Question

Which expression is equivalent to

Answer 1

Answer:

$22x^2-x+17$

Step-by-step explanation:

We have

$10-(2x^3+11x^2-2x-35) \div (x+5)+6(4x^2).$

We must simplify this in the order specified by BIDMAS: brackets, indices, division, multiplication, addition, subtraction.

Because there is nothing in brackets which can be simplified and there are no indices (powers) either our first step is to simplify the division. That is, to simplify $(2x^3+11x^2-2x-35) \div (x+5).$

My favourite way to solve this is by "reverse-factorisation." Observe that if

$(2x^3+11x^2-2x-35) \div (x+5) = y$ then $y(x+5) = (2x^3+11x^2-2x-35)$. That means we are looking for a polynomial y that makes that equation true.

$2x^3+11x^2-2x-35 = (\ \ \ \ \ \ )(x+5)$ where y is the "blank" bracket we need to (and can) fill in. Imagine we are expanding the brackets, the first terms multiplied together must give $2x^3$ and we know one of them is $x$ which means the missing term (at the start of the y bracket) must be $2x^2$. But if $2x^2$ is in y then after we have multiplied the first terms we have to multiply the $2x^2$ by the next term in $(x+5)$, that is the $+5$ term. That is $2x^2 \times 5$ which is $10x^2$. We want it to be $11x^2$ however so we need to add one more $x^2$ to the left hand side. What can we add into the empty y bracket to be multiplied by $(x+5)$ to give $x^2$? We add another $x$ and multiply out by the $(x+5)$ bracket to see we achieve the required $11x^2$ but also obtain an "extra" $5x$ by multiplying the new $x$ by the $5.$ Repeat the process one final time, we have +5x we need -2x so add $-7$ to the bracket. When multiplied out, this yields $-7x$ which brings us to the required $-2x$ and also when multiplied by the +5 yields -35. This completes the reverse factorisation and we see the missing y bracket is $(2x^2+x-7)$. Hence this is the result of the division $(2x^3+11x^2-2x-35) \div (x+5).$

The expression then becomes $10-(2x^2+x-7)+6(4x^2)$. To multiply out the last bracket is simple and we see the expression is equivalent to $10-(2x^2+x-7)+24x^2$. Finally notice the minus sign is in front of a bracket which means the minus applies to all terms within the bracket so flipping the signs inside the bracket shows us the expression is $10-2x^2-x+7+24x^2 = 22x^2-x+17$.

Most of the hard work came from the polynomial division. I hope I explained it clearly enough, if not, leave  message in the comments and I'll try again. If you don't like my "reverse-factorisation" there are other methods to do polynomial division which I can try demonstrate but they are difficult to show on Brainly.