Dividing Polynomials

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, $$2x^2 + 3x + 5$$ is a polynomial.

Dividing polynomials involves finding how many times one polynomial (the divisor) can fit into another polynomial (the dividend), resulting in a quotient and possibly a remainder.

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in $$3x^4 + 2x^3 + x$$ , the degree is 4.

Synthetic division is a simplified method of dividing a polynomial by a linear divisor of the form $$x - c$$ . It is faster than long division and is used when the divisor is a first-degree polynomial.

The Remainder Theorem states that if a polynomial $$f(x)$$ is divided by $$x - c$$ , the remainder of that division is equal to $$f(c)$$ .

The Factor Theorem states that a polynomial $$f(x)$$ has a factor $$x - c$$ if and only if $$f(c) = 0$$ .

To factor $$a^2 + 15a + 44$$ , look for two numbers that multiply to 44 and add to 15. The factors are 11 and 4, so it can be factored as $$(a + 11)(a + 4)$$ .

Using synthetic division or polynomial long division, the result is $$a + 11$$ with a remainder of 0.

Long division of polynomials is a method similar to long division with numbers, where you divide the leading term of the dividend by the leading term of the divisor, multiply, and subtract.

A zero of a polynomial is a value of $$x$$ for which the polynomial evaluates to zero. For example, if $$f(x) = x^2 - 4$$ , then the zeros are $$x = 2$$ and $$x = -2$$ .