To simplify the polynomial expression \((2x−3)(x^2−4x+5)\), we will use the distributive property (often called the FOIL method for binomials). We will distribute \(2x\) and \(-3\) to each term inside the second polynomial.
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Distributing \(2x\):
- \(2x \cdot x^2 = 2x^3\)
- \(2x \cdot (-4x) = -8x^2\)
- \(2x \cdot 5 = 10x\)
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Distributing \(-3\):
- \(-3 \cdot x^2 = -3x^2\)
- \(-3 \cdot (-4x) = 12x\)
- \(-3 \cdot 5 = -15\)
Now we combine all these results together: \[ 2x^3 - 8x^2 + 10x - 3x^2 + 12x - 15 \]
Now combine like terms:
- The \(x^3\) term is \(2x^3\).
- The \(x^2\) terms: \(-8x^2 - 3x^2 = -11x^2\).
- The \(x\) terms: \(10x + 12x = 22x\).
- The constant term is \(-15\).
So, the simplified polynomial expression is: \[ 2x^3 - 11x^2 + 22x - 15 \]
Based on the format asked in the question, the answer breaks down as follows:
- \(x^3\) term: \(2\)
- \(x^2\) term: \(-11\)
- \(x\) term: \(22\)
- Constant term: \(-15\)
Thus, the ultimately simplified polynomial is: \[ 2x^3 - 11x^2 + 22x - 15 \]