polynomials integrated math 4 mrs. tyrpak. definition
TRANSCRIPT
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Polynomials
Integrated Math 4
Mrs. Tyrpak
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Definition
Let n be a nonnegative integer and let be real numbers and exponents be positive.
This is called a polynomial function of degree n.
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Key Terms
Let
Let’s label the degree, leading coefficient, and constant term. (MP)
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You try one…
Find a partner.
If you’re the tallest, say the degree.
If you’re the shortest, say the constant term.
Say the leading coefficient at the same time.
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Relationships?
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The Leading Coefficient Test
L.C.EVEN
ODD
Same
Different
L.C.
DEGREE
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Real Zeros
The following are equivalent statements about real zeros of a polynomial function:
1. x = a is a zero (solution or root) of the function
2. (a, 0) is an x-intercept of the graph
3. (x – a) is a factor of the polynomial
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Finding All Zeros
First we are going to use the rational zeros theorem: If a polynomial has integer coefficients, every rational zero has the form , where p is a factor of the constant term, and q is a factor of the leading coefficient.
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Rational Zeros Theorem
Possible rational zeros:
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Synthetic Division
Second, we are going to use synthetic division to test each possible zero.
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The Remainder and Factor Theorems
• Remainder Theorem: If a polynomial f(x) is divided by x – k, the remainder is r = f(k).
• Factor Theorem: A polynomial f(x) has a factor (x - k) if and only if f(k)=0.
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Factor the Polynomial
Thirdly, we are going to factor using the remainder and factor theorems.
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Find all the zeros
• Lastly, we will solve each factor for each zero.
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Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n, where n > 0, f has at least one zero in
the complex number system.
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Multiplicity
If is a factor of a polynomial …
k is odd, then the graph crosses the x-axis at (c, 0)
k is even, then the graph is tangent to the x-axis at (c,0)
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𝑓 (𝑥 )=(𝑥+5 )3 (𝑥−4 ) (𝑥+1 )2
Find the zeros of the polynomial function and state the multiplicity of each and what happens at each zero.
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Finding All Zeros and Factors
• Find all the zeros of
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Take 5 minutes and try this one..
𝑔 (𝑥 )=𝑥4+6 𝑥3+10 𝑥2+6 𝑥+9
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Let’s try and work backwards
• Find a polynomial with integer coefficients that has the given zeros:
1, 5i, -5i
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Thanks for your attention!
Don’t forget to complete both the extension and enrichment assignments
before you move on.
Keep up the hard work!