HiSET · Mathematics · 6 min read
Simplifying Expressions and Solving Equations
- Simplify algebraic expressions and polynomials using the distributive property, combining like terms, and rules of exponents.
- Solve multistep linear equations in one variable, including clearing fractions or decimals and classifying equations by their solution types.
- Rearrange and solve literal equations for a specified variable using inverse operations and factoring techniques.
Simplifying Algebraic Expressions and Polynomials
In algebra, expressions are built from individual units called terms. A term in an algebraic expression consists of a number, a variable, or the product of numbers and variables. When simplifying expressions, you look for like terms, which are terms that share the exact same variables raised to the exact same exponents. For instance, just as you would sort and tally matching items in daily life rather than mixing different items together, you group matching algebraic terms. Unlike terms cannot be combined through addition or subtraction into a single term. To combine like terms, add or subtract their numerical coefficients while leaving the variable part unchanged.
To clear groupings and expand expressions, use the distributive property. The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and adding the products. Algebraically, the distributive property is written as a(b + c) = ab + ac. Pay close attention when signs are involved: distributing a negative value across parentheses reverses the sign of every term inside the parentheses. In fact, a negative sign directly preceding parentheses is mathematically equivalent to multiplying each internal term by negative one. When working with two binomials, the FOIL method (First, Outer, Inner, Last) serves as an application of the distributive property used to multiply two binomials.
Working with polynomials follows these same structural principles. A polynomial in one variable is in standard form when its terms are arranged in descending order of exponent. The degree of a nonzero term in one variable is the exponent on its variable, while a nonzero constant term is a number without variables and has degree zero. When combining polynomials, subtraction requires particular care: subtracting a polynomial requires changing the signs of all terms in the subtracted polynomial before combining like terms.
Common mistakes
- Treating -x^2 as equivalent to (-x)^2: The expression -x^2 means the negative of x squared, which is distinct from (-x)^2 where the negative sign is also squared.
- Failing to reverse every sign when distributing a negative: Distributing a negative value across parentheses reverses the sign of every term inside, because a negative sign directly preceding parentheses is mathematically equivalent to multiplying each internal term by negative one.
- Forgetting to exclude values that make an original denominator zero: When clearing fractions by multiplying by a common denominator in any equation, you must exclude values that make an original denominator zero.
- Confusing an identity with a contradiction: An equation simplifying to a true statement like 0 = 0 has infinitely many solutions (an identity), whereas an equation simplifying to a false statement like 0 = 5 has no solution (a contradiction).
Key points
- A term consists of a number, variable, or product of numbers and variables; like terms share the exact same variables and exponents and are combined by adding or subtracting their numerical coefficients.
- The distributive property states that a(b + c) = ab + ac; distributing a negative value reverses the sign of every term inside parentheses, and the FOIL method applies this property to multiply two binomials.
- A polynomial in one variable is in standard form when terms are in descending order of exponent; nonzero constant terms have degree zero, and subtracting a polynomial requires changing the signs of all its terms before combining like terms.
- Key exponent rules include (a^m)(a^n) = a^(m+n), a^m/a^n = a^(m−n) for nonzero base a, (a^m)^n = a^(m*n), a^0 = 1 for non-zero real numbers, and a^(−n) = 1/a^n for a ≠ 0.
- Rational exponents represent roots, where a^(1/n) is the n-th root of a and a^(m/n) is the n-th root of a raised to the m-th power, wherever defined in the real numbers.
- Linear equations are solved using inverse operations and properties of equality; they may have one real solution (conditional), infinitely many solutions (an identity, like 0 = 0), or no solution (a contradiction, like 0 = 5).
- Literal equations contain two or more distinct variables; when solving for a target variable, treat other variables as constants, clear denominators, factor out the target variable if it appears in multiple terms, and divide by any nonzero remaining factor.
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