Simplification of Algebraic Expressions

Algebraic Expression and Polynomial Simplification

The provided expression is an algebraic problem from Page 1, labeled as item (b), which requires the simplification of a polynomial expression: $3x(2x+3)-x(3x+5)$. This mathematical statement consists of two distinct products of monomials and binomials, separated by a subtraction operator. To simplify this expression, one must apply the distributive property to eliminate the parentheses and then combine like terms to achieve the most concise form of the polynomial.

Expansion via the Distributive Property

The first step in resolving the expression $3x(2x+3)-x(3x+5)$ is to apply the distributive property, which is a fundamental algebraic principle defined by the formula $a(b + c) = ab + ac$. This requires multiplying the term outside the parentheses to every term located within the parentheses.

For the first part of the expression, $3x(2x+3)$, the monomial $3x$ is distributed. First, we calculate the product of $3x$ and $2x$. Multiplying the coefficients $3$ and $2$ results in $6$, while multiplying the variables $x$ and $x$ (by adding their exponents according to the product rule $x^a \times x^b = x^{a+b}$) results in $x^{2}$. Thus, the first product is $6x^{2}$. Next, we multiply $3x$ by the constant $3$. Multiplying the coefficient $3$ by the constant $3$ gives $9$, and the variable $x$ is retained, resulting in $9x$. The fully expanded form of the first section is $6x^{2} + 9x$.

Managing Negatives in Algebraic Distribution

The second part of the expression is $-x(3x+5)$. It is crucial in algebra to treat the subtraction sign preceding the variable as a negative sign attached to that variable (essentially $-1x$). This negative monomial must be distributed across the binomial $(3x+5)$.

When distributing $-x$ to the term $3x$, we multiply $-1$ by $3$ to get $-3$, and $x$ by $x$ to get $x^{2}$, resulting in the term $-3x^{2}$. Following this, we distribute $-x$ to the constant $+5$. Multiplying a negative by a positive yields a negative value: $-1 \times 5 = -5$, and the variable $x$ is appended, resulting in $-5x$. Combining these results, the expansion of the second part of the original expression is $-3x^{2} - 5x$.

Final Simplification through Combining Like Terms

After completing the distribution for both parts of the expression, we are left with the expanded polynomial string: $6x^{2} + 9x - 3x^{2} - 5x$. To simplify this further, we must identify and combine "like terms," which are terms that share identical variable bases raised to the same powers.

The quadratic terms in this expression are those containing $x^{2}$. We group $6x^{2}$ and $-3x^{2}$. By performing the subtraction of the coefficients, we get $(6-3)x^{2}$, which simplifies to $3x^{2}$.

The linear terms in this expression are those containing the variable $x$ to the power of 1. We group $+9x$ and $-5x$. By subtracting the coefficients, we get $(9-5)x$, which simplifies to $4x$.

Concatenating these Two simplified parts yields the final, irreducible form of the expression: $3x^{2} + 4x$.