# (a)To calculate: The value of P(2) without a calculator using both forms of the polynomial. The value of P (2) is 236. (b)The least amount of arithmetic operations was performed in (c) using Horner's method.

Question
Polynomial arithmetic
(a)To calculate: The value of P(2) without a calculator using both forms of the polynomial. The value of P (2) is 236. (b)The least amount of arithmetic operations was performed in (c) using Horner's method.

2021-03-19
(a) Given: A fourth-degree polynomial in x such as 3x^4 + 5x^3 + 4x^2 + 3x + 1 contains all of the powers of x from the first through the fourth. However, any polynomial can be written without powers of x. Evaluating a polynomial without powers of x (Horner's method) is somewhat easier thanevaluating a polynomial with powers. Calculation: The polynomial, $$P(x) = 6x^5 — 3x^4 + 9x^3 + 6x^2 — 8x + 12,$$ can be written as, $${[((6x — 3)x + 9)x + 6]x — 8) x + 12$$ without its power (Horner's method), and value of P(2) is given by,
$$P(2) = {[((6.2 - 3)2 + 9)2 + 6]2 - 8} 2 + 12$$
$$= {[(18 + 9)2+6]2—8}2 + 12$$
$$= {[54 + 6]2 - 8}2 + 12$$
$$= (120 - 8)2 + 12$$ Further, $$P(2) = 6(2)^5 - 3(2)^4 + 9(2)^3 + 6(2)^2 - 8(2) + 12$$
$$= 6.32 - 3.16 + 9.8 + 6.4 - 16 + 12$$
$$= 192 - 48 + 72 + 24 - 4$$
$$=236$$ Therefore, the value of P(2)is 236. (b) Given: A fourth-degree polynomial in x such as $$3x^4 + 5x^3 + 4x^2 + 3x + 1$$ contains all of the powers of xfrom the first through the fourth. However, any polynomial can be written without powers of x. Evaluating a polynomial without powers of x (Horner's method) is somewhat easier than evaluating a polynomial with powers. Calculation: Horner's method is the method of writing a polynomial without the powers of x and in its simplest form which makes the calculation much easier. therefore, the least amount of airhmetic operations was performed in (c) using Horner's method.

