Simplify (15x^2+6x)/(3x)*(2x-1)

Brief Explanation of Question:

The question asks you to perform algebraic simplification on the given expression. The expression consists of a fraction where the numerator is a polynomial (15x^2 + 6x) and the denominator is a monomial (3x), which is further multiplied by a binomial (2x - 1). You are expected to apply the rules of algebraic operations, which include distributing the multiplication over addition, cancellation of common factors in the numerator and denominator, and simplification of the resulting expressions to reach the simplest form.

$\frac{15 x^{2} + 6 x}{3 x} \cdot (2 x - 1)$

Answer

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Solution:

Step 1: Simplify the expression.

Step 1.1: Extract the common factor from the numerator.

Step 1.1.1: Take out $3 x$ from $15 x^{2}$ .

$\frac{3 x (5 x) + 6 x}{3 x} \cdot (2 x - 1)$

Step 1.1.2: Take out $3 x$ from $6 x$ .

$\frac{3 x (5 x) + 3 x (2)}{3 x} \cdot (2 x - 1)$

Step 1.1.3: Complete the factoring.

$\frac{3 x (5 x + 2)}{3 x} \cdot (2 x - 1)$

Step 1.2: Eliminate the common $3$ .

Step 1.2.1: Strike out the $3$ .

$\frac{3 x (5 x + 2)}{3 x} \cdot (2 x - 1)$

Step 1.2.2: Rewrite without the $3$ .

$\frac{x (5 x + 2)}{x} \cdot (2 x - 1)$

Step 1.3: Remove the common $x$ .

Step 1.3.1: Strike out the $x$ .

$\frac{x (5 x + 2)}{x} \cdot (2 x - 1)$

Step 1.3.2: Simplify the expression.

$(5 x + 2) \cdot (2 x - 1)$

Step 2: Expand using the FOIL method.

Step 2.1: Distribute the first term.

$5 x (2 x - 1) + 2 (2 x - 1)$

Step 2.2: Distribute the second term.

$5 x (2 x) + 5 x (- 1) + 2 (2 x - 1)$

Step 2.3: Distribute the third term.

$5 x (2 x) + 5 x (- 1) + 2 (2 x) + 2 (- 1)$

Step 3: Combine like terms and simplify.

Step 3.1: Simplify each term.

Step 3.1.1: Apply commutative property.

$5 \cdot 2 x^{2} + 5 x (- 1) + 2 \cdot 2 x + 2 (- 1)$

Step 3.1.2: Multiply powers of $x$ .

Step 3.1.2.1: Combine $x$ terms.

$5 \cdot 2 (x \cdot x) + 5 x (- 1) + 2 \cdot 2 x + 2 (- 1)$

Step 3.1.2.2: Apply exponent rules.

$5 \cdot 2 x^{2} + 5 x (- 1) + 2 \cdot 2 x + 2 (- 1)$

Step 3.1.3: Multiply constants.

$10 x^{2} + 5 x (- 1) + 2 \cdot 2 x + 2 (- 1)$

Step 3.1.4: Multiply by $- 1$ .

$10 x^{2} - 5 x + 2 \cdot 2 x + 2 (- 1)$

Step 3.1.5: Multiply constants.

$10 x^{2} - 5 x + 4 x + 2 (- 1)$

Step 3.1.6: Multiply by $- 1$ .

$10 x^{2} - 5 x + 4 x - 2$

Step 3.2: Add and subtract like terms.

$10 x^{2} - x - 2$

Knowledge Notes:

To solve the given problem, we employed several algebraic techniques:

Factoring: This involves finding a common factor in terms and expressing them as a product of that factor and the remaining terms. In this case, we factored out $3 x$ from the numerator.
Simplifying Fractions: When a common factor appears in both the numerator and denominator, it can be canceled out. We did this with both $3$ and $x$ to simplify the fraction.
Distributive Property: This property states that $a (b + c) = a b + a c$ . It was used to expand the expression $(5 x + 2) (2 x - 1)$ .
FOIL Method: This is a specific case of the distributive property used for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, referring to the terms in each binomial that are multiplied together.
Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power. For example, $- 5 x$ and $4 x$ were combined to give $- x$ .
Exponent Rules: When multiplying powers with the same base, we add the exponents. This is why $x \cdot x$ becomes $x^{2}$ .

By applying these concepts in a structured manner, we simplified the given algebraic expression to its simplest form.