Transcript
Sign Charts of Factorable Formulas
http://www.lahc.edu/math/precalculus/math_260a.html
In this section we give a theorem about sign-charts of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas
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
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)
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
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
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
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
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.
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.
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,
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.
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.
Sign Charts of Factorable FormulasAn important property of a root is whether it is an even-ordered root or an odd-ordered root.
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,
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)
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).
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)
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.
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.
Sign Charts of Factorable FormulasExample B. Make the sign-chart of x3(x + 3)2 (x – 3)
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.
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.
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)
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.
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
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
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
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
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
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
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
Sign Charts of Factorable FormulasThe theorem may be generalized to rational formulasthat are factorable, that is, both the numerator and the denominator are factorable.
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
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:
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
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)
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).
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).
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
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
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
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
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
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
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