Download - 2.3 Real Zeros of Polynomial Functions
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2.3 Real Zeros of Polynomial Functions
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Long Division
Write the dividend and divisor in descending powers of variable and insert placeholders with zeros for missing powers of the variable
1) Divide x2 +5x+6 by x+2
2) Divide x3 – 1 by x – 1
3)Divide 2x4 + 4x3 – 5x2 + 3x – 2 by x2 + 2x - 3
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The Division Algorithm
If f(x) and d(x) are polynomials such that d(x) ≠ 0, and the degree of d(x) is less than or equal to the degree of f(x), there exist unique polynomials q(x) and r(x) such that
f(x) = d(x)q(x) + r(x)
Where r(x) = 0 or the degree of r(x) is less than the degree of d(x).
Dividend
Divisor
Quotient
Remainder
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Long Division of Polynomials
f(x) = 6x3 – 19x2 + 16x – 4 Divide by 2x2 – 5x + 2
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Remainder
Divide x2 + 3x + 5 by x + 1
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Synthetic Division
Only can be used for linear divisors: (x – k) Divide by the zero!!! Not the factor
1) Divide x4 – 10x2 -2x + 4 by x + 3
2) Divide -2x3 + 3x2 + 5x – 1 by x + 2
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The Remainder Theorem
If a polynomial f(x) is divided by x – k, the remainder is
r = f(k)
Use the remainder theorem to evaluate the following function at x = -2
f(x) = 3x3 + 8x2 + 5x – 7
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Factor Theorem
• A polynomial f(x) has a factor (x – k) if and only if f(k) = 0
• You can use synthetic division to test whether a polynomial has a factor (x – k) – If the last number in your answer line is a zero,
then (x – k) is a factor of f(x)
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Repeated Division
Show that (x – 2) and (x + 3) are factors of f(x) = 2x4 + 7x3 – 4x2 – 27x – 18
Then find the remaining factors of f(x)
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Using the Remainder in Synthetic Division
In summary, the remainder r, obtained in the synthetic division of f(x) by (x – k), provides the following information.
1) The remainder r gives the value of f at x = k. That is, r = f(k)
2) If r = 0, (x – k) is a factor of f(x)
3) if r = 0, (k, 0) is an x-intercept of the graph f
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Examples!!
f(x) = 2x3 + x2 – 5x + 2 (x + 2) and (x – 1)
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Rational Zero TestIf the polynomial
f(x) = anxn + an-1xn-1 + …a2x2 + a1x + a0
Has integer coefficients, every rational zero of f has the form
Rational Zero = p/q
Where p and q have not common factor other than 1, p is a factor of the constant term a0, and q is a factor of the leading coefficient an
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Finding the possible real zeros
1) f(x) = x3 + 3x2 – 8x + 3
2) f(x) = 2x 4 – 5x2 + 3x – 8
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Descartes’ Rule of SignsLet f(x) = anxn + an-1xn-1 + …a2x2 + a1x + a0 be a polynomial with real coefficients and a0 ≠ 0
1) The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer
2) The number of negative real zeros of f is either equal to the number of variations in sign of f(-x) or less than that number by an even integer.
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Using Descartes’ Rule of Signs
Describe the possible real zeros of f(x) = 3x3 – 5x2 + 6x - 4
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Upper and Lower Bound RulesLet f(x) be a polynomial with real coefficients and a positive leading coefficient. Suppose f(x) is divided by x – c, using synthetic division.
1. if c > 0 and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f
2. If c < 0 and the numbers in the last row are alternately positive and negative (zeros count as either), c is a lower bound for the real zeros of f
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Finding the Zeros of a Polynomial Function
f(x) = 6x3 – 4x2 + 3x - 2