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9.4: Exercises

Last updated

May 2, 2022
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- 9.3: Optional Section- Graphing Polynomials by Hand
- 10: Roots of Polynomials

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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Exercise $\PageIndex{1}$

Which of the graphs below could be the graphs of a polynomial?

$clipboard_eccd4976a4db98c81906d1411e9de1c2d.png$
$clipboard_ed06a093aa7fb90025ab36bbecd8dc67e.png$
$clipboard_e179b42dfd4f4478f997f38745199e6a0.png$
$clipboard_ea1dbd3f00355895daf82d6d615a9ffd4.png$
$clipboard_eaa388b50a4f49148ea8adcf824a57989.png$
$clipboard_ed6576ca05b12277bf6bfd1d76fce89ba.png$

Answer

yes
no (due to the discontinuity)
no (due to horizontal asymptote)
no (due to corner)
yes (polynomial of degree 1)
yes

Exercise $\PageIndex{2}$

Identify each of the graphs (a)-(e) with its corresponding assignment from (i)-(vi) below.

$clipboard_ef52a00bcb5d083a3e9861c73f17f2ef7.png$
$clipboard_eae5247acf7ce48db71576f139230ecff.png$
$clipboard_e03c102e59fada6f4816b51f91dc3a527.png$
$clipboard_e3516c1d5c2a05f6618448181985cbe9a.png$
$clipboard_efe1203f1e8c7e22f48cc5aee5bd510f8.png$

$f(x)=-x^2$
$f(x)=-0.2x^2+1.8$
$f(x)=-0.6 x +3.8$
$f(x)=-0.2 x^3+0.4x^2+x-0.6$
$f(x)=x^3-6x^2+11x-4$
$f(x)=x^4$

Answer

corresponds to (iii)
corresponds to (v)
corresponds to (vi)
corresponds to (ii)
corresponds to (iv)

Exercise $\PageIndex{3}$

Identify the graph with its assignment below.

$clipboard_e937e195944465f3182f7aae76eb09784.png$
$clipboard_ee90939bbdd8a45549af47046eceda3e0.png$
$clipboard_ea63c3680b937f4abd162a7c6f3bb0ab8.png$

$f(x)=x^6-14x^5+78.76 x^4-227.5 x^3+355.25x^2-283.5x+93$
$f(x)=-2x^5+30x^4-176x^3+504x^2-704x+386$
$f(x)=x^5-13x^4+65x^3-155x^2+174x-72$

Answer

corresponds to (iii)
corresponds to (i)
corresponds to (ii)

Exercise $\PageIndex{4}$

Sketch the graph of the function with the TI-84, which includes all extrema and intercepts of the graph.

$f(x)=0.002 x^3+0.2 x^2-0.05x-5$
$f(x)=x^3+4x+50$
$f(x)=0.01 x^4-0.101 x^3-3 x^2+50.3x$
$f(x)=x^3-.007x$
$f(x)=x^3+.007x$
$f(x)=0.025 x^4+0.0975 x^3-1.215 x^2+2.89 x-22$

Answer

$clipboard_e601918625705d412d198c78b85bcc8d1.png$
$clipboard_e634dd9243ea604d148bcf3630b7006f7.png$
$clipboard_ed3984d1c063d72c0d604473c78edf2fe.png$
$clipboard_e0218a4e575d060060b3e27a7e48a61c0.png$
$clipboard_e45559a0b1e714df5d63b26c4da709979.png$
$clipboard_e4d1e503dff49a2690c18558ec2a32316.png$

Exercise $\PageIndex{4}$

Find the exact value of at least one root of the given polynomial.

$f(x)=x^3-10x^2+31x-30$
$f(x)=-x^3-x^2+8x+8$
$f(x)=x^3-11x^2-3x+33$
$f(x)=x^4+9x^3-6x^2-136x-192$
$f(x)=x^2+6x+3$
$f(x)=x^4-6x^3+3x^2+5x$

Answer

$x = 2$ , $x = 3$ , or $x = 5$
$x = −1$
$x = 11$
$x = −8$ , $x = −3$ , $x = −2$ , or $x = 4$
$x=-3 \pm \sqrt{6}$ (use the quadratic formula)
$x = 0$

Exercise $\PageIndex{5}$

Graph the following polynomials without using the calculator.

$f(x)=(x+4)^2(x-5)$
$f(x)=-3(x+2)^3 x^2 (x-4)^5$
$f(x)=2(x-3)^2(x-5)^3(x-7)$
$f(x)=-(x+4)(x+3)(x+2)^2(x+1)(x-2)^2$

Answer

$clipboard_ef4f92d6e177498b749141401ce6bcb3c.png$
$clipboard_ede182b55ff389b4e633a1b348b839e19.png$
$clipboard_e658546592435a7fe5467a50092a50fbe.png$
$clipboard_e415fac322053d29c89691f9f54b735e4.png$

Exercise 9.4.1\PageIndex{1}

Exercise 9.4.2\PageIndex{2}

Exercise 9.4.3\PageIndex{3}

Exercise 9.4.4\PageIndex{4}

Exercise 9.4.4\PageIndex{4}

Exercise 9.4.5\PageIndex{5}

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Exercise $\PageIndex{1}$

Exercise $\PageIndex{2}$

Exercise $\PageIndex{3}$

Exercise $\PageIndex{4}$

Exercise $\PageIndex{4}$

Exercise $\PageIndex{5}$