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So, I want to simplify an equation : $(2x^2 + x)^2$.

I thought this would simplify to $2x^4 + x^2$

But, if you input a value for $x$, the answers do not equal. For example, if you input $x = 3$, then: $$(2x^2+x)^2 = 21^2 = 441$$

AND: $$2x^4 + x^2 = 2(82) + 9 = 173$$ Can anyone explain why this is the case?

Jordan Ruetz
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    "(2x^2 + x)^2" is an expression not an equation. Also, see this Wikipedia Article: http://en.wikipedia.org/wiki/FOIL_method – JimmyK4542 Jun 02 '15 at 20:28

5 Answers5

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The commutativity property states that:

  • For all $\color{red}{a},\color{green}{b}$ we have $\color{red}{a}+\color{green}{b} = \color{green}{b}+\color{red}{a}$
  • For all $\color{red}{a},\color{green}{b}$ we have $\color{red}{a}\times\color{green}{b} = \color{green}{b}\times\color{red}{a}$

distributivity property of multiplication over addition states that:

  • For all $\color{red}{a},\color{green}{b},\color{blue}{c}$ you have: $(\color{red}{a}+\color{green}{b})\times \color{blue}{c} = \color{red}{a}\times \color{blue}{c} + \color{green}{b}\times \color{blue}{c}$

$(2x^2 + x)^2 = (\color{red}{2x^2}+\color{green}{x})\color{blue}{(2x^2+x)}$

For the moment, let us refer to the blue parenthesis as a single piece, and use the distributivity property above:

$=\color{red}{2x^2}\color{blue}{(2x^2+x)}+\color{green}{x}\color{blue}{(2x^2+x)}$

Now, using commutativity and distributivity, and reassigning colors, we see that this is:

$=(\color{red}{2x^2}+\color{green}{x})\color{blue}{2x^2} + (\color{red}{2x^2}+\color{green}{x})\color{blue}{x} = \color{red}{2x^2}\times \color{blue}{2x^2}+\color{green}{x}\times\color{blue}{2x^2}+\color{red}{2x^2}\times\color{blue}{x}+\color{green}{x}\times\color{blue}{x}$

$=4x^4+2x^3+2x^3+x^2=4x^4+4x^3+x^2$

JMoravitz
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  • Distributive property can be visualized using rectangle areas (see this picture) (not only for positive reals, e.g. imagine $((a+b)-b)c=(a+b)c-bc$ in the picture). – user26486 Jun 02 '15 at 20:54
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$\left(2x^2+x\right)^2=\left(2x^2+x\right)\left(2x^2+x\right)=4x^4+4x^3+x^2$

Karl
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Hint: $$ (A+B)^2= (A+B)(A+B)=A^2+AB+BA+B^2=A^2+B^2+2AB $$ Use: $A=2x^2$ and $B=x$ and you find the right result.

Emilio Novati
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In general, $(a + b)^2 = (a+b)(a+b) = a^2 + 2ab + b^2$. So, in your case,

$$(2x^2 + x)^2 = (2x^2)^2 + 2(2x^2\cdot x) + x^2 = 4x^4 + 4x^3 + x^2$$

Ken
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Think of it like this.

$a^{2}$ means $a \times a$.

More generally, $a^{n}$ means $\underbrace{a \times a \times a \times \dots \times a}_{n \text{ times}}$.

So $2^{3}$ means $2 \times 2 \times 2$. That's $2 \times 2$, $3$ times.

Therefore $(2x^{2} + x)^{2}$ means $(2x^{2} + x) \times (2x^{2} + x)$.

Now, using your number, we see that:

$(2 \times 3^{2}) + 3 = 18 + 3 = 21$

And $21 \times 21$ is very different to $(2 \times 3^{4}) + 3^{2} = 162 + 9 = 171$.

What you have learned the hard way is that exponentiation is not distributive over addition. Which means that $(a + b)^{2} \neq a^{2} + b^{2}$. Why is this? Because $(a + b)^{2}$ means:

Multiply $(a + b)$ by $(a + b)$

Not multiply $a$ by $a$ and multiply $b$ by $b$ and add them together.

For a more simple example than yours, let's look at $(3 + 2)^{2}$

$(3 + 2)^{2} = (3 + 2) \times (3 + 2) = 5 \times 5 = 25$

But:

$(3 + 2)^{2} \neq 3 \times 3 + 2 \times 2 = 9 + 4 = 13$

N.B.: Here we are talking about the general case. There is certainly one specific example of where $(a + b)^{2} = a^{2} + b^{2}$ but this is basically just by coincidence.

Au101
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