/41.jpg)
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat 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
![Page 42: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/42.jpg)
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat 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
![Page 43: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/43.jpg)
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat 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
![Page 44: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/44.jpg)
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat 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
![Page 45: 1.5 sign charts and inequalities ii](https://reader036.vdocuments.us/reader036/viewer/2022081603/5583f7b7d8b42aa82c8b4857/html5/thumbnails/45.jpg)
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat 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