Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am not sure how I would factor this. The $x^4$ and $x^2$ are really throwing me off. Can someone explain how I would factor this?

share|improve this question
6  
Let $y=x^2$, then factor the expression in $y$. –  Joel Reyes Noche Apr 19 '13 at 4:33
1  
I've heard this method colloquially referred to as 'chunking'. It works in other situations which might throw you off, e.g. $e^{2x}+ae^x+b$ becomes $y^2+ay+b$ with $y=e^x$. Solving the quadratic equation in $y$ and substituting back in is far easier than any alternative. –  Ian Coley Apr 19 '13 at 4:35
    
Even if you don't recognize immediately that you can substitute $y=x^2$, you can work to that as follows. Note that the polynomial is even in $x$: replace $x$ with $-x$ and the polynomial stays the same. So, if $a$ is a root, then $-a$ is a root. So, it factors to $(x-a)(x+a)(x-b)(x+b)$ for some complex $a$ and $b$. Collect related factors to get $(x^2 - a^2)(x^2 - b^2)$. Give $a^2$ and $b^2$ simpler names, say $c$ and $d$, where these are possibly complex. It should then be clear. –  Eric Jablow Apr 19 '13 at 5:23

5 Answers 5

Let $y=x^2$. You then get $y^2-7y-18$. Can you factor it now?

share|improve this answer

Since all of the powers of $x$ in this polynomial are even ($18$ counts as $18 \cdot x^0$), you would make a substitution of $ t = x^2 $ . Since $x^4 = (x^2)^2$ , you can write your polynomial as $t^2 - 7t - 18$ . How would you factor that?

share|improve this answer

Solution 1. \begin{eqnarray*} x^4-7x^2-18&=&(x^4+2x^2)-(9x^2+18)\\ &=&x^2(x^2+2)-9(x^2+2)\\ &=&(x^2+2)(x^2-9)\\ &=&(x^2+2)(x-3)(x+3) \end{eqnarray*}

Solution 2. \begin{eqnarray*} x^4-7x^2-18&=&(x^4-9x^2)+(2x^2-18)\\ &=&x^2(x^2-9)+2(x^2-9)\\ &=&(x^2-9)(x^2+2)\\ &=&(x-3)(x+3)(x^2+2) \end{eqnarray*}

Solution 3. \begin{eqnarray*} x^4-7x^2-18&=&(x^4-81)-(7x^2-63)\\ &=&(x^2+9)(x^2-9)-7(x^2-9)\\ &=&(x^2-9)(x^2+9-7)\\ &=&(x-3)(x+3)(x^2+2) \end{eqnarray*}

share|improve this answer

Hint: For this one, note that $x$ only appears as an even power. Substitute $y$ for $x^2$ and see if you can do it.

share|improve this answer

Hint: If the $x^4$ and $x^2$ are confusing, a very useful trick is to replace them.

More precisely, if we let "$y$" mean $x^2$, then the polynomial is $$y^2-7y-18.$$ Can you factor this? After you have done that, you can replace $y$ with $x^2$ and keep going.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.