A polynomial is a mathematical expression constructed from one or more variables and coefficients, utilizing only the operations of addition, subtraction, multiplication, and non-negative integer exponents. These expressions can appear complex due to the presence of grouping symbols that dictate the order of operations. The primary objective of simplification is to rewrite the expression into its shortest, most manageable form without changing its overall mathematical value. Simplification involves systematically removing these grouping symbols through specific algebraic rules.
Removing Parentheses via Distribution
The most direct method for removing parentheses involves distribution, which applies when a single term (a monomial or a constant coefficient) immediately precedes the grouping symbol. This property dictates that the term outside must be multiplied by every individual term located inside the parentheses. For instance, in the expression $4(x+2y-3)$, the coefficient 4 must be multiplied by $x$, by $2y$, and by $-3$ to produce the expanded form $4x + 8y – 12$.
In cases where the parentheses are preceded only by a positive sign or by no operation symbol at all, the grouping symbols can be immediately removed without altering the signs or coefficients of the internal terms. This reflects an implicit distribution of the number positive one, where multiplying any term by one leaves the term unchanged. For example, the expression $(5x^2 – 3x) + (2x^2 + 4x)$ becomes $5x^2 – 3x + 2x^2 + 4x$ directly upon removal of the parentheses.
A different rule applies when the parentheses are preceded by a subtraction sign, which is algebraically equivalent to distributing a negative one. This operation requires changing the sign of every term contained within the parentheses to maintain the expression’s equivalence. If the expression is $-(4a – 5b + 7)$, the terms inside must be treated as being multiplied by the negative coefficient.
The term $4a$ becomes $-4a$, the term $-5b$ becomes $+5b$, and the term $+7$ becomes $-7$, resulting in the expression $-4a + 5b – 7$. Failure to apply the sign change across all enclosed terms is a frequent source of computational error. This careful application ensures that the original relationship between the terms is accurately represented after the removal of the grouping symbols.
Multiplying Two Polynomials
When two distinct sets of parentheses containing multiple terms are multiplied together, the process expands beyond the scope of simple monomial distribution. This requires a comprehensive application of the distributive property, where every term in the first polynomial must be multiplied by every term in the second polynomial. The resulting number of individual products will always equal the product of the number of terms in each polynomial being multiplied.
For the specific case of multiplying two binomials—a polynomial with exactly two terms—a mnemonic device known as FOIL is often used to ensure all four required multiplications are performed. FOIL is an acronym standing for First, Outer, Inner, and Last, which maps the pairs of terms that must be multiplied together. In the expression $(x+3)(x-5)$, the procedure involves multiplying the First terms ($x \cdot x$), the Outer terms ($x \cdot -5$), the Inner terms ($3 \cdot x$), and the Last terms ($3 \cdot -5$).
Executing the FOIL steps yields the expanded form $x^2 – 5x + 3x – 15$, which still requires the final step of simplification. The utility of FOIL is that it provides a systematic method to avoid missing any of the necessary pairwise products during the expansion phase. Although this method is specific to binomials, it is fundamentally a structured way to apply the general distributive principle accurately.
Certain combinations of binomial multiplication lead to special products, which can often be identified and calculated more quickly than through the standard expansion. The difference of squares pattern arises when two binomials are identical except for the sign connecting their terms, such as $(a+b)(a-b)$. In this case, the resulting Outer and Inner terms always have opposite signs and equal magnitude, causing them to cancel each other out and resulting in the simplified form $a^2 – b^2$.
Another type of special product involves squaring a binomial, represented as $(a+b)^2$ or $(a-b)^2$. Expanding this expression always results in a perfect square trinomial, a three-term polynomial with a defined structure. The structure involves squaring the first term, squaring the second term, and then including a middle term that is precisely twice the product of the first and second terms, for example $(a+b)^2 = a^2 + 2ab + b^2$.
Combining Like Terms After Removal
Once all parentheses have been successfully removed, whether through simple distribution or through polynomial multiplication, the final step in simplification is to combine the resulting like terms. This process ensures the polynomial is written in its most condensed form, fulfilling the goal of simplification by eliminating redundancy. Combining terms is an algebraic cleanup that must follow all prior expansion and distribution.
Like terms are defined as terms within a polynomial that share the exact same variable component, meaning they must have the identical variable or variables raised to the identical power. For example, $5x^3$ and $-2x^3$ are considered like terms because both contain the variable $x$ raised to the power of three, but $5x^3$ and $5x^2$ cannot be combined. Only the coefficients, the numerical parts of the terms, are permitted to differ between like terms.
The act of combining involves identifying these matching variable components, grouping them together, and then performing the indicated addition or subtraction solely on their coefficients. When $5x^3$ and $-2x^3$ are combined, the operation is $5 – 2 = 3$, resulting in the single term $3x^3$. This process effectively consolidates multiple instances of the same variable component into a single, comprehensive term.
It is a fixed rule in algebra that the variable component, including its exponent, remains completely unchanged during the combination of like terms. The operation acts only on the count or magnitude represented by the coefficients, not on the nature of the variable itself. This ensures the algebraic identity of the original expression is preserved in its final, simplified state, representing the shortest possible form of the polynomial.