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I can't figure out how to simplify this polynominal $5x^2+3x^4-7x^3+5x+8+2x^2-4x+9-6x^2+7x$

I tried combining like terms $5x^2+3x^4-7x^3+5x+8+2x^2-4x+9-6x^2+7x$ $(5x^2+5x)+3x^4-(7x^3+7x)+2x^2-4x-6x^2+(8+9)$ $5x^3+3x^4-7x^4+2x^2-4x-6x^2+17$

It says the answer is $3x^4-7x^3+x^2+8x+17$ but how did they get it?

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    Sadly, no. You have to choose the best one. Occasionally, I use enny-meany-minie-mo.2012-12-05

3 Answers 3

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Group the terms with the same power of $x$ together.

$5x^2+3x^4−7x^3+5x+8+2x^2−4x+9−6x^2+7x$

$=3x^4−7x^3+5x^2+2x^2−6x^2+5x−4x+7x+8+9$

$=3x^4−7x^3+x^2+8x+17$

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Observe the magical power of color:

$\color{blue}{5}x^\color{blue}{2}+3x^4-7x^3+\color{green}{5}x+\color{orange}{8}+\color{blue}{2}x^\color{blue}{2}+(\color{green}{-4})x+\color{orange}{9}+(\color{blue}{-6})x^\color{blue}{2}+\color{green}{7}x.$

Instead of Color-Me-Elmo, we have Color-Me-Like-Terms-And-Combine (not as catchy, I know): $3x^4-7x^3+(\color{blue}{5}+\color{blue}{2}+(\color{blue}{-6}))x^\color{blue}{2}+(\color{green}{5}+(\color{green}{-4})+\color{green}{7})x+(\color{orange}{8}+\color{orange}{9}).$

Presto-simplification-o!


Combining Like Terms

In a polynomial $p(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0$ and $q(x)=b_nx^n+b_{n-1}x^{n-1}+\dots+b_1x+b_0$, they are added thusly: $ \begin{align} p(x)+q(x)&=a_nx^n+b_nx^n+a_{n-1}x^{n-1}+b_{n-1}x^{n-1}+\cdots+a_1x+b_1x+a_0+b_0\\ &=(a_n+b_n)x^n+(a_{n-1}+b_{n-1})x^{n-1}+\cdots+(a_1+b_1)x+(a_0+b_0). \end{align} $

In other words, add the coefficients of terms with the same power.

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    @JasperLoy I am getting you a _Color-Me-Elmo_ for Christmas.2012-12-05
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You cannot combine terms like that, you have to split your terms by powers of $x$.

So for example $5x^2+5x+2x^2 = (5+2)x^2+5x = 7x^2+5x$ and not $5x^3+2x^2$. Using this, you should end up with your answer.