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Unformatted text preview: /' ' 6 “may 0% MA 180 — Precalculus
Professor Terry Section 3.4: Complex and Rational Zeros of Polynomials E Complex Zeros Theorem on Conjugate Pair Zeros of a Polynomial: If a polynomial f (x) of degree n >1 has real Coefﬁcients and if z : a +bi with b at 0 is a complex zero of f (x) , then the conjugate Eza—biis also azero of f(x). 4 ﬁndg polynomial f (x) with real coefﬁcients and leading coefﬁcient 1 that has zero<—§ and
f _ '\ ‘—"—__‘—'_'_‘"— , V‘
LYL (1—71 nddegreeB. “:2; dpﬁW33 {mﬂ63_3‘ if“ I (+7L  59a} = (1(x4cXX—c—JJCM—Ce) .
_ §§§_3__=ctC__><f (§3?]__,Cf7(_ltt_1_i)] gm % l (mzﬂUﬂUCxlmg T polynomial with real coefﬁcients and positive degree n can be expressed as a product of linear
and quadratic polynomials with real coefﬁcients such that the quadratic factors are irreducible over SR. Proof: Since f(x) have precisely n complex zeros c1, 02, C", we can write f(x) = 8(x — 01)(X  C2) . .  (x — on). where a is the leading coefﬁcient of f(x). Of course, some of the zeros may be real. In such cases, we obtain the linear factors referred to in the statement.
If a zero ck is not real, then by the theorem of conjugate pair zeros of a polynomial, the conjugate Ek is also a zero of f(x) and hence most be one of the numbers 01, c2, cn_This
implies that both x — ck and x  Ek appear in the factorization of f(x) if those factors are multiplied, we obtain
(x_ck)(x—Ek) = ’52 ‘(ck +Ek)x+ckak) which has real coefﬁcients since ck +Ek and CE,( are real numbers. Thus, if ck is a complex
zero, then the product (x—ck)(x—Ek) is a quadratic polynomial that is irreducible over SR. This completes the proof. MA180 Sec3.4 SpringO3 ewt Express the polynomial as a product of linear and quadratic factors: x5 —4x3 —x1 +4 :‘Sewixzqul 2 XBKQQH (35:43 , m m_jlégtq>(£éfi>_t_ ,,,,, _W
* 4*“???le E'Céfié?(w3@"m‘2“@ Rational Zeros Theorem on Rational Zeros of a Pol nomial: if the polynomial has integer coefﬁcients and if did is a rational zero of f(x) such that c and d have no common prime factor, then
1) the numerator c of the zero is a factor of the common term a; 2) the denominator d of the zero is a factor of the leading coefﬁcient an Show that the equatioréla —4x2 + 7x @ O has no rational root.
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¥on2 Bygﬁqx +7y+gf Finding the Rational Solutions of an Eguation Find all solutions of the equation 3 +83:2 —3x@ 0. J+o<2 (355523 —i (and?) :4 O Use a calculator to ﬁnd the rational solutions for 12::3 + 8x2 7 3s— 2 = 0.
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AL} M 6 Constructing a crate: The frame for a shipping crate is to be constructed from 24 feet of 2 X 2
lumber. Assuming that crate is to have square ends of length x feet, determine the value(s) of x that result(s) in a volume of .4 ft3. V W a V .3 H gﬁ 3 F’J am 9% 09 93a? M ’é . W MA180 Sec3.4 SpringD3 ewt X 3  Co K 2+ L} 5: O ...
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 Spring '08
 Terry

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