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 \(x=2\)
    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 \(x=\frac{1}{3}-\frac{\sqrt{2}}{3}\), \(x=\frac{1}{3}+\frac{\sqrt{2}}{3}.\)
    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. \(-375x^{11}\)

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