7. Exponents & Polynomials

An exponent is repeated multiplication: \(2^4 = 2 \times 2 \times 2 \times 2 = 16\). The laws of exponents let you simplify quickly:
\[ a^m \cdot a^n = a^{m+n}, \qquad \frac{a^m}{a^n} = a^{m-n}, \qquad (a^m)^n = a^{mn}, \]
\[ a^0 = 1, \qquad a^{-n} = \frac{1}{a^n}. \]
Example: \(x^3 \cdot x^2 = x^5\) and \(\frac{x^6}{x^2} = x^4\).

A polynomial is a sum of terms like \(2x^2 + 3x - 5\). Add or subtract polynomials by combining like terms:
\[ (2x^2 + 3x) + (x^2 - 5x) = 3x^2 - 2x. \]

Multiply two binomials with FOIL (First, Outer, Inner, Last):
\[ (x + 3)(x + 2) = \underbrace{x^2}_{F} + \underbrace{2x}_{O} + \underbrace{3x}_{I} + \underbrace{6}_{L} = x^2 + 5x + 6. \]

⚠️ Common mistake: \(a^m \cdot a^n\) is \(a^{m+n}\) — you ADD exponents when multiplying, not multiply them.

💡 Tip: only like terms combine. \(x^2\) and \(x\) are different and cannot be added together.

Enriched GED® problem-solving lab

Learning targets

• apply exponent rules only when their conditions are met
• classify polynomial terms by degree
• add and multiply polynomial expressions

Visual explanation

How to read this SVG. Scan each term in standard form. Terms with the same power belong in one vertical column when adding or subtracting. Degree describes the greatest exponent after simplification; it does not come from adding every exponent visible in the expression.

Reliable problem-solving method

1. Identify the operation: product, quotient, power, or sum.
2. Apply an exponent law only to factors with the same base.
3. Write polynomial powers in descending order and align like powers.
4. Estimate the expected degree before multiplying as an error check.

Worked example 1: Combine exponent operations

Problem. Simplify \(\frac{x^4\cdot x^3}{x^2}\) for \(x\neq0\).

Solution.

1. Multiply same bases by adding exponents: \(x^4x^3=x^7\).
2. Divide same bases by subtracting exponents: \(x^{7-2}=x^5\).
3. The restriction \(x\neq0\) comes from the original denominator.

Answer. \(x^5\)

Worked example 2: Multiply monomials

Problem. Simplify \((3a^2b)(-2ab^3)\).

Solution.

1. Multiply coefficients: \(3(-2)=-6\).
2. Add a-exponents: \(a^{2+1}=a^3\).
3. Add b-exponents: \(b^{1+3}=b^4\).

Answer. \(-6a^3b^4\)

Worked example 3: Add polynomials

Problem. Add \((2x^2-3x+4)+(5x^2+x-9)\).

Solution.

1. Combine squared terms: \(2x^2+5x^2=7x^2\).
2. Combine linear terms: \(-3x+x=-2x\).
3. Combine constants: \(4-9=-5\).

Answer. \(7x^2-2x-5\)

Independent self-check

Mastery habit. Say what the answer means, include units when relevant, and verify it in the original condition.