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Section 4.2 Polynomial Functions of Higher Degree
Polynomial Function P(x) Graph
P(x) = a degree 0 Horizontal line through (0,a)
P(x) = ax +b (degree 1) line with yintercept (0,b) and slope a
P(x) = ax2+bx+c (degree 2) parabola with vertex (b/2a, P(b/2a) )
Polynomials degree 3 or higher
P(x) = anxn + an1xn1 + . . . + a1x + a0
The leading term anxn is the dominating term
Farright and farleft behavior of the graph of a polynomial can be determined by looking at the leading coefficient an.
n is even n is odd
an > 0 Up to left and up to right Down to left and up to right
an < 0 Down to left and down to right Up to left and down to right
GraphP(x) = 2x4 + 3x + 1 R(x) = 3x3 2x + 1
S(x) = 4x6 + 2x2 x + 1 T(x) = 2x5 + 1
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turning points
Turning Points
The graph of a polynomial function of degree n has at most n1 turning points
Relative Minimum and Relative Maximum
If there is an open interval I containing c on whichf(c) ≤ f(x) for all x in I, then f(c) is a relative minimum of f.f(c) ≥ f(x) for all x in I, then f(c) is a relative maximum of f.
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Use the calculator to find the relative max and min of P(x) = x4 6x2 2
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The Zero Location Theorem
Let P(x) be a polynomial and let a and b be two distinct real numbers. If P(a) and P(b) have opposite signs, then there is at least one real number c between a and b such that P(c) = 0.
(b,P(b))
P
(a,P(a))
ba c
P(a)
P(b)
P(c)=0
Use the Zero Location Theorem to verify that P has a zero between a and b.
P(x) = 2x3 + 3x2 23x 42; a=3, b=4
3 2 3 23 42
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Even and Odd Powers of (x c) TheoremIf c is a real number and the polynomial function P(x) has (xc) as a factor exactly k times, then the graph of P will
• if k is even, intersect but not cross the xaxis at (c,0)
• if k is odd, cross the xaxis at (c,0)
Graph: P(x) = (x+1)3(x2)2 : Crosses at 1, intersects at 2
P(x) = (x 1)3(x 3)(x +1)4 :
Application: Advertising Expenses
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The rational zero theorem helps us find zeros when we have large numbers that are hard to factor
Rational Zero Theorem If P(x) = anxn + …+ a1x + a0, has integer
coefficients (an≠ 0) and p/q is a rational zero of P, then• p is a factor of the constant term a0 and• q is a factor of the leading term coefficient an.
Example : Use the Rational Zero Theorem to list all possible zeros of
P(x) = 3x3 + 11x2 6x 8
p: ±1, ±2, ±4, ±8q: ±1, ±3
possible rational zeros are ±1, ±2, ±4, ±8, ±1/3, ±2/3, ±4/3, ±8/3
other real zero is not rational
an
ana0
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Rational Numbers can be represented as p/q where p, q are integers.
Some real numbers that are not : π, e, √2
Example: x2 + 2x 2 = 0
p: ±1, ±2q: ±1
p/q = ±1, ±2
Solve x2 + 2x 2 = 0x2 + 2x = + 2
+ 1 +1(x + 1)2 = 3
x = 1 ±√3
Use the Rational Zero Theorem to list the possible rational zeros.
P(x) = x3 19x 30
p = ±1,±2, ±3, ±5, ±6, ±10, ±15, ±30q = ±1
p/q = ±1,±2, ±3, ±5, ±6, ±10, ±15, ±30
Graph x = 5 is a zero
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Upper and LowerBound theoremLet P(x) be a polynomial function with real coefficients and a positive
leading coefficient. Use synthetic division to divide P(x) by x b, where b is a nonzero real number:
Upper bound If b > 0 and all the numbers in the bottom row of the synthetic division of P by x b are either positive or zero, then b is an upper bound for the real zeros of P.
Lower bound If b < 0 and all the numbers in the bottom row of the synthetic division of P by x b alternate sign (zero either pos. or neg.), then b is a lower bound for the real zeros of P.
Example: P(x) = 3x3 + 11x2 6x 8
3 11 6 86 3 11 6 85
Not uniquelower bound
Upper and LowerBound theoremLet P(x) be a polynomial function with real coefficients and a positive
leading coefficient. Use synthetic division to divide P(x) by x b, where b is a nonzero real number:
Upper bound If b > 0 and all the numbers in the bottom row of the synthetic division of P by x b are either positive or zero, then b is an upper bound for the real zeros of P.
