Chapter 3.1 Practice

Questions

  1. For each quadratic polynomial \(f\), determine the vertex of \(f\) and whether \(f\) has a maximum or minimum \(y\)-value.
    1. \(f(x)=-2(x+5)^2-9\)
    2. \(f(x)=(x-2)^2-4\)
    3. \(f(x)=5(x-1)^2-9\)
    4. \(f(x)=-4(x-1)^2+4\)
  2. Determine of all the zeros of the quadratic polynomial \(f\).
    1. \(f(x)=-2(x+5)^2-9\)
    2. \(f(x)=(x-2)^2-4\)
    3. \(f(x)=5(x-1)^2-9\)
    4. \(f(x)=-4(x-1)^2+4\)
  3. For each quadratic polynomial \(f\), identify coefficients \(A,h\), and \(k\) so that \(f(x)=A(x-h)^2+k\) and use this expression for \(f\) to determine all zeros of \(f.\)
    1. \(f(x)=2x^2+5x+3\)
    2. \(f(x)=4x^2-16x+21\)
    3. \(f(x)=-2 x^2 + 12 x - 22\)
    4. \(f(x)=-9 x^2 + 6 x + 1\)
  4. For each polynomial \(f\) with the prescribed features, determine all possible equations for \(f.\)
    1. \(f\) is a degree \(3\) polynomial that intersects the \(x\)-axis at \(-4\), \(2\), and \(5\).
    2. \(f\) is a degree \(2\) polynomial with leading coefficient \(-5\) that has zeros \(-1\) and \(4\).
    3. \(f\) is a degree \(4\) polynomial with leading coefficient \(-5\) that has zeros \(-1\) and \(4\) and no other zeros.
  5. Identify polynomials \(q\) and \(r\) with the given properties.
    1. \(\frac{r(x)}{x+1}\) is a proper fraction and \(2x^3+2x^2+5x+9=q(x)(x+1)+r(x)\)
    2. \(\frac{r(x)}{x^2+4x+2}\) is a proper fraction and \(x^5 + 4 x^4 + 2 x^3 + x^2 + 8 x + 3=q(x)(x^2+4x+2)+r(x)\)
    3. \(\frac{r(x)}{x-4}\) is a proper fraction and \(x^3 - 3 x^2 - 4 x + 5=q(x)(x-4)+r(x)\)
  6. Identify the following reminders.
    1. \(\frac{3x^2+5x+1}{x-2}\)
    2. \(\frac{10x^5+4x^2+1}{x+1}\)
    3. reminder of \(4x^3+2x^2-x-5\) divided by \(x-3\)
  7. For each polynomial \(f\), identify the zeros and the orders of each zero.
    1. \(f(x)=2(x+3)^2(x+2)^3(x-5)^3(x-6)^8\)
    2. \(f(x)=5x^3(x+5)(x+1)^6(2x-3)^6(x-4)^7\)
    3. \(f(x)=4(x+5)^4(5x+2)^3(5-2x)^5(x-6)\)
    4. \(f(x)=-5x(x+5)^4(5x+2)^2(2x-5)(x-6)^3\)
  8. For each polynomial \(f\), identify the leading term.
    1. \(f(x)=2(x+3)^2(x+2)^3(x-5)^3(x-6)^8\)
    2. \(f(x)=5x^3(x+5)(x+1)^6(2x-3)^6(x-4)^7\)
    3. \(f(x)=4(x+5)^4(5x+2)^3(5-2x)^5(x-6)\)
    4. \(f(x)=-5x(x+5)^4(5x+2)^2(2x-5)(x-6)^3\)
  9. For each polynomial \(f\), identify the asymptotic behavior.
    1. \(f(x)=2(x+3)^2(x+2)^3(x-5)^3(x-6)^8\)
    2. \(f(x)=5x^3(x+5)(x+1)^6(2x-3)^6(x-4)^7\)
    3. \(f(x)=4(x+5)^4(5x+2)^3(5-2x)^5(x-6)\)
    4. \(f(x)=-5x(x+5)^4(5x+2)^2(2x-5)(x-6)^3\)
  10. Graph each polynomial \(f\) by using the asymptotic behavior of \(f\) and the behavior of \(f\) near its zeros.
    1. \(f(x)=2(x+3)^2(x+2)^3(x-5)^3(x-6)^8\)
    2. \(f(x)=5x^3(x+5)(x+1)^6(2x-3)^6(x-4)^7\)
    3. \(f(x)=4(x+5)^4(5x+2)^3(5-2x)^5(x-6)\)
    4. \(f(x)=-5x(x+5)^4(5x+2)^2(2x-5)(x-6)^3\)

