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How to Simplify the Polynomial Expression $(9x^{5}+2x^{3}+4x)-(8x^{4}-6x^{3}+6x^{2})$
Mathematics
Grade 9 (Junior High School)
Question Content
Simplify the polynomial expression: $(9x^{5}+2x^{3}+4x)-(8x^{4}-6x^{3}+6x^{2})$
Correct Answer
$9x^{5}-8x^{4}+8x^{3}-6x^{2}+4x$
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 $(8x^{4}-6x^{3}+6x^{2})$, resulting in: $9x^{5}+2x^{3}+4x -8x^{4}+6x^{3}-6x^{2}$
2
Step 2: Group like terms (terms with the same exponent of $x$) together: $9x^{5} -8x^{4} + (2x^{3}+6x^{3}) -6x^{2} +4x$
3
Step 3: Combine the coefficients of like terms. Add $2x^{3}$ and $6x^{3}$ to get $8x^{3}$, and leave the other terms as they have no matching like terms: $9x^{5}-8x^{4}+8x^{3}-6x^{2}+4x$
Knowledge Points Involved
1
Distributive Property of Subtraction for Polynomials
When subtracting a polynomial from another, the negative sign outside the parentheses must be distributed to every term inside the parentheses, flipping the sign of each term. This follows the rule $a-(b-c+d)=a-b+c-d$, and it is used to remove parentheses in polynomial subtraction problems.
2
Like Terms Identification
Like terms are terms in an algebraic expression that have the same variable(s) raised to the same power(s). For example, $2x^{3}$ and $6x^{3}$ are like terms, while $x^{5}$ and $x^{4}$ are not. Identifying like terms is essential for simplifying polynomial expressions.
3
Combining Like Terms
To combine like terms, add or subtract the numerical coefficients of the like terms while keeping the variable and its exponent unchanged. For example, $2x^{3}+6x^{3}=(2+6)x^{3}=8x^{3}$. This is used to simplify polynomial expressions to their standard form.
4
Standard Form of a Polynomial
A polynomial is written in standard form when its terms are ordered from the highest degree (the sum of the exponents of the variables in a term) to the lowest degree. For the simplified result, we order the terms as $9x^{5}-8x^{4}+8x^{3}-6x^{2}+4x$ since the degrees are 5, 4, 3, 2, 1 respectively.
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