# Tag Info

Accepted

### Is there a method to factor this equation?

This is just factoring out the common factor of $x^2$ from the first two terms, and the common factor of $-4$ from the last two terms. However, the correct factorization would be $$x^2(x+1) - 4(x+1)$$ ...
• 36.7k
Accepted

### How to tell if a polynomial is square

The coefficients of $x_i^4$ in $p$ will be the squares of the coefficients of $x_i^2$ in $q$. Because of signs, there are $2^6=64$ very similar possibilities. Then use the coefficients of $x_i^3x_j$ ...
• 48.7k

### Is there an easy way to tell whether a cubic or quartic polynomial is factorable over the integers?

There isn't a one-size-fits-all solution. There are strategies that one could try, for example reduction modulo a prime or a prime power. There are some well known criteria for detecting whether a ...
• 2,587
Accepted

### Terminology. Can I say that $t^2 - 9$ is factorable by a "product of conjugates"?

The answer to whether or not terminology is appropriate is very often; depends. If I had to give a direct answer, I would say no. Here is my reasoning. In number theory, your terminology at least has ...
• 1,278

### Is there a method to factor this equation?

Observe that for $p(x) = x^3+x^2 - 4x - 4$, then $p(-1) = 0$. Thus as a polynomial, $p(x)$ has a factor $x+1$. You can then use synthetic division to obtain the quotient and can factor $p(x)$ into a ...
• 5,085
1 vote

### Prove that the polynomial $f(x)= x^4 + 2x^2 + 2x + 1998$ cannot be writen as the product of two quadratic polynomials with integer coefficients

Easiest way: use a calculator like Wolfram Alpha to compute the four roots, then check every possible product $(x-\alpha)(x-\beta)$, where $\alpha$ and $\beta$ are roots.
• 6,361
1 vote
Accepted

### Prove that the polynomial $f(x)= x^4 + 2x^2 + 2x + 1998$ cannot be writen as the product of two quadratic polynomials with integer coefficients

Use only $a+c=o, ad+bc=2\Leftrightarrow a(d-b)=2\Rightarrow d=b\pm1, \pm2$ while bd=1998. Then try to solve the resulting quadratic in b.
• 1,918
1 vote
Accepted

### How has this been simplified?

In the above expression, you have a confusing expression: $$36x\left(9x^2+3\right)\left(2x-8\right)-2\left(9x^2+3\right)^2$$ One way to see what happened is to simplify it by letting $y=9x^2+3$, and ...
• 6,094
1 vote

• 1,412
1 vote
Accepted

### Two combinatorics questions, one on product of combinations, the other on why factoring heuristic isn't applicable

In a group of $14$ students, there are $8$ girls and $6$ boys. Determine the number of ways that a committee of $4$ students which has at least $1$ boy can be chosen from the group. Let's examine two ...

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