To simplify the polynomial expression \((r - 1)(r^2 - 2r + 3)\), we can use the distributive property (also known as the FOIL method for binomials).
We'll distribute \( (r - 1) \) to each term in \( (r^2 - 2r + 3) \):
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Multiply \( r \) by each term in \( r^2 - 2r + 3 \): \[ r \cdot r^2 = r^3 \] \[ r \cdot (-2r) = -2r^2 \] \[ r \cdot 3 = 3r \]
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Multiply \( -1 \) by each term in \( r^2 - 2r + 3 \): \[ -1 \cdot r^2 = -r^2 \] \[ -1 \cdot (-2r) = 2r \] \[ -1 \cdot 3 = -3 \]
Now combine all these results: \[ r^3 + (-2r^2) + 3r + (-r^2) + 2r + (-3) \]
This simplifies to: \[ r^3 + (-2r^2 - r^2) + (3r + 2r) - 3 \] \[ r^3 - 3r^2 + 5r - 3 \]
So the final polynomial expression is: \[ r^3 - 3r^2 + 5r - 3 \]
Filling in the blanks: __r^3 + __r^2 + __r + __ will be: 1, -3, 5, -3
So the completed expression is:
- \( r^3 + (-3)r^2 + 5r - 3 \) or more simply,
- \( 1, -3, 5, -3 \)
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