5.3E Exercises

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Identifying Polynomial Functions

Tell whether the functions are polynomial functions, and if necessary, write them in standard form. If they are polynomial functions, identify the degree and the leading coefficient.

$ f(x) = 2x + 4x^2 + 8 x^3 + 16 x^4 + 32 x^5 $
$ g(x) = \sqrt{ x^2 - 3x + 5 }$
$ h(x) = x^{-2} + x + x^7 $
$ p(x) = x(x+1) + x^2(1-x) $
$ q(x) = (x-3)(x+1)(x-4)^2 $
$ r(x) = \dfrac{ x}{x^2+2} $

Answer

Yes, degree 5, leading coefficient 32.
No.
No.
Yes, degree 3, leading coefficient $-1$. Standard form: $ p(x) = -x^3 + 2x^2 + x $.
Yes, degree 4, leading coefficient 1. Standard form: $q(x) = x^4 - 10x^3+29x^2 - 8x - 48 $.
No.

Describing End Behavior

Describe the end behavior of the polynomial functions.

$ f(x) = -x^3 + x $
$ f(x) = x^2 + 2x + 1 $
$ f(x) = 2(x+1)^5 $
$ f(x) = -x(x+7) $
$ f(x) = -2 x^{8} + x^4 + 16 $
$ f(x) = 12x + 12 $

Answer

As $ x \rightarrow -\infty$, $ f(x) \rightarrow \infty $. As $ x \rightarrow \infty$, $ f(x) \rightarrow -\infty $.
As $ x \rightarrow -\infty$, $ f(x) \rightarrow \infty $. As $ x \rightarrow \infty$, $ f(x) \rightarrow \infty $.
As $ x \rightarrow -\infty$, $ f(x) \rightarrow -\infty $. As $ x \rightarrow \infty$, $ f(x) \rightarrow \infty $.
As $ x \rightarrow -\infty$, $ f(x) \rightarrow -\infty $. As $ x \rightarrow \infty$, $ f(x) \rightarrow -\infty $.
As $ x \rightarrow -\infty$, $ f(x) \rightarrow -\infty $. As $ x \rightarrow \infty$, $ f(x) \rightarrow -\infty $.
As $ x \rightarrow -\infty$, $ f(x) \rightarrow -\infty $. As $ x \rightarrow \infty$, $ f(x) \rightarrow \infty $.

Finding Real Roots With Easy Factoring

Find all real roots of the polynomial functions, with multiplicities.

$ p(x) = x^3 - x $
$ p(x) = (x+1)(x^2 + 2x + 1) $
$ p(x) = 3x^3 - 12x - x^2 + 4 $
$ p(x) = x^6 - 3x^5 $
$ p(x) = (x+2)(x^2 - 4) $
$ p(x) = (x-4)(x-3)(x^2 - x - 2 )$

Answer

$ x = -1, 0 , 1 $ (all mult 1).
$ x = -1$ (mult 3).
$ x =\frac{1}{3}, -2, 2 $ (all mult 1).
$ x = 0$ (mult 5), $x = 3 $ (mult 1).
$ x = -2$ (mult 2), $ x = 2$ (mult 1).
$ x = 4, 3, 2, -1 $ (all mult 1).

Quadratic Factors

The following polynomial functions are partially factored. Determine if the quadratic factors are irreducible. List all real roots.

$ f(x) = (x-2)(x^2 + 7x + 6 ) $
$ g(x) = (x^2 - 1) (x^2 + 1) $
$ h(x) = x(x^2 - x + 12) $
$ p(x) = (x + 2)(x - 1) ( x^2 + 9) $
$ q(x) = (x^3 - 1) (x^2 + 2x + 10) $

Answer

The quadratic factors further as $ (x+1)(x+6)$. Roots: $x = -6, -1, 2$.
$ (x^2 + 1)$ is irreducible, but $ (x^2 - 1) = (x+1)(x-1) $ gives roots $x = \pm 1$.
The discriminant $ b^2 - 4ac = 1 - 48 $ is negative, so the quadratic is irreducible. Only real root is 0.
$ (x^2 + 9)$ is irreducible, so roots are $x = -2, 1$.
The discriminant $ b^2 - 4ac = 4 - 400 $ is negative, so it's irreducible. The other factor does give a real root of $x = 1$.

Matching Polynomial Functions to Graphs

Match the following polynomial functions to the graphs given below.

$ p(x) = (x+1)^4(x-1) $
$ p(x) = 0.1(x+1)^2(x-4) $
$ p(x) = -(x-3)^4 $
$ p(x) = 0.3x(x+2)(x-2)^3 $

$aa (2).png$	$b4.png$	$aa.png$	$aa (3).png$
A.	B.	C.	D.

Answer

The funky leading coefficients are necessary to make the graphs fit nicely, lol.

Deducing Information From Graphs

Based on the graph given, determine a possible set of real roots with their multiplicities, write the factors that must appear in the factorization of the polynomial function, and use end behavior to determine the sign of the leading coefficient.

$c1.png$	$c2.png$	$c3.png$	$c4.png$
1.	2.	3.	4.

Answer

Roots: $ x = 0, 1, 2$ (mult 1). Factors: $ x(x-1)(x-2) $. Degree-3, negative leading coeff.
Roots: $ x = -2, -1, 0, 1, 2$ (mult 1). Factors: $ x(x-1)(x+1)(x-2)(x+2) $. Degree-5, positive leading coeff.
Roots: $ x = \pm 2$ (both mult 2). Factors: $ (x-2)^2(x+2)^2 $. Degree-4, negative leading coeff.
Roots: $ x = -1$ (mult 3), $x = 2$ (mult 5). Factors: $ (x+1)^3 (x-2)^5 $. Degree-8, positive leading coeff.

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