November 3, 2017 3.3 - Zeroes of Polynomials Fundamental Theorem of Algebra -Every polynomial of degree n will have n zeroes (real and complex/imaginary) Linear Factorization Theorem Every polynomial p(x) with degree n can be written as product of linear factors where c are complex numbers: p(x)=a(x-c 1 )(x-c 2 )....(x-c n ) The real numbers are a subset of the complex (when b = 0) complex = a + bi Example 1: Write p(x) = x 4 - 3x 2 - 4 as a product of linear factors and list all the zeroes.
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Fundamental Theorem of Algebra - Washington-Liberty...3.3 - Zeroes of Polynomials Fundamental Theorem of Algebra-Every polynomial of degree n will have n zeroes (real and complex/imaginary)
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November 3, 2017
3.3 - Zeroes of PolynomialsFundamental Theorem of Algebra
-Every polynomial of degree n will have n zeroes (real and complex/imaginary)
Linear Factorization Theorem
Every polynomial p(x) with degree n can be written as product of linear factors where c are complex numbers:
p(x)=a(x-c1)(x-c2)....(x-cn)
The real numbers are a subset of the complex (when b = 0)
complex = a + bi
Example 1: Write p(x) = x4 - 3x2 - 4 as a product of linear factors and list all the zeroes.
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Example 2: Write p(x) = 5x3 + 2x2 - 10x - 4 as a product of linear factors and list all the zeroes.
Factor by group tip:
It works if (first)(fourth) = (second)(third)
Multiplicity of Zeroes:
Multiplicity is how many times a zero is repeated. You may have an even or odd multiplicity
(x-3)2(x+3)4(x-1)(x+2)5
Example 3: State the zeroes and their multiplicity:
a)
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b) (x2+10x+25)(x2+x-20)(x-4)(x-5)
We saw in the last notes that zeroes that are complex (or use square roots) always come in conjugate pairs.
That is if -8i is a zero, then ____ is also one.
If 7+2i is a zero, then ______ is also one.
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How to multiply complex/radical roots the easier way to produce the original polynomial:
Remember (a-b)(a+b) = a2 - b2
Example 4:
a) zeroes are 4i and -4i
b) zeroes are ⎷3 and -⎷3
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c) zeroes are 5+i and 5-i
d) zeroes are 3+2⎷5 and 3-2⎷5
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Example 5: Write the equation of a degree 4 polynomial when it is known that 2 is the only real zero and -3i is also a zero.
Example 6: Write the equation of a degree 3 polynomial knowing that it has zeros of 1 and (1+i⎷2 )
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Note: If a problem doesn't ask for linear factors, then it is okay to leave (x2+4) as a factor instead of writing (x-2i)(x+2i).
They call (x2+4) an irreducible factor since it produces no real zeroes.
You will often have to do this if a problem asks for real coefficients/numbers only.
Intermediate Value Theorem (IVT)
Given P is a polynomial with real coefficients, if P(a) and P(b) have opposite signs then there is at least one value c between a and b such that P(c) = 0
In other words if you switch from positive y values to negative y-values (or vice versa), then there is a place where y = 0
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Example 7: Use the Intermediate Value Theorem to show P(x) = x3 - 9x + 6 has at least one zero in the given interval:
a) [0, 1] b) [-4, -3]
If P(a) and P(b) are both positive, does that mean there are no zeroes in the interval [a, b]?
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We've seen two theorems so far:
1. The Fundamental Theorem tells us that the zeroes exist.
2. IVT tells how to check intervals for a zero.
The next theorem will help give us a list of possible rational zeroes for a polynomial
Rational Zeroes TheoremGiven a polynomial with integer coefficients, and (p/q) is a rational number in lowest terms, the rational zeroes (if they exist) must be in the form of (p/q) where p is a factor of the constant term, and q is a factor of the leading coefficient.
Keep in mind this will not give complex or radical zeroes!
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Example 8: List all the possible rational zeroes for 3x3+7x2-4
Let's use a calculator to find the zeroes of the y = 3x3+7x2-4
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Example 9: List all the possible rational zeroes for 12x2-5x-2
Steps for finding the real zeroes of a polynomial function
Step 1: Use the degree of the polynomial to determine maximum number of zeroes
Step 2: Use the Remainder Theorem, synthetic division, and/or long division to test potential zeroes. Each time a zero is found continue to do step 2 on the new equation.
Remember that you can still use factoring techniques!
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Example 10: Write the polynomial as product of linear factorsa) 4x4-15x3+9x2+16x-12