Simplify (8x^2-3x)-(5x-5-8x^2)
The problem provided is an algebraic expression that involves basic operations such as addition, subtraction, and multiplication, as well as variables and exponents. The expression consists of two parts which are enclosed in parentheses. The task requires you to combine like terms and simplify the expression by performing the subtraction operation across the two sets of terms in parentheses, keeping in mind the correct order of operations and the rules for combining like terms and distributive property if necessary. The result should be a simplified algebraic expression with no parentheses and all like terms combined.
$\left(\right. 8 x^{2} - 3 x \left.\right) - \left(\right. 5 x - 5 - 8 x^{2} \left.\right)$
Answer
Solution:
Step 1: Break down the expression
Start by distributing the negative sign across the terms in the parentheses.
Step 1.1: Apply the distributive property
Use the distributive property to remove parentheses: $8x^2 - 3x - (5x) - (-5) - (-8x^2)$
Step 1.2: Execute simplification
Proceed to simplify the terms.
Step 1.2.1: Handle the multiplication by -1
Multiply $5$ by $-1$: $8x^2 - 3x - 5x - (-5) - (-8x^2)$
Step 1.2.2: Simplify the negative of a negative
Change $-(-5)$ to $+5$: $8x^2 - 3x - 5x + 5 - (-8x^2)$
Step 1.2.3: Simplify the negative of a negative for the last term
Change $-(-8x^2)$ to $+8x^2$: $8x^2 - 3x - 5x + 5 + 8x^2$
Step 2: Combine like terms
Add and subtract like terms to simplify the expression.
Step 2.1: Add the $x^2$ terms
Combine $8x^2$ and $8x^2$: $16x^2 - 3x - 5x + 5$
Step 2.2: Combine the $x$ terms
Combine $-3x$ and $-5x$: $16x^2 - 8x + 5$
Knowledge Notes:
The problem involves simplifying a polynomial expression by combining like terms and applying the distributive property. The distributive property states that for any real numbers $a$, $b$, and $c$, the equation $a(b + c) = ab + ac$ holds true. This property is used to eliminate parentheses in algebraic expressions.
When simplifying expressions, it is important to:
Apply the distributive property correctly, especially when dealing with negative signs in front of parentheses.
Combine like terms, which are terms that have the same variable raised to the same power.
Keep track of positive and negative signs to ensure accurate addition and subtraction of terms.
In the given solution, the following steps are taken:
Distributive property is applied to eliminate parentheses.
Negative signs are distributed across the terms within the parentheses.
Like terms are combined by addition or subtraction to simplify the expression to its simplest form.
The final simplified form of the given polynomial expression is obtained by following these steps, resulting in $16x^2 - 8x + 5$.
