If $\left(x^2-4\right) \div(x+2)=x-2$, which polynomial should fill in the blank?
$(x+2) \times \_ = x^2-4$
To find the polynomial that fills the blank in the equation \((x + 2) \times \text{____} = x^2 - 4\), we can use the fundamental relationship between multiplication and division. If we have an equation of the form \(A \times B = C\), then we know that \(B = C \div A\). In our case, \(A = (x + 2)\), \(C = (x^2 - 4)\), and we want to find the expression \(B\) such that \((x + 2) \times B = x^2 - 4\).
The problem provides the division result: \((x^2 - 4) \div (x + 2) = x - 2\). Since we established that the missing polynomial \(B\) is equivalent to \((x^2 - 4) \div (x + 2)\), we can substitute the given value into our expression:
\[B = (x^2 - 4) \div (x + 2) = x - 2\]
Thus, the polynomial that fills the blank is \(x - 2\).
We compare our result, \(x - 2\), with the given multiple-choice options:
The expression \(x - 2\) corresponds to option (b). Therefore, the polynomial that fills the blank is \(x - 2\), and the correct option is (b).
In algebra, understanding the inverse relationship between operations is essential for solving equations. For example, if you know that a certain amount of items, when split into 5 groups, results in 4 items per group, you can find the total amount by multiplying: \(5 \times 4 = 20\). This mirrors how we solve for unknown factors in polynomials by performing the inverse operation, effectively "undoing" the multiplication to reveal the hidden factor. This skill is foundational for simplifying complex algebraic expressions and solving for variables in various scientific and real-world applications.