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Multiply to simplify the polynomial expression (r−1)(r2−2r+3) .

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To simplify the expression \((r - 1)(r^2 - 2r + 3)\), we'll use the distributive property (also known as the FOIL method for binomials).

Step 1: Distribute \(r\) from the first polynomial to each term in the second polynomial: \[ r(r^2) + r(-2r) + r(3) = r^3 - 2r^2 + 3r \]

Step 2: Distribute \(-1\) from the first polynomial to each term in the second polynomial: \[ -1(r^2) + (-1)(-2r) + (-1)(3) = -r^2 + 2r - 3 \]

Step 3: Now combine all the terms together: \[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]

Step 4: Combine like terms: \[ r^3 + (-2r^2 - r^2) + (3r + 2r) - 3 = r^3 - 3r^2 + 5r - 3 \]

Thus, the simplified form of the polynomial expression \((r - 1)(r^2 - 2r + 3)\) is: \[ \boxed{r^3 - 3r^2 + 5r - 3} \]

do this again

Sure! Let's simplify the expression \((r - 1)(r^2 - 2r + 3)\) step by step.

Step 1: Distribute \(r\) from the first polynomial to each term in the second polynomial.

\[ r(r^2) + r(-2r) + r(3) = r^3 - 2r^2 + 3r \]

Step 2: Distribute \(-1\) from the first polynomial to each term in the second polynomial.

\[ -1(r^2) + (-1)(-2r) + (-1)(3) = -r^2 + 2r - 3 \]

Step 3: Combine all the terms together:

Now we put together the results from Step 1 and Step 2: \[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]

Step 4: Combine like terms:

Combine the like terms:

  • The \(r^2\) terms: \(-2r^2 - r^2 = -3r^2\)
  • The \(r\) terms: \(3r + 2r = 5r\)

Putting it all together, we get: \[ r^3 - 3r^2 + 5r - 3 \]

So, the simplified expression for \((r - 1)(r^2 - 2r + 3)\) is: \[ \boxed{r^3 - 3r^2 + 5r - 3} \]

can u do this only with the numbers

Sure! Let's simplify the expression \((r - 1)(r^2 - 2r + 3)\) using only the coefficients and constants without detailing the letters.

Step 1: Distribute the first term

First, distribute \(r\):

  • \(r \cdot 1 = r\)
  • \(r \cdot (-2) = -2r\)
  • \(r \cdot 3 = 3r\)

Adding these gives: \[ 1r - 2r + 3r \]

Step 2: Distribute the second term

Now, distribute \(-1\):

  • \(-1 \cdot 1 = -1\)
  • \(-1 \cdot (-2) = +2\)
  • \(-1 \cdot 3 = -3\)

Adding these gives: \[ -1 + 2 - 3 \]

Step 3: Combine both results

Now combine all the terms together:

  • Terms from distributing \(r\): \(r - 2r + 3r\) simplifies to \(2r\)
  • Terms from distributing \(-1\): \(-1 + 2 - 3\) simplifies to \(-2\)

Final Result:

So we have: \[ 1r + 0r - 2 \Rightarrow r^3 - 3r^2 + 5r - 3 \]

Thus, the simplified expression is: \[ \boxed{r^3 - 3r^2 + 5r - 3} \]

I realize my explanation may still sound complex. Here is a truly numerical form:

Direct Approach with Numerical Representation:

  1. From \((1)(1) = 1\)
  2. From \((r)(-2) = -2\)
  3. From \((r)(3) = 3\)
  4. From \((-1)(1) = -1\)
  5. From \((-1)(-2) = 2\)
  6. From \((-1)(3) = -3\)

Finally, combine:

  • Coefficient of \(r^2\): \(-2 - 1 = -3\)
  • Coefficient of \(r\): \(3 + 2 = 5\)

The constant term:

  • Total = \(-1 + 2 - 3 = -2\)

Thus the final form stays: \[ r^3 - 3r^2 + 5r - 2 \] Which was a direct accounting resulted in maintaining the logical terms and please consider rectifying the distribution conditions or mentions. Let me know if you have another numerical focus or format inquiry!