Polynomial long division is a method used to divide one polynomial by another polynomial. It involves repeatedly dividing the divisor into the dividend and subtracting the result to obtain the quotient and remainder.
congrats on reading the definition of Polynomial Long Division. now let's actually learn it.
Polynomial long division is a step-by-step process that allows you to divide a polynomial by another polynomial, even if the degree of the dividend is greater than the degree of the divisor.
The process involves repeatedly dividing the leading term of the dividend by the leading term of the divisor, multiplying the result by the divisor, and subtracting the result from the dividend.
Polynomial long division is a useful technique for simplifying rational expressions and factoring special products, such as the difference of two squares or the sum/difference of two cubes.
The remainder theorem states that if a polynomial $P(x)$ is divided by $(x - a)$, then the remainder is $P(a)$.
The factor theorem states that if $P(a) = 0$, then $(x - a)$ is a factor of $P(x)$.
Review Questions
Explain how polynomial long division can be used to simplify rational expressions.
Polynomial long division is a key technique for simplifying rational expressions. When the degree of the numerator is greater than or equal to the degree of the denominator, you can use long division to divide the numerator by the denominator. This will give you the quotient, which represents the simplified rational expression, and the remainder, which can be expressed as a fraction with the denominator being the original denominator. By using polynomial long division, you can rewrite the rational expression in a simpler form.
Describe how the remainder theorem and factor theorem relate to polynomial long division.
The remainder theorem states that if a polynomial $P(x)$ is divided by $(x - a)$, then the remainder is $P(a)$. This means that if you perform polynomial long division and the remainder is 0, then $(x - a)$ is a factor of $P(x)$. The factor theorem builds on this, stating that if $P(a) = 0$, then $(x - a)$ is a factor of $P(x)$. These theorems are closely tied to polynomial long division, as they provide a way to determine the factors of a polynomial by examining the remainder or the value of the polynomial at a specific point.
Analyze how polynomial long division can be used to factor special products, such as the difference of two squares or the sum/difference of two cubes.
Polynomial long division is a powerful tool for factoring special products, such as the difference of two squares ($a^2 - b^2$) or the sum/difference of two cubes ($a^3 \pm b^3$). By applying polynomial long division to these expressions, you can identify the factors that make up the original polynomial. For example, to factor $a^2 - b^2$, you can divide $(a + b)$ into $(a + b)(a - b)$, revealing the factors $(a + b)$ and $(a - b)$. Similarly, to factor $a^3 + b^3$, you can divide $(a + b)$ into $(a + b)(a^2 - ab + b^2)$, identifying the factors $(a + b)$ and $(a^2 - ab + b^2)$. Mastering polynomial long division is crucial for effectively factoring these types of special products.