### Relevant Questions

Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same $$\displaystyle{P}{\left({x}\right)}={3}{x}^{{4}}-{5}{x}^{{3}}+{x}^{{2}}-{3}{x}+{5}{N}{S}{K}{Q}{\left({x}\right)}={\left({\left({\left({3}{x}-{5}\right)}{x}+{1}\right)}{x}-{3}\right)}{x}+{5}$$ Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial $$\displaystyle{R}{\left({x}\right)}={x}^{{5}}—{2}{x}^{{4}}+{3}{x}^{{3}}—{2}{x}^{{3}}+{3}{x}+{4}$$ in “nested” form, like the polynomial Q. Use the nested form to find R(3) in your head. Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value ofa polynomial using synthetic division?
DISCOVER: Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same $$P(x) = 3x^{4} - 5x^{3} + x^{2} - 3x +5$$
$$Q(x) = (((3x - 5)x + 1)x 3)x + 5$$
Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial
R(x) =x^{5} - 2x^{4} + 3x^{3} - 2x^{2} + 3x + 4\) in “nested” form, like the polynomial Q. Use the nested form to find R(3) in your head.
Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value ofa polynomial using synthetic division?
Given the following function: $$\displaystyle{f{{\left({x}\right)}}}={1.01}{e}^{{{4}{x}}}-{4.62}{e}^{{{3}{x}}}-{3.11}{e}^{{{2}{x}}}+{12.2}{e}^{{{x}}}-{1.99}$$ a)Use three-digit rounding frithmetic, the assumption that $$\displaystyle{e}^{{{1.53}}}={4.62}$$, and the fact that $$\displaystyle{e}^{{{n}{x}}}={\left({e}^{{{x}}}\right)}^{{{n}}}$$ to evaluate $$\displaystyle{f{{\left({1.53}\right)}}}$$ b)Redo the same calculation by first rewriting the equation using the polynomial factoring technique c)Calculate the percentage relative errors in both part a) and b) to the true result $$\displaystyle{f{{\left({1.53}\right)}}}=-{7.60787}$$
Given the following function:
$$\displaystyle{f{{\left({x}\right)}}}={1.01}{e}^{{{4}{x}}}-{4.62}{e}^{{{3}{x}}}-{3.11}{e}^{{{2}{x}}}+{12.2}{e}^{{{x}}}$$
a) Use three-digit rounding frithmetic, the assumption that $$\displaystyle{e}^{{{1.53}}}={4.62}$$, and the fact that $$\displaystyle{e}^{{{n}{x}}}={\left({e}^{{{x}}}\right)}^{{{n}}}$$ to evaluate $$\displaystyle{f{{\left({1.53}\right)}}}$$
b) Redo the same calculation by first rewriting the equation using the polynomial factoring technique
c) Calculate the percentage relative errors in both part a) and b) to the true result $$\displaystyle{f{{\left({1.53}\right)}}}=-{7.60787}$$
(a)To calculate: The following equation {[(3x + 5)x + 4]x + 3} x + 1 = 3x^4 + 5x^3 + 4x^2 + 3x + 1 is an identity, (b) To calculate: The lopynomial P(x) = 6x^5 - 3x^4 + 9x^3 + 6x^2 -8x + 12 without powers of x as in patr (a).
1. Explain with numerical examples what Real Numbers and Algebraic Expressions are. 2. Explain with numerical examples Factoring and finding LCMs (least common multiples). Explain factoring of a larger number. 3. Explain with numerical examples arithmetical operations (addition, subtraction, multiplication, division) with fractions 4, Explain with numerical examples arithmetical operations (addition, subtraction, multiplication, division) with percentages 5. Explain with numerical examples exponential notation 6. Explain with numerical examples order (precedence) of arithmetic operations 7. Explain with numerical examples the concept and how to find perimeter, area, volume, and circumference (use related formulas)
a) Use base b = 10, precision k = 4, idealized, chopping floating-point arithmetic to show that fl(g(1.015)) is inaccurate, where $$\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}^{{\frac{{1}}{{4}}}}-{1}}}{{{x}-{1}}}}$$ b) Derive the second order (n = 2) quadratic Taylor polynomial approximation for $$\displaystyle{f{{\left({x}\right)}}}={x}^{{\wedge}}\frac{{1}}{{4}},$$ expanded about a = 1, and use it to get an accurate approximation to g(x) in part (a). c) Verify that your approximation in (b) is more accurate.
A random sample of $$\displaystyle{n}_{{1}}={16}$$ communities in western Kansas gave the following information for people under 25 years of age.
$$\displaystyle{X}_{{1}}:$$ Rate of hay fever per 1000 population for people under 25
$$\begin{array}{|c|c|} \hline 97 & 91 & 121 & 129 & 94 & 123 & 112 &93\\ \hline 125 & 95 & 125 & 117 & 97 & 122 & 127 & 88 \\ \hline \end{array}$$
A random sample of $$\displaystyle{n}_{{2}}={14}$$ regions in western Kansas gave the following information for people over 50 years old.
$$\displaystyle{X}_{{2}}:$$ Rate of hay fever per 1000 population for people over 50
$$\begin{array}{|c|c|} \hline 94 & 109 & 99 & 95 & 113 & 88 & 110\\ \hline 79 & 115 & 100 & 89 & 114 & 85 & 96\\ \hline \end{array}$$
(i) Use a calculator to calculate $$\displaystyle\overline{{x}}_{{1}},{s}_{{1}},\overline{{x}}_{{2}},{\quad\text{and}\quad}{s}_{{2}}.$$ (Round your answers to two decimal places.)
(ii) Assume that the hay fever rate in each age group has an approximately normal distribution. Do the data indicate that the age group over 50 has a lower rate of hay fever? Use $$\displaystyle\alpha={0.05}.$$
(a) What is the level of significance?
State the null and alternate hypotheses.
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}<\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}>\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}\ne\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}>\mu_{{2}},{H}_{{1}}:\mu_{{1}}=\mu_{{12}}$$
(b) What sampling distribution will you use? What assumptions are you making?
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations,
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations,
The Student's t. We assume that both population distributions are approximately normal with known standard deviations,
What is the value of the sample test statistic? (Test the difference $$\displaystyle\mu_{{1}}-\mu_{{2}}$$. Round your answer to three decimalplaces.)
What is the value of the sample test statistic? (Test the difference $$\displaystyle\mu_{{1}}-\mu_{{2}}$$. Round your answer to three decimal places.)
(c) Find (or estimate) the P-value.
P-value $$\displaystyle>{0.250}$$
$$\displaystyle{0.125}<{P}-\text{value}<{0},{250}$$
$$\displaystyle{0},{050}<{P}-\text{value}<{0},{125}$$
$$\displaystyle{0},{025}<{P}-\text{value}<{0},{050}$$
$$\displaystyle{0},{005}<{P}-\text{value}<{0},{025}$$
P-value $$\displaystyle<{0.005}$$
Sketch the sampling distribution and show the area corresponding to the P-value.
P.vaiue Pevgiue
P-value f P-value
First, construct the sixth degree Taylor polynomial $$\displaystyle{P}_{{6}}{\left({x}\right)}$$ for function $$\displaystyle{f{{\left({x}\right)}}}={\sin{{\left({x}^{{2}}\right)}}}$$ about $$\displaystyle{x}_{{0}}={0}$$ The use $$\displaystyle{\int_{{{0}}}^{{{1}}}}{P}_{{6}}{\left({x}\right)}{\left.{d}{x}\right.}$$ to approximate the integral $$\displaystyle\ {\int_{{{0}}}^{{{1}}}}{\sin{{\left({x}^{{2}}\right)}}}{\left.{d}{x}\right.}.$$ Use 4-digit rounding arithmetic in all calculations. What is the approximate value?