Answers

    1. vertex at \((-5,-9)\), maximum \(y\)-value at \(-9\).
    2. vertex at \((2,-4)\), minimum \(y\)-value at \(-4\)
    3. vertex at \((1,-9)\), minimum \(y\)-value at \(-9\)
    4. vertex at \((1,4)\), maximum \(y\)-value at \(4\).
    1. no zeros
    2. \(x=4\) and \(x=0\)
    3. \(x=1-\frac{3}{\sqrt{5}}\) and \(x=1+\frac{3}{\sqrt{5}}\)
    4. \(x=0\) and \(x=1\)
    1. \(f(x)=2\left(x+\frac{5}{4}\right)^2-\frac{1}{8}\), its zeros are \(x=-1\), \(x=-\frac{3}{2}\)
    2. \(f(x)=4(x-2)^2+5\), it has no zeros
    3. \(f(x)=-2(x-3)^2-4\), it has no zeros
    4. \(f(x)=-9\left(x-\frac{1}{3}\right)^2+2\), its zeros are \(x=\frac{1}{3}-\frac{\sqrt{2}}{2}\), \(x=\frac{1}{3}+\frac{\sqrt{2}}{2}.\)
    1. \(f(x)=A(x+4)(x-2)(x-5)\) where \(A\) is a non-zero real number
    2. \(f(x)=-5(x+1)(x-4)\)
    3. \(f(x)=-5(x+1)^3(x-4)\), \(f(x)=-5(x+1)^2(x-4)^2\) or \(f(x)=-5(x+1)(x-4)^3.\)
    1. \(q(x)=2x^2+5\) and \(r(x)=4\)
    2. \(q(x)=x^3+1\) and \(r(x)=4x+1\)
    3. \(q(x)=x^2+x\) and \(r(x)=5.\)
    1. \(23\)
    2. \(-5\)
    3. \(118\)
    1. zero at \(x=-3\) with order \(2\), zero at \(x=-2\) with order \(3\), zero at \(x=5\) with order \(3\), zero at \(x=6\) with order \(8\)
    2. zero at \(x=-5\) with order \(1\), zero at \(x=-1\) with order \(6\), zero at \(x=0\) with order \(3\), zero at \(x=\tfrac{3}{2}\) with order \(6\), zero at \(x=4\) with order \(7\)
    3. zero at \(x=-5\) with order \(4\), zero at \(x=-\frac{2}{5}\) with order \(3\), zero at \(x=\frac{5}{2}\) with order \(5\), zero at \(x=6\) with order \(1\).
    4. zero at \(x=-5\) with order \(4\), zero at \(x=-\frac{2}{5}\) with order \(2\), zero at \(x=\frac{5}{2}\) with order \(1\), zero at \(x=6\) with order \(3\).
    1. \(2x^{16}\)
    2. \(320x^{23}\)
    3. \(-16000x^{18}\)
    4. \(-375x^{11}\)
    1. \(2x^{16}\)
    2. \(320x^{23}\)
    3. \(-16000x^{18}\)
    4. \(-375x^{11}\)

Return

© Copyright 2024 by the POC Writing Team: Bryan Carrillo, Yat Sun Poon, and David Weisbart.

All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the POC Writing Team.