2.5E: Exercises - Polynomial Functions

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Exercise

In Exercises 1-8, arrange each polynomial in descending powers of x, state the degree of the polynomial, identify the leading term, then make a statement about the coefficients of the given polynomial.

Exercise $\PageIndex{1}$

$p(x) = 3x−x^2+4−x^3$

Answer

$p(x) = −x^3−x^2+3x+4$, degree = 3, leading term = $−x^3$, “p is a polynomial with integer coefficients, polynomial with rational coefficients,” or “p is

a polynomial with real coefficients.”

Exercise $\PageIndex{2}$

$p(x) = 4+3x^2−5x+x^3$

Exercise $\PageIndex{3}$

$p(x) = 3x^2+x^4−x−4$

Answer

$p(x) = x^4+3x^2−x−4$, degree = 4, leading term = $x^4$, “p is a polynomial with integer coefficients,” “p is a polynomial with rational coefficients,” or “p is

a polynomial with real coefficients.”

Exercise $\PageIndex{4}$

$p(x) = −3+x^2−x^3+5x^4$

Exercise $\PageIndex{5}$

$p(x) = 5x−\frac{3}{2}x^3+4−\frac{2}{3}x^5$

Answer: $p(x) = −\frac{2}{3}x^5−\frac{3}{2}x^3+5x+4$, degree = 5, leading term = $−\frac{2}{3}x^5$, “p is a polynomial with rational coefficients,” or p is a polynomial with real coefficients.”

Exercise $\PageIndex{6}$

$p(x) = −\frac{3}{2}x+5−\frac{7}{3}x^5+\frac{4}{3}x^3$

Exercise $\PageIndex{7}$

$p(x) = −x+\frac{2}{3}x^3−\sqrt{2}x^2+\pi x^6$

Answer: $p(x) = \pi x^6+\frac{2}{3}x^3−\sqrt{2}x^2−x$, degree = 6, leading term = $\pi x^6$ , “p is a polynomial with real coefficients.”

Exercise $\PageIndex{8}$

$p(x) = 3+\sqrt{2}x^4+\sqrt{3}x−2x^2+\sqrt{5}x^6$

In Exercises 9-14, you are presented with the graph of $y = ax^{n}$. In each case, state whether the degree is even or odd, then state whether a is a positive or negative number.

Exercise $\PageIndex{9}$

$Screen Shot 2019-08-30 at 8.30.28 AM.png$

Answer: $y=ax^{n}$, n odd, a < 0.

Exercise $\PageIndex{10}$

$Screen Shot 2019-08-30 at 8.32.10 AM.png$

Exercise $\PageIndex{11}$

$Screen Shot 2019-08-30 at 8.35.40 AM.png$

Answer: $y=ax^{n}$, n even, a > 0.

Exercise $\PageIndex{12}$

$Screen Shot 2019-08-30 at 8.37.26 AM.png$

Exercise $\PageIndex{13}$

$Screen Shot 2019-08-30 at 8.38.27 AM.png$

Answer: $y=ax^{n}$, n odd, a < 0.

Exercise $\PageIndex{14}$

$Screen Shot 2019-08-30 at 8.39.17 AM.png$

In Exercises 15-20, you are presented with the graph of the polynomial $p(x) = a_{n}x^n +···+a_{1}x+a_{0}$. In each case, state whether the degree of the polynomial is even or odd, then state whether the leading coefficient $a_{n}$ is positive or negative.

Exercise $\PageIndex{15}$

$Screen Shot 2019-08-30 at 8.40.55 AM.png$

Answer: odd, positive

Exercise $\PageIndex{16}$

$Screen Shot 2019-08-30 at 8.42.09 AM.png$

Exercise $\PageIndex{17}$

$Screen Shot 2019-08-30 at 8.43.09 AM.png$

Answer: even, negative

Exercise $\PageIndex{18}$

$Screen Shot 2019-08-30 at 8.43.58 AM.png$

Exercise $\PageIndex{19}$

$Screen Shot 2019-08-30 at 8.44.52 AM.png$

Answer: odd, positive

Exercise $\PageIndex{20}$

$Screen Shot 2019-08-30 at 8.46.27 AM.png$

For each polynomial in Exercises 21-30, perform each of the following tasks.

Predict the end-behavior of the polynomial by drawing a very rough sketch of the polynomial. Do this without the assistance of a calculator. The only concern here is that your graph show the correct end-behavior.
Draw the graph on your calculator, adjust the viewing window so that all “turning points” of the polynomial are visible in the viewing window, and copy the result onto your homework paper. As usual, label and scale each axis with xmin, xmax, ymin, and ymax. Does the actual end-behavior agree with your predicted end-behavior?

Exercise $\PageIndex{21}$

$p(x) = −3x^3+2x^2+8x−4$

Answer

Note that the leading term $−3x^3$ (dashed) has the same end-behavior as the polynomial p.

$Screen Shot 2019-08-30 at 9.06.09 AM.png$

Exercise $\PageIndex{22}$

$p(x) = 2x^3−3x^2+4x−8$

Exercise $\PageIndex{23}$

$p(x) = x^3+x^2−17x+15$

Answer

Note that the leading term $x^3$ (dashed) has the same end-behavior as the polynomial p.

$Screen Shot 2019-08-30 at 9.07.12 AM.png$

Exercise $\PageIndex{24}$

$p(x) = −x^4+2x^2+29x−30$

Exercise $\PageIndex{25}$

$p(x) = x^4−3x^2+4$

Answer

Note that the leading term $x^4$ (dashed) has the same end-behavior as the polynomial p.

$Screen Shot 2019-08-30 at 9.09.00 AM.png$

Exercise $\PageIndex{26}$

$p(x) = −x^4+8x^2−12$

Exercise $\PageIndex{27}$

$p(x) = −x^5+3x^4−x^3+2x$

Answer

Note that the leading term $−x^5$ (dashed) has the same end-behavior as the polynomial p.

$Screen Shot 2019-08-30 at 9.10.12 AM.png$

Exercise $\PageIndex{28}$

$p(x) = 2x^4−3x^3+x−10$

Exercise $\PageIndex{29}$

$p(x) = −x^6−4x^5+27x^4+78x^3+4x^2+376x−480$

Answer

Note that the leading term $−x^6$ (dashed) has the same end-behavior as the polynomial p.

$Screen Shot 2019-08-30 at 9.11.10 AM.png$

Exercise $\PageIndex{30}$

$p(x) = x^5−27x^3+30x^2−124x+120$