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Simplify Polynomial Expression (4x³ - 2x² + 5x - 1) - (x³ + 3x² - 2x + 4)
Mathematics
Grade 8
Question Content
Simplify the following polynomial expression: \((4x^3 - 2x^2 + 5x - 1) - (x^3 + 3x^2 - 2x + 4)\)
Correct Answer
\(3x^3 - 5x^2 + 7x - 5\)
Detailed Solution Steps
1
Step 1: Apply the distributive property (subtraction rule) to remove the parentheses. Since there is a minus sign before the second polynomial, change the sign of each term inside the second parentheses: \((4x^3 - 2x^2 + 5x - 1) - x^3 - 3x^2 + 2x - 4\)
2
Step 2: Rewrite the expression by distributing the negative sign: \(4x^3 - 2x^2 + 5x - 1 - x^3 - 3x^2 + 2x - 4\)
3
Step 3: Combine like terms for \(x^3\): \(4x^3 - x^3 = 3x^3\)
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Step 4: Combine like terms for \(x^2\): \(-2x^2 - 3x^2 = -5x^2\)
5
Step 5: Combine like terms for \(x\): \(5x + 2x = 7x\)
6
Step 6: Combine constant terms: \(-1 - 4 = -5\)
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Step 7: Combine all simplified terms: \(3x^3 - 5x^2 + 7x - 5\)
Knowledge Points Involved
1
Polynomial Subtraction
The process of subtracting one polynomial from another, which requires changing the sign of each term in the subtrahend (the polynomial being subtracted) and then adding it to the minuend (the polynomial being subtracted from).
2
Distributive Property (Subtraction)
For any real numbers \(a\), \(b\), and \(c\), \(a - (b + c) = a - b - c\). This means when a negative sign precedes a parenthetical expression, each term inside the parentheses has its sign flipped.
3
Combining Like Terms
Like terms are terms with the same variable(s) raised to the same power(s). To combine them, add or subtract their coefficients while keeping the variable part unchanged (e.g., \(3x^2 + 5x^2 = 8x^2\)).
4
Like Terms Definition
Terms that have identical variable parts (same variables with the same exponents) are called like terms. For example, \(2x^3\) and \(5x^3\) are like terms, but \(2x^3\) and \(5x^2\) are not.
5
Polynomial Definition
A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, combined using only addition, subtraction, multiplication, and non-negative integer exponents of variables (e.g., \(4x^3 - 2x^2 + 5x - 1\) is a polynomial in \(x\)).
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