Simplify 5x(x + 2) + 4x A.5x 2+ 10 B.9x + 10 C.5x 2- 14x
D
Simplifying Algebraic Expressions
The question asks us to simplify the algebraic expression: \(5x(x + 2) + 4x\).
To simplify this expression, we need to follow the standard rules of algebra, which involve the distributive property and combining like terms.
Applying the Distributive Property
First, we apply the distributive property to the term \(5x(x + 2)\). The distributive property states that \(a(b+c) = ab + ac\). In our case, \(a = 5x\), \(b = x\), and \(c = 2\).
So, \(5x(x + 2)\) becomes:
\(5x \times x + 5x \times 2\)
This simplifies to:
\(5x^2 + 10x\)
Combining Like Terms
Now we substitute this back into the original expression:
\(5x^2 + 10x + 4x\)
Next, we combine the like terms. Like terms are terms that have the same variables raised to the same power. In this expression, \(10x\) and \(4x\) are like terms because they both have the variable \(x\) raised to the power of 1.
We combine them by adding their coefficients:
\(10x + 4x = (10 + 4)x = 14x\)
The term \(5x^2\) is not a like term with \(10x\) or \(4x\), so it remains as is.
Putting it all together, the simplified expression is:
\(5x^2 + 14x\)
Final Simplified Expression
The simplified form of \(5x(x + 2) + 4x\) is \(5x^2 + 14x\).
Let's compare this result with the given options:
| Option | Expression |
|---|---|
| A | \(5x^2 + 10\) |
| B | \(9x + 10\) |
| C | \(5x^2 - 14x\) |
| D | \(5x^2 + 14x\) |
Our simplified expression \(5x^2 + 14x\) matches option D.
Steps to Simplify
- Distribute the term outside the parenthesis: \(5x(x+2) = 5x \times x + 5x \times 2 = 5x^2 + 10x\).
- Rewrite the expression: \(5x^2 + 10x + 4x\).
- Combine like terms (\(10x\) and \(4x\)): \(10x + 4x = 14x\).
- The simplified expression is \(5x^2 + 14x\).
Revision Table: Simplifying Algebraic Expressions
| Concept | Description | Example |
|---|---|---|
| Distributive Property | Multiply a term outside parentheses by each term inside. | \(a(b+c) = ab + ac\) |
| Like Terms | Terms with the same variable(s) raised to the same power(s). | \(3x\) and \(7x\); \(2y^2\) and \(-5y^2\) |
| Combining Like Terms | Add or subtract the coefficients of like terms. | \(3x + 7x = (3+7)x = 10x\) |
Additional Information: Polynomials
Algebraic expressions like \(5x^2 + 14x\) are examples of polynomials. A polynomial is an expression consisting of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
- Term: A single number, variable, or the product of numbers and variables (e.g., \(5x^2\), \(10x\), \(4x\)).
- Coefficient: The numerical factor of a term (e.g., 5 in \(5x^2\), 10 in \(10x\), 4 in \(4x\)).
- Variable: A symbol representing an unknown value (e.g., \(x\)).
- Exponent: A number that indicates how many times the base is multiplied by itself (e.g., 2 in \(x^2\)).
Simplifying polynomials involves combining like terms to reduce the expression to its simplest form.
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