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How to Simplify the Polynomial Expression $(4x^{3}-2x^{2}+5x-1)-(x^{3}+3x^{2}-2x+4)$
Mathematics
Grade 9 of Junior High School
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: Distribute the negative sign to each term inside the second parentheses. This changes the sign of every term in $(x^{3}+3x^{2}-2x+4)$, resulting in: $4x^{3}-2x^{2}+5x-1 -x^{3}-3x^{2}+2x-4$
2
Step 2: Group like terms together. Like terms are terms with the same variable raised to the same power: $(4x^{3}-x^{3}) + (-2x^{2}-3x^{2}) + (5x+2x) + (-1-4)$
3
Step 3: Combine the coefficients of each group of like terms: $(4-1)x^{3} + (-2-3)x^{2} + (5+2)x + (-1-4)$
4
Step 4: Calculate the numerical values of the coefficients and constants to get the simplified polynomial: $3x^{3}-5x^{2}+7x-5$
Knowledge Points Involved
1
Distributive Property of Subtraction for Polynomials
This property states that when subtracting a polynomial from another, you can distribute the negative sign to every term in the subtracted polynomial, which is equivalent to adding the additive inverse of each term. It is written as $a-(b+c+d)=a-b-c-d$, and it is used to remove parentheses when subtracting polynomials.
2
Like Terms Identification and Combination
Like terms are algebraic terms that have the same variables raised to the exact same exponents (constants are also like terms with each other). To combine like terms, you add or subtract their numerical coefficients while keeping the variable part unchanged. This is the core step in simplifying polynomial expressions.
3
Polynomial Simplification Rules
Simplifying a polynomial involves eliminating parentheses (using distribution rules) and combining all like terms to get the polynomial in its standard form, where terms are ordered from the highest degree to the lowest degree of the variable.
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