Simplify the polynomial expression 4x 3 2x 2 5x 1 x 3 3x 2 2...

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--- ## Question 1 **Question content**: Simplify the polynomial expression \((4x^3 - 2x^2 + 5x - 1) - (x^3 + 3x^2 - 2x + 4)\) **Discipline**: Mathematics **Grade**: Eighth grade (or first year of junior high school) ### Correct answer \(3x^3 - 5x^2 + 7x - 5\) ### Detailed problem-solving steps 1. **Distribute the negative sign**: Apply the distributive property to the second polynomial by multiplying each term inside \((x^3 + 3x^2 - 2x + 4)\) by \(-1\): \[ (4x^3 - 2x^2 + 5x - 1) - x^3 - 3x^2 + 2x - 4 \] 2. **Combine like terms**: Group and simplify terms with the same power of \(x\): - **\(x^3\) terms**: \(4x^3 - x^3 = (4 - 1)x^3 = 3x^3\) - **\(x^2\) terms**: \(-2x^2 - 3x^2 = (-2 - 3)x^2 = -5x^2\) - **\(x\) terms**: \(5x + 2x = (5 + 2)x = 7x\) - **Constant terms**: \(-1 - 4 = -5\) 3. **Write the simplified polynomial**: Combine all simplified terms: \[ 3x^3 - 5x^2 + 7x - 5 \] ### Knowledge points involved 1. **Distributive Property** - **Interpretation**: For any real numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). When simplifying polynomials, this property is used to distribute negative signs (or coefficients) to all terms inside parentheses (e.g., \(-(b + c) = -b - c\)). 2. **Combining Like Terms** - **Interpretation**: Terms with the same variable raised to the same power (e.g., \(3x^2\) and \(5x^2\)) are "like terms" and can be combined by adding or subtracting their coefficients (e.g., \(3x^2 + 5x^2 = (3 + 5)x^2 = 8x^2\)). 3. **Polynomial Subtraction** - **Interpretation**: To subtract two polynomials, distribute the negative sign to all terms in the second polynomial, then combine like terms. This is equivalent to adding the opposite of the second polynomial. 4. **Polynomial Structure** - **Interpretation**: A polynomial is an expression of the form \(a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0\), where \(a_n, a_{n-1}, \dots, a_0\) are coefficients and \(n\) is a non-negative integer. Simplifying a polynomial involves reducing it to the form with the fewest like terms.

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