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Simplify Polynomial Expression (4x³ - 2x² + 5x - 1) - (x³ + 3x² - 2x + 4)
Mathematics
Junior High School (e.g., Grade 7-9)
Question Content
Simplify the polynomial expression: \\((4x^3 - 2x^2 + 5x - 1) - (x^3 + 3x^2 - 2x + 4)\\)
Correct Answer
The simplified form is \( 3x^3 - 5x^2 + 7x - 5 \)
Detailed Solution Steps
1
Step 1: Distribute the subtraction sign (-) to all terms in the second polynomial. This means 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
Step 2: Combine like terms (terms with the same power of \( x \)): - For \( x^3 \)-terms: \( 4x^3 - x^3 = 3x^3 \) - For \( x^2 \)-terms: \( -2x^2 - 3x^2 = -5x^2 \) - For \( x \)-terms: \( 5x + 2x = 7x \) - For constant terms: \( -1 - 4 = -5 \)
3
Step 3: Combine all simplified terms: \( 3x^3 - 5x^2 + 7x - 5 \)
Knowledge Points Involved
1
Polynomial Subtraction
To subtract polynomials, distribute the negative sign to every term in the subtrahend (the polynomial being subtracted) and then combine like terms. This follows the distributive property of multiplication over addition/subtraction.
2
Combining Like Terms
Like terms are terms with the same variable(s) raised to the same power(s). When combining them, add or subtract their coefficients while keeping the variable part unchanged. For example, \( 4x^3 - x^3 = (4 - 1)x^3 = 3x^3 \).
3
Distributive Property
The distributive property states that \( a(b + c) = ab + ac \). In polynomial subtraction, this is used to distribute the subtraction sign (equivalent to multiplying by -1) across all terms in the subtrahend: \( -(a + b) = -a - b \).
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