Learn – Algebra/Precalculus – Polynomials

Definition

Exercises

Problem set

Determine which of the following are polynomials. And, for each polynomial, identify its degree.

$2x^{99} - 2$
$2x^{-1} - 2$
$\pi x - \frac{1}{\pi}$
$x - \frac{1}{x}$
$x$
$-5$
$x^{1/2}$
$5^x-4$

Problem set

Determine which of the following are polynomials. And, for each polynomial, identify its degree.

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

Problem set

Determine which of the following are polynomials. And, for each polynomial, identify its degree.

$x^{-1}(3x^2+2x)$
$(x^{-3}+x^{-2})(7x^4-5x^3)$
$3x^{1/2}(5x^{3/2}+2x^{1/2})$
$(4x^{1/2}-x^{3/2})(x^{3/2}-3x^{1/2})$

Graphing

Exercises

Problem set

Draw the graphs of the following polynomial functions.

$y = x^2$
$y = x^3$
$y = x^4$
$y = x^5$
$y = -x^3$
$y = -x^4$

Problem set

Determine the end behavior of the following polynomials

$4x^5-2x^4+3x^3-2x^2+6x-5$
$3x^4-50x^3-10x^2-11x-5$
$-5x^6+40x^5+10x^3+11x$
$4x^5-2x^4-8x^3-4x^2-10x-2$
$-4x^5+50x^4+4x^2+2$

Problem set

Determine the zeros, y-intercepts and end behavior of the following functions. Use those to draw the approximate graphs of the following functions.

$(x+5)(3x-7)(x-10)$
$(x+5)(3x-7)(10-x)(x+10)$
$(x+5)(2x-7)^2(10-x)$
$(x+6)^3(2x-5)^2(10-x)$
$(x-5)^3x^2(x+7)$
$(-x-4)^3x^2(x+7)^2$
$(x+6)^3(2-x)^2(10-x)$

Problem set

Determine the zeros, y-intercepts and end behavior of the following functions. Use those to draw the approximate graphs of the following functions.

$(x-1)(x^2 - 1)$
$(x-1)(x^3 + 1)$
$(-x+5)^3(x+8)^2$
$(-x+5)^2(-2x^3+16)$

Problem set

Determine the zeros, y-intercepts and end behavior of the following functions. Use those to draw the approximate graphs of the following functions.

$(-x^2+2x-1)(-x^2+3x-2)$
$(2x+7)(3x^2-5x+2)$
$x^2+1$
$(2x-3-\sqrt{5})(2x-3+\sqrt{5})+4$
$(x-1)(x-2)+1$
$(x-1)(x-2)-1$

Multiplication and division

Exercises

Problem set

In each of the following problems, multiply the two polynomials.

$3x^3-2x^2-5x-1,\,\, -4x^3-x^2+3x+2$

Problem set

In each of the following, divide the first polynomial by the second polynomial.

$(4x^3+3x^3+2x+1),\,\,(x-1)$
$4x^3+2x^2+x+11 ,\,\, x^2+1$
$4x^4-51 ,\,\,2x^2+x+1$
$4x^4+3x^2+11 ,\,\, 2x^2-1$
$4x^4+1,\,\,x^2+1$
$4x^4,\,\, x-1$

Problem set

In each of the following, divide the first polynomial by the second polynomial.

$27x^3+8,\,\, 9x^2-6x+4$
$16x^4,\,\, 4x-1$
$2^5x^5,\,\, x-\frac{1}{2}$
$1251x^5,\,\, x - 5$ .
$x^6 - x^3,\,\, 2x^2 - 2x - 2$ .
$x^8,\,\, x^3 + x^2 + x + 1$ .

Problem set

In each of the following, estimate the quotient and remainder when the first polynomial is divided by the second polynomial.

$x^{100} + 1,\,\, x - 1$
$2^{2000}x^{2000}+1,\,\, x-1$

Problem set

In each of the following, use the Polynomial remainder theorem to find the remainder when the first polynomial is divided by the second polynomial.

$1000x^{1000} - 1000x^{501} - 200,\,\,x-1$ .
$1000x^{1000} - 1000x^{501} - 200,\,\,x+1$ .
$3x^{9} +2,\,\,2x+1$ .

Zeros, Factoring

Exercises

Problem set

In each of the following problems, find the zeros of the polynomial.

$-5x^2+8x+4$
$27x^3+8$
$2x^2-2$

Problem set

In each of the following problems, find the zeros of the polynomial.

$(2x+3)(3x^2-4x+1)$
$(4-5x)(2x-3)(x^2-x+1)$
$(x+1)^3(x^2+4x+3)$
$(3x-4)^2(4-3x)^2(x^2-1)(x^3-27)$

Problem set

In each of the following, write the polynomial as a product of factors.

$x^3 + 1$
$a^3 - 8$
$2x^3 + 16$
$\sqrt{8}x^3 - \sqrt{27}$
$\sqrt{3}x^3+\sqrt{3}$

Miscellaneous

Exercises

Problem set

How many zeros (including complex zeros) does the polynomial $x^{100}+x^{50}+1$ have?
Zeros of the polynomial $x^6-14x^4+49x^2-36$ are $-3,-2,-1,1,2,3$ . What is the biggest zero of the polynomial $(x+4)^6-14(x+4)^4+49(x+4)^2-36$ ?
If $a$ is one of the zeros of the polynomial $x^{99}-1000x^{41}+51x^2$ , what is the smallest possible value of $|a|$ ?