Answered

Use algebra tiles to find $\left(-4 x^2-x-7\right)-\left(7 x^2+5\right)$.




$x^2$		$x^2$$x^2$$x^2$

$x^2$		$x^2$$x^2$
\begin{tabular}[t]{l}
1

1

1
\\

\hline

\end{tabular}



Combine like terms. For example, write 3 instead of 1 + 1 + 1.

Verified Answer :

  • Distribute the negative sign across the second polynomial: \(-4x^2 - x - 7 - 7x^2 - 5\).
  • Group like terms by their powers: \((-4x^2 - 7x^2) - x + (-7 - 5)\).
  • Calculate the combined coefficients: \(-11x^2 - x - 12\).
  • The final simplified expression is \(\boxed{-11x^2 - x - 12}\).

Explanation

  1. Distributing the Negative Sign

To simplify the expression \((-4x^2 - x - 7) - (7x^2 + 5)\), we first recognize that the negative sign in front of the second set of parentheses applies to every term within those parentheses. We distribute the negative sign:
\[ - (7x^2 + 5) = -7x^2 - 5 \]
Now, rewrite the entire expression without the parentheses:
\[ -4x^2 - x - 7 - 7x^2 - 5 \]

  1. Grouping and Combining Like Terms

Next, we group the like terms together. Like terms are those that have the same variable raised to the same power. Here, we group the \(x^2\) terms, the \(x\) term, and the constant terms:
\[ (-4x^2 - 7x^2) + (-x) + (-7 - 5) \]
Now, combine the coefficients of each group:

  • For the \(x^2\) terms: \(-4 - 7 = -11\), so we have \(-11x^2\).
  • For the \(x\) term: There is only \(-x\).
  • For the constant terms: \(-7 - 5 = -12\).

Combining these results gives us the simplified polynomial:
\[ -11x^2 - x - 12 \]

Examples

In everyday life, algebra tiles and polynomial simplification can represent managing inventory or budget planning. For example, if you have a budget of \(-4x^2 - x - 7\) (representing a base debt of \(x^2\) units and fixed costs) and you need to subtract additional expenses of \(7x^2 + 5\), calculating the result \(-11x^2 - x - 12\) helps you determine your new total balance. Mastering these operations allows you to accurately track changes in complex systems or financial scenarios.

Related Questions