Question

the polynomial $8x^2 - 8x + 2 - 5 + x$ is simplified to $8x^2 - gx - h$. what are the value of $g$ and $h$?

• $g = -9$ and $h = 7$
• $g = 9$ and $h = -3$
• $g = -7$ and $h = 7$
• $g = 7$ and $h = 3$

Explanation:

Step1: Combine like terms for x

Original polynomial: \(8x^2 - 8x + 2 - 5 + x\)
Combine the x - terms: \(-8x + x=-7x\)

Step2: Combine constant terms

Combine the constant terms: \(2 - 5=-3\)
So the simplified polynomial is \(8x^2-7x + 3\) (we can rewrite \(-7x+3\) as \(-7x-(- 3)\), but wait, the target form is \(8x^2 - gx - h\). Let's re - express our simplified polynomial:
\(8x^2-7x + 3=8x^2-7x-(-3)\). Wait, no, let's do it again. Wait, the original simplification target is \(8x^2 - gx - h\). Let's correctly simplify the original polynomial:
\(8x^2-8x + 2-5 + x=8x^2+(-8x + x)+(2 - 5)=8x^2-7x-3\)? Wait, no, \(2-5=-3\), so \(8x^2-7x-(-3)\) is wrong. Wait, \(8x^2-7x-3\) is equal to \(8x^2-7x-3\), but the target is \(8x^2 - gx - h\). So if we have \(8x^2-7x-3=8x^2 - gx - h\), then by equating coefficients:
For the x - term: \(-g=-7\), so \(g = 7\)
For the constant term: \(-h=-3\), so \(h = 3\)

Wait, let's check again:
Original polynomial: \(8x^2-8x + 2-5 + x\)
Combine x terms: \(-8x+x=-7x\)
Combine constants: \(2 - 5=-3\)
So the polynomial is \(8x^2-7x-3\)
We want it in the form \(8x^2 - gx - h\)
So, \(-gx=-7x\) implies \(g = 7\)
And \(-h=-3\) implies \(h = 3\)

Answer:

\(g = 7\) and \(h = 3\) (the option is "g = 7 and h = 3")