# Find a second-degree polynomial P such that P(4)=5, P'(4)=3, and P"(4)=3

Question
Polynomials
Find a second-degree polynomial P such that P(4)=5, P'(4)=3, and P"(4)=3

2021-02-03
Consider second degree polynomial $$P(x)=a+bx+cx^{2}$$ such that
$$P(4)=5, P′(4)=3$$, and $$P′′ (4)=3.$$
$$P(4)=5P(4)=5$$ gives us
$$a+4b+16c=5$$.
Now differentiating P, we get $$P′=b+2cx$$. Now putting x=4, we get
$$b+8c=3. Now differentiating P′, we get \(P''=2cP$$. Now putting x=4, we get
$$b+12=3\Rightarrowb=−9.$$
Now putting $$c=\frac{3}{2} and \(b= -9 in {eq1}$$ we get
$$a−36+48=5\Rightarrowa=−7.$$
So required second degree polynomial is $$P(x)=−7−9x+(\frac{3}{2})x2$$

### Relevant Questions

Consider the polynomial function $$p(x)=(3x^{2}-5x-2)(x-5)(x^{2}-4)$$.
A) what is the degree of the polynomial? What is the y intercept of the function?
B)What are the zeros and their multiplicities?
C) what is the leading term of the polynomial and what power function has a graph most similiar to the graph of p?

Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same $$\displaystyle{P}{\left({x}\right)}={3}{x}^{{4}}-{5}{x}^{{3}}+{x}^{{2}}-{3}{x}+{5}\ {Q}{\left({x}\right)}={\left({\left({\left({3}{x}-{5}\right)}{x}+{1}\right)}{x}-{3}\right)}{x}+{5}$$ 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?

For the following polynomial, P(x) =x^3 – 2x^2 + 4x^5 + 7, what is: 1) The degree of the polynomial, 2) The leading term of the polynomial, 3) The leading coefficient of the polynomial.
Find all rational zeros of the polynomial, and write the polynomial in factored form.
$$\displaystyle{P}{\left({x}\right)}={3}{x}^{{{5}}}-{14}{x}^{{{4}}}-{14}{x}^{{{3}}}+{36}{x}^{{{2}}}+{43}{x}+{10}$$
Find a polynomial of the specified degree that has the given zeros. Degree 4, zeros -2, 0, 2, 4
Polynomial calculations Find a polynomial f that satisfies the following properties. Determine the degree of f. then substitute a polynomial of that degree and solve for its coefficients.
$$\displaystyle{f{{\left({f{{\left({x}\right)}}}\right)}}}={9}{x}-{8}$$
$$\displaystyle{f{{\left({f{{\left({x}\right)}}}\right)}}}={x}^{{{4}}}-{12}{x}^{{{2}}}+{30}$$
$$\displaystyle{T}=\mathbb{R}^{{2}}\rightarrow\mathbb{R}^{{4}}\ {s}{u}{c}{h} \ {t}\hat \ {{T}}{\left({e}_{{1}}\right)}={\left({7},{1},{7},{1}\right)},{\quad\text{and}\quad}{T}{\left({e}_{{2}}\right)}={\left(-{8},{5},{0},{0}\right)},{w}{h}{e}{r}{e}{\ e}_{{1}}={\left({1},{0}\right)},{\quad\text{and}\quad}{e}_{{2}}={\left({0},{1}\right)}$$.