Lower bound If b < 0 and all the numbers in the bottom row of the synthetic division of P by x b alternate sign (zero either pos. or neg.), then b is a lower bound for the real zeros of P.
Example: P(x) = 3x3 + 11x2 6x 8
3 11 6 83 3 11 6 82
Upperbound
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If P (x) = anxn + an−1xn−1 + . . . + a2x2 + a1x + a0 be a polynomial with real coefficients. 1. The number of positive real zeros of P(x) is either equal to the number
of sign changes of P (x) or is less than that number by an even integer. If there is only one variation in sign, there is exactly one positive real zero.
2. The number of negative real zeros of P(x) is either equal to the number of sign changes of P (−x) or is less than that number by an even integer. If P (−x) has only one variation in sign, then f has exactly one negative real zero.
Descartes' Rule of Signs
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Determine the possible number of positive and negative real zeros of P(x) = x3 + 2x2 + 5x + 4.
EXAMPLE: Using Descartes’ Rule of Signs
1. To find possibilities for positive real zeros, count the number of sign changes in the equation for P (x). Because all the terms are positive, there are no variations in sign. Thus, there are no positive real zeros.
There are three sign changes, so there are either three or one negative real zero
EXAMPLE 2: Using Descartes’ Rule of Signs
Determine the possible number of positive and negative real zeros of P(x) = 2x4 − 5x2 − 12.
1. P(x) sign changes (pos zeros):
2. P(x) sign changes (neg. zeros) :
Factor to solve :
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Find the Real Zeros of P(x) = 3x3 + 11x2 6x 8
1. Degree is 3. At most 3 zeros.
2. Descartes Rule of Signs: 1 positive real zero
P(x) = 3x3 + 11x2 + 6x 82 or 0 neg. real zeros
3. Rational Zero Theorem : p factors of 8 = ±1, ±2, ±4, ±8q factors of 3 = ±1, ±3
p/q = ±1, ±2, ±4, ±8, ±1/3, ±2/3, ±4/3, ±8/3
4. Synthetic Division Reduce polynomial:
Find the Real Zeros of P(x) = 3x3 x2 6x + 2
1. Degree is 3. At most 3 zeros.
2. Descartes Rule of Signs: positive real zero
P(x) = neg. real zeros
3. Rational Zero Theorem : p factors of 2 = q factors of 3 =
p/q =
4. Synthetic Division Reduce polynomial:
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P(x) = 0.03x3 + 0.02x2 + 1.5x 2.1
Advertising Expenses A company manufactures computers. the company estimates that the profit from computer sales is
where P is profit in millions of dollars and x is the amount, in hundredthousands of dollars, spent on advertising.
Determine the minimum amount, rounded to the nearest thousand dollars, the company needs to spend on advertising if it is to receive a profit of $2,000,000.
Section 4.4 The Fundamental Theorem of Algebra
The Fundamental Theorem of AlgebraIf P(x) is a polynomial function of degree n≥1 with complex coefficients, then P(x) has at least one complex zero.
Linear Factor TheoremIf P(x) is a polynomial function of degree n≥1 with leading coefficient an≠ 0.
P(x) = anxn + …+ a1x + a0then P(x) has exactly n linear factors
P(x) = an(x c1)(x c2) .... (x cn)where c1, c2, ..., cn are complex numbers.
The Number of Zeros TheoremIf P(x) is a polynomial function of degree n≥1 with complex coefficients, then P(x) has exactly n complex zeros, provided each zero is counted according to its multiplicity.
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Find the Zeros and linear factors of a Polynomial Function
P(x) = x3 3x2 + 7x 5
p: ±1, ±5q: ±1
p/q: ±1, ±5
Graph: Try x = 1.
1 1 3 7 5
P(x) = (x 1)(x 1 2i)(x 1 + 2i)
The Conjugate Pair TheoremIf a + bi is a complex zero of a polynomial function with real coefficients, then the conjugate a bi is also a complex zero of the polynomial function.
Find all the zeros of P(x) = 2x3 5x2 + 6x 2, given that (1 + i) is a zero.
1 + i 2 5 6 2
remember i2 = 1
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Find all the zeros of P(x) = 8x4 2x3 +199x2 50x 25, given that 5i is a zero.
5i 8 2 199 50 25
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40i
2 40i
5i, 5i, 1/4, 1/2
Find a polynomial function P(x) with real coefficients that has the indicated zeros
Zeros : 3, 2i, 2i
P(x) = (x 3)(x 2i)(x + 2i) Multiply
= (x 3) (x2 + 4)
Zeros : 2 + 3i, 2 3i, 5, 2