Algebra Word Problem Solutions

Explain what is wrong with the “obvious” approach to polynomial division using a point-value representation, i.e., dividing the corresponding y values. Discuss separately the case in which the division comes out exactly and the case in which it doesn’t.

An engineer traveled 170 mi by car and then an additional 750 mi by plane. The rate of the plane was three times the rate of the car. The total trip took 7 h. Find the rate of the car.

Consider the following system of linear equations:
$x_1+2x_2+x_3-<!--\; -\; -->2x_4=10$
$-<!--\; -\; -->x_1+2x_2+x_3-<!--\; -\; -->x_4=6$
$+x_2+x_3=2$
Find all its canonical forms and basic solutions.

Is it possible to intuitively explain, how the three irrational numbers $e,i$ and $\mathrm{\pi <!--\; \pi \; -->}$ are related?

Find these missing terms [__,8,__,128,....]

To find, wether '1' lies in the range of f, where $f(x)=[ln(\frac{7x-<!--\; -\; -->x^2}{12}){]}^{\frac{3}{2}}$?
For the given function, the question is whether, f(x) can equal 1 for some real value of x?

Write an nth term rule for ${\displaystyle r=\frac{1}{3}}$ and ${\displaystyle a_1=45}$

Find the 6th term if the first term of a geometric sequence is 2 and the common ratio is 4

How do you check your solutions?

Integral roots for $f(x)=41$ if $f(x)=37$ has 5 distinct integral roots.
Given a polynomial f(x) with integral coefficients and $f(x)=37$ has 5 distinct integral roots, find the number of integral roots of $f(x)=41$?
My Approach: Say $f(x)=(x-<!--\; -\; -->r_1)(x-<!--\; -\; -->r_2)(x-<!--\; -\; -->r_3)(x-<!--\; -\; -->r_4)(x-<!--\; -\; -->r_5)g(x)+37$, where ri are the distinct integers.
Now for $f(x)=41$ we have $(x-<!--\; -\; -->r_1)(x-<!--\; -\; -->r_2)(x-<!--\; -\; -->r_3)(x-<!--\; -\; -->r_4)(x-<!--\; -\; -->r_5)g(x)=4$, so the factors can be $\pm <!--\; \pm \; -->1,\pm <!--\; \pm \; -->2$ or $\pm <!--\; \pm \; -->4$. Given $r_i$ are distinct at most two of them will give $\pm <!--\; \pm \; -->1$, then there can be both of $\pm <!--\; \pm \; -->2$ or one of $\pm <!--\; \pm \; -->4$. This is where I get lost, since even if I use all of $\pm <!--\; \pm \; -->1,\pm <!--\; \pm \; -->2$, I will be still be left with one $x-<!--\; -\; -->r_i$ factor. What about that? Does it matter that 37 and 41 are primes, or is it just a coincidence?

Tell whether the sequence with the terms ${\displaystyle a_3=21,a_5=27,a_{12}=48}$ is arithmetic

Find ${\displaystyle a_8}$ for the geometric sequence 4,−12,36,...

Prove or disprove: For all positive integers $n$ , $\sqrt[3]{n}+\sqrt[3]{n+1}$ are irrational numbers.

${\displaystyle x{(x-3)}^2\ge 0}$

Sum of $2$ irrational numbers, rational or irrational, more?

Write an equation through point (6, -8), perpendicular to y= -6x - 9

Find the 10th term of this geometric sequence 1, 2, 4, 8, …

Prove that ${\log }_a<!--\; \; -->(1/x)=-<!--\; -\; -->{\log }_a<!--\; \; -->(x)$
But it has two problems: when x=0 and on the other problem it doesn't mention any condition. How should I solve it in each of them?

You invest $50,000, part at 3% simple interest and part at 4% simple interest for one year. How much money did you invest in each account if the interest earned in the 3% account was $30 greater than the interest earned in the 4% account?

Is $1+\lg <!--\; \; -->i=\lg <!--\; \; -->(i+i)$?
I've been studying Sedgewick's "Algorithms" book and in proof of one proposition he writes the following:
the property is preserved because
$1+\lg <!--\; \; -->i=\lg <!--\; \; -->(i+i)\le <!--\; \le \; -->\lg <!--\; \; -->(i+j)=\lg <!--\; \; -->k$
I cannot wrap my brain around the first part of this inequation, namely $1+\lg <!--\; \; -->i=\lg <!--\; \; -->(i+i)$. Can anyone offer an explanation? Thanks in advance!