Polynomial operations
Polynomials are sums of terms built from numbers and whole-number powers of variables. Adding, subtracting, and multiplying them uses the properties of real numbers you already know.
Vocabulary
A monomial is a number, a variable, or a product of them, such as \(-5x^3\). A polynomial is a sum of monomials with whole-number exponents, like \(4x^3 - 2x^2 + x - 7\). The degree is the highest exponent, here 3. The leading coefficient is the coefficient of that highest power, here 4, and -7 is the constant term.
Naming polynomials
Polynomials with one, two, and three terms are monomials, binomials, and trinomials. By degree: 1 is linear, 2 quadratic, 3 cubic. Write polynomials in standard form, with exponents in decreasing order. Expressions such as \(\frac{1}{x}\) or \(\sqrt{x}\) are not polynomials, since their exponents are negative or fractional.
Adding and subtracting
Combine like terms. \((3x^2 + 2x - 5) + (x^2 - 4x + 1) = 4x^2 - 2x - 4\). To subtract, change the sign of every term in the second polynomial: \((3x^2 + 2x - 5) - (x^2 - 4x + 1) = 2x^2 + 6x - 6\).
Multiplying monomials and distributing
Multiply coefficients and add exponents: \((3x^2)(4x^5) = 12x^7\). To multiply a monomial by a polynomial, distribute: \(2x(x^2 - 3x + 4) = 2x^3 - 6x^2 + 8x\).
Multiplying binomials
Multiply every term of one by every term of the other. For binomials, FOIL (first, outer, inner, last) organises this: \((x + 3)(x - 5) = x^2 - 5x + 3x - 15 = x^2 - 2x - 15\). For longer polynomials, a grid or vertical layout keeps track of all the products.
Special products
Three patterns save time: \((a + b)^2 = a^2 + 2ab + b^2\), \((a - b)^2 = a^2 - 2ab + b^2\), and \((a + b)(a - b) = a^2 - b^2\). A common mistake is \((x + 4)^2 = x^2 + 16\); the correct expansion is \(x^2 + 8x + 16\).
Dividing polynomials
Divide a polynomial by a monomial term by term: \(\frac{6x^3 - 9x^2}{3x} = 2x^2 - 3x\). Dividing by a binomial uses long division, like numerical long division: \((x^2 + 5x + 6) \div (x + 2) = x + 3\). Synthetic division is a shortcut when dividing by \(x - c\).
Worked example
A rectangular garden is \(x + 5\) metres long and \(x + 2\) metres wide. Its area is \((x + 5)(x + 2) = x^2 + 7x + 10\) square metres. With x = 3, that is \(9 + 21 + 10 = 40\) square metres, matching \(8 \times 5\). Expanding gives a formula for every possible size.
Key takeaways
A polynomial is a sum of terms with whole-number exponents; its degree is the highest exponent and its leading coefficient multiplies that term. Add and subtract by combining like terms, changing every sign when subtracting. Multiply by distributing every term over every term, with FOIL for two binomials. Learn the special products: (a + b) squared, (a - b) squared, and (a + b)(a - b) = a squared - b squared; (x + 4) squared is not x squared + 16. Divide by monomials term by term and by binomials with long or synthetic division.
Flashcard
Number, variable, or their product
Check your understanding
Expand \((x + 4)^2\).