Use algebra tiles to find $\left(-10 x^2-8 x+9\right)-(-3 x+7)$.
Combine like terms. For example, write 3 instead of $1+1+1$.
To simplify the expression \((-10x^2 - 8x + 9) - (-3x + 7)\), we analyze the polynomial subtraction. Subtracting a polynomial is equivalent to adding its additive inverse. Using algebra tiles, this means that instead of subtracting three negative \(x\)-tiles and seven positive unit tiles, we add three positive \(x\)-tiles and subtract seven positive unit tiles (which is the same as adding seven negative unit tiles).
First, distribute the negative sign to each term inside the second set of parentheses. This transforms the subtraction into an addition problem:
\[\left(-10x^2 - 8x + 9\right) - \left(-3x + 7\right) = -10x^2 - 8x + 9 + 3x - 7.\]
By removing the parentheses and changing the signs, we have effectively added the additive inverse of the second polynomial.
Now, we group the like terms together based on their powers of \(x\). The terms are organized as follows:
\[-10x^2 + (-8x + 3x) + (9 - 7).\]
Combining the coefficients:
Thus, the expression simplifies to \(-10x^2 - 5x + 2\).
Algebra tiles are a great visual tool for understanding how variables behave in equations. Imagine you have a garden where you represent different vegetable patches as blocks: \(x^2\) blocks for large areas, \(x\) blocks for rows, and unit blocks for individual plants. When you 'subtract' a group of plants, you are effectively taking them out of your garden or removing their markers. Algebraically, this helps you maintain an inventory of what you have left, ensuring your calculations reflect the physical reality of the objects being managed.