Review: Synthetic Division
Factor Theorem
Find (x2 - 5x - 5x3 + x4) ÷ (5 + x).
Solve 2x3 - 5x2 + x + 2 =0 given that 2 is a
zero of f(x) = 2x3 - 5x2 + x + 2.
Introduction
Zeros of Polynomial Functions Objective: To find a polynomial with specified zeros, rational zeros, and other zeros.
Rational Zero Theorem If the polynomial f(x) = anxn + an-1xn-1 + . . . + a1x + a0 has integer coefficients, then every rational _____ of f(x) is of the form
Polynomial Type of Coefficient 5x3 + 3x2 + (2 + 4i) + i complex 5x3 + 3x2 + √2x – π
real
5x3 + 3x2 + ½ x – ⅜
rational
5x3 + 3x2 + 8x – 11
integer
Rational Root (Zero) Theorem (in other words) If “q” is the leading coefficient and “p” is the
constant term of a polynomial, then the only possible rational roots are + factors of “p” divided by + factors of “q”. (p / q)
p q where p is a factor of the ________ coefficient a0 and q is a factor of the ________ coefficient an.
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Rational Root (Zero) Theorem (in other words) 5
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Example: f (x) = 6x −4x −12x + 4 To find the POSSIBLE rational roots of f(x), we need the FACTORS of the leading coefficient (6 for this example) and the factors of the constant term (4, for this example). Possible rational roots are
Example List all possible rational zeros of
f(x) = x3 + 2x2 – 5x – 6.
± factors of p ±1, ±2, ±4 1 1 1 2 4 = = ± 1, 2, 4, , , , , ± factors of q ±1, ±2, ±3, ±6 2 3 6 3 3
Another example
How do we know which possibilities are really zeros (solutions)?
List all possible rational zeros of
Use trial and error and ________ division to
f(x) = 4x5 + 12x4 – x – 3.
see if one of the possible zeros is actually a zero. Remember: When dividing by x – c, if the ________ is 0 when using synthetic division, then c is a zero of the polynomial. If c is a zero, then solve the polynomial resulting from the synthetic division to find the other zeros.
Example
Finding the Rational Zeros of a Polynomial
Find all zeros of f(x) = x3 + 8x2 + 11x – 20.
1.
List all ________ rational zeros of the polynomial using the Rational Zero Theorem.
2.
Use synthetic division on each possible rational zero and the polynomial until one gives a remainder of ________. This means you have found a zero, as well as a factor. Write the polynomial as the ________ of this factor and the quotient.
3. 4.
Repeat procedure on the quotient until the quotient is ________
5.
Once the quotient is quadratic, factor or use the quadratic formula to find the remaining real and imaginary zeros.
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Example
List all possible zeros, and use synthetic division to test and find an actual zero. Then use the quotient to find the remaining zeros.
Find all zeros of f(x) = x3 + x2 - 5x – 2.
f(x) = x3 – 4x2 + 8x - 5
More review -- List all possible zeros. Use synthetic division to test and find an actual zero. Then use the resulting quotient to find the remaining zeros.
f(x) = x3 + 4x2 - 3x - 6
How many zeros, not necessarily rational, does a polynomial with rational coefficients have? An nth degree polynomial has a total of n ________.
Some may be rational, irrational or complex. Because all coefficients are RATIONAL, irrational roots
exist in ________ (both the irrational # and its conjugate). ________ roots also exist in pairs (both the complex # and its conjugate). If a + bi is a root, a – bi is a root If a + b is a root, a − b is a root. NOTE: Sometimes it is helpful to graph the function and find the x-intercepts (zeros) to narrow down all the possible zeros.
Example Solve: x4 - 6x3 + 22x2 - 30x + 13 = 0.
Fundamental Theorem of Algebra If f(x) is a polynomial function of degree n,
≥
where n > 1, then the equation f(x) = 0 has at least one complex zero, real or imaginary. Note: This theorem just guarantees a zero
exists, but does not tell us how to find it.
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Remember…
Linear Factorization Theorem
Complex zeros come in pairs as
complex conjugates: a + bi, a – bi Irrational zeros come in pairs.
a+ c b , a− c b
More Practice
Practice
Find a polynomial function, in factored form, of degree 5 with -1/2 as a zero with multiplicity 2, 0 as a zero of multiplicity 1, and 1 as a zero of multiplicity 2.
Find a polynomial function of degree 3 with 2 and i as zeros.
Example Find a third-degree polynomial function f(x)
Solve the given polynomial equation. Use the Rational Zero Theorem, or graph as an aid to obtaining the first zero. x4 – x3 + 2x2 – 4x – 8 = 0.
with real coefficients that has -3 and i as zeros and such that f(1) = 8.
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Extra Example
Extra Example
Suppose that a polynomial function of degree 4 with rational coefficients has i and (-3 +√3) as zeros. Find the other zero(s).
Find a polynomial of degree 3 where 4 and 2i are zeros, and f(-1) = -50.
Extra Example Use the Rational Zero Theorem to list all the
possible zeros for f(x) = 4x5 – 8x4 – x + 2.
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