# Simplification process with exponent in a polynomial function input

During the factorization process from the point of view of a function, I have difficulty to see where some part gets converted.

Assuming we have the function

$$f(x)=2x^2+3x-1$$

When I want to evaluate as...

$$f(a+h)$$

Taken from my course textbook outlining this process to replace all $x$ to $a+h$.

The simplification procedure seems unclear to me:

$$f(a+h) = 2(a+h)^2+3(a+h)-1$$ $$f(a+h) = 2(a^2+2ah+h^2)+3(a+h)-1$$ $$f(a+h) = 2a^2+4ah+2h^2+3a+3h-1$$

The second line where Mostly when ... $2ah$ ... gets me confused.

What could be the explanation?

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Let's change letters. We want to show that $$(x+y)^2=x^2+2xy+y^2.$$ With these changed letters the result may already be familiar to you. But if you want algebraic details, $$(x+y)^2=(x+y)(x+y)=x(x+y)+y(x+y)=x^2+xy+yx+y^2=x^2+2xy+y^2.$$

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The second line here is very confusing for me. Mostly when 2ah.

I am assuming you want to know where did the 2ah come from. The diagram below is meant to show how we computer $(a+h)(a+h)$ and where did the $2ah$ came from. Each arrow is labeled with a number. There are 4 multiplications required. Each multiplication's result is show besides the arrow number.

I hope this helps.

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Wow. That is very clear explanation. This actually helped a lot. – renoirb Sep 23 '12 at 19:33
Thank you for your feedback, I will be happy to help you with any such questions you might have. – NoChance Sep 23 '12 at 20:27

We have,$$f(a + h) =2(a + h)^2 + 3(a + h) -1$$Most students confuse $2(a + h)^2$ with $(2a + 2h)^2$, and I hope that is not the case with you. Remember that $2(a + h)^2$ is actually $2 \times (a + h)^2$ and you have to simplify the exponent before multiplying according to order of operations.

Simplifying $(a + h)^2$ again, is an elementary process involving intuition and distributive property. Continuing,\begin{align}(a + h)^2 & = &(a + h)(a +h) \\ & = & a(a + h) + h(a + h)\\& = &a^2 + ah + ah + h^2\\ & = &a^2 + 2ah + h^2 \end{align}Coming back, we had $2(a + h)^2$ which simplifies to $2(a^2 + 2ah + h^2)$ or $2a^2 + 4ah + 2h^2$. For completing the answer, you have to add $3(a + h)$ to $2a^2 + 4ah + 2h^2$ and subtract $1$ from the result. Note that $3(a + h) = 3a + 3h$.

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