SECTION 36: ZEROS OF POLYNOMIAL FUNCTIONS 3.6 SECTION EXERCISES VERBAL 1. Descri
ID: 3114351 • Letter: S
Question
SECTION 36: ZEROS OF POLYNOMIAL FUNCTIONS 3.6 SECTION EXERCISES VERBAL 1. Describe a use for the Remainder Theorem 2. Explain why the Rational Zeno Theorem does not guarantee finding zeros of a polynomial function. a What is the dafference between ratioeal and real I 4 If Descartes' Rale of Signs reveals a no change of signs or one sign of changes, what specific conclusion can be drawn? zeros? 5. If synthetic division reveals a zero, why should we try that value again as a possible solution ALGEBRAIC & i)32x+124-336x+80x, answer the following h What is the leading coefficient of fCx)? a. What is the degree of Cx) e 1f2+ V3,4i, 5, and 0 are zeros of fx), give the other zero(s) of fx). 7. Ifn«)-108r _ 60 + 20.. 40r'-4-24, answer the following: a What is the degree of fCx)? cIfv3,1, and 2 - are zeros of flx), give the other zerols) of flx). b. What is the leading coeficient of flx)? For the following exercises, use synthetic division and the Remainder Theorem to evaluate f at the given value. 11. 1/4) if(x)-8x'-18x' + 12x-7 For the folowing exercises, use the given zero to find the remaining zeros for each of the following polynomial functions 13. fx) +5x-2x-10,-5 15.f(x)=x'+4e+16e + 36x + 63; 3i 7. flx)-12+51x-78x+585+2i For the following exercises, find a polynomial of lowest degree in general form with rational coefficients that has the given numbers as some of its zeros. In addition to the theorems introduced in this section, the below theorem may also be used. irrational conjugate theorem Ife+ cvS is a zero ofa polynomial functionf(x) with rational coefficients, then-cvS is also b is not a perfect square. zero, c. 164i,-3 19. 1 + 2i,9Explanation / Answer
1. The Remainder Theorem starts with an unnamed polynomial p(x), where "p(x)" just means "some polynomial p whose variable is x". Then the Theorem talks about dividing that polynomial by some linear factor (x – a), where a is just some number. Then, as a result of the long polynomial division, you end up with some polynomial answer q(x) (the "q" standing for "the quotient polynomial") and some polynomial remainder r(x).
The reamainder will always be atleast one degree lesser than the dividend. If the dividend has a degree of 6 (like x^6) then the degree of the remainder will be less than or equal to 5.
The Remainder Theorem then points out the connection between division and multiplication.
If p(x) / (x – a) = q(x) with remainder r(x),
then p(x) = (x – a) q(x) + r(x).
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.