Chapter 12, Polynomial Algorithms and Fast Fourier Transforms Video Solutions, Applied Algebra: Codes, Ciphers and Discrete Algorithms

Problem 1

Find the content, and the corresponding primitive polynomial, of the following polynomials.
a. $2 x^2-\frac{1}{3} x$
b. $\frac{4}{15} x^3-\frac{6}{7} x^2+10$
c. $\frac{1}{3} x+\frac{1}{2}$
d. $14 x^2-21 x+35$

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01:07

Problem 2

Use the Lagrange interpolation formula to find an equation of the line through the points $(1,2)$ and $(-3,5)$.

Linh Vu

Numerade Educator

01:34

Problem 3

Use the Lagrange interpolation formula to find an equation of the line through the points $\left(x_0, y_0\right)$ and $\left(x_1, y_1\right)$. Show that the equation reduces to the two-point form
$$
y=y_0+\frac{y_1-y_0}{x_1-x_0}\left(x-x_0\right)
$$
for the line through the points $\left(x_0, y_0\right)$ and $\left(x_1, y_1\right)$.

Yuva S

Numerade Educator

02:52

Problem 4

Use the Lagrange interpolation formula to find an equation of the parabola that goes through the three points $(0,1),(1,-2),(2,2)$.

Ashley Volpe

Numerade Educator

Problem 5

Find the polynomial of lowest degree that goes through the points $(2,7)$, $(4,10),(7,23),(11,5)$ by defining a generic polynomial
$$
p(x)=a+b x+c x^2+d x^3
$$
and solving the system
$$
p(2)=7, p(4)=10, p(7)=23, p(11)=5
$$
for the unknown coefficients $a, b, c, d$.

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Problem 6

Find a polynomial over the integers modulo 101 of lowest degree that goes through the points $(1,3),(2,51),(4,78),(7,23)$ by defining a generic polynomial
$$
p(x)=a+b x+c x^2+d x^3
$$
solving the system
$$
p(1)=3, p(2)=51, p(4)=78, p(7)=23
$$
for the unknown coefficients $\{a, b, c, d\}$ and reducing the coefficients modulo 101 .

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04:40

Problem 7

Find the polynomial of lowest degree that goes through the points $(-2,5)$, $(-1,-10),(0,8),(1,-5),(2,10)$.

Nyah Kshatriya

Numerade Educator

10:58

Problem 8

Prove the rational root theorem directly by plugging in a root $p / q$ in lowest terms and clearing fractions.

Monique Rousselle Maynard

Numerade Educator

Problem 9

Verify that the polynomial
$$
p(x)=\sum_{i=0}^k d_i \prod_{j \neq i} \frac{x-r_j}{r_i-r_j}
$$
satisfies $p\left(r_i\right)=d_i$ for $i=0,1, \ldots, k$.

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