# Write the polynomials as the product of linear factors and list all the zeros of the function.h(x) = x^{3}-3x^{2}+4x-2

Question
Polynomial arithmetic
Write the polynomials as the product of linear factors and list all the zeros of the function.$$h(x) = x^{3}-3x^{2}+4x-2$$

2020-10-21
Step 1 $$x^{3}-3x^{2}+4x-2=0$$ Step 2 $$x-1| \frac{x^{2}-2x+2}{x^{3}-3x^{2}+4x-2} \frac{x^{3}-x^{2}}{-2x^{2}+4x-2}|\frac{-2x^{2}+2x}{2x-2}|(2x-2)$$ Step 3 $$x-1|\frac{x^{2}-2x+2}{x-1}(x^{2}-2x+2)=0$$
$$x=\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$$
$$x=\frac{-(-2)\pm \sqrt{(-2)^{2}-4(1)(2)}}{2(1)}$$
$$x = 1.1 \pm i$$
$$f(x) = (x-(1+i)) (x-(1-i)) (x-1)$$

### 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?
We need find:
The real zeros of polynomial function form an arithmetic sequence
$$f(x) = x^{4} - 4x^{3} - 4x^{2} + 16x.$$
Determine whether the following state-ments are true and give an explanation or counterexample.
a) All polynomials are rational functions, but not all rational functions are polynomials.
b) If f is a linear polynomial, then $$\displaystyle{f}\times{f}$$ is a quadratic polynomial.
c) If f and g are polynomials, then the degrees of $$\displaystyle{f}\times{g}$$ and $$\displaystyle{g}\times{f}$$ are equal.
d) To graph $$\displaystyle{g{{\left({x}\right)}}}={f{{\left({x}+{2}\right)}}}$$, shift the graph of f 2 units to the right.
(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).
An experiment on the probability is carried out, in which the sample space of the experiment is
$$S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.$$
Let event $$E={2, 3, 4, 5, 6, 7}, event$$
$$F={5, 6, 7, 8, 9}, event G={9, 10, 11, 12}, and event H={2, 3, 4}$$.
Assume that each outcome is equally likely. List the outcome s in For G.
Now find P( For G) by counting the number of outcomes in For G.
Determine P (For G ) using the General Addition Rule.
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 $$\displaystyle{f{{\left({1.53}\right)}}}=-{7.60787}$$
$$\displaystyle{f{{\left({x}\right)}}}={1.01}{e}^{{{4}{x}}}-{4.62}{e}^{{{3}{x}}}-{3.11}{e}^{{{2}{x}}}+{12.2}{e}^{{{x}}}$$
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 $$\displaystyle{f{{\left({1.53}\right)}}}$$
c) Calculate the percentage relative errors in both part a) and b) to the true result $$\displaystyle{f{{\left({1.53}\right)}}}=-{7.60787}$$
Indicate whether the expression defines a polynomial function. $$\displaystyle{P}{\left({x}\right)}=−{x}{2}+{3}{x}+{3}$$ polynomial or not a polynomial If it is a polynomial function, identify the following. (If it is not a polynomial function, enter DNE for all three answers.) (a) Identify the leading coefficient. (b) Identify the constant term. (c) State the degree.
Indicate whether the expression defines a polynomial function. $$\displaystyle{g{{\left({x}\right)}}}=−{4}{x}^{{5}}−{3}{x}^{{2}}+{x}−{2}$$ polynomial or not a polynomial If it is a polynomial function, identify the following. (If it is not a polynomial function, enter DNE for all three answers.) (a) Identify the leading coefficient. (b) Identify the constant term. (c) State the degree.