10C

Polynomials and division

Polynomial language (degree, leading coefficient, constant term), evaluating P(a), adding and multiplying polynomials, and long division by a linear factor.

Language

For $P(x) = 2x^3 - 5x^2 + x - 7$: the degree is $3$, the leading coefficient is $2$ and the constant term is $-7$.

Division

Example 1●●●

Divide $P(x) = x^3 - 2x^2 + x - 5$ by $x - 2$.

$x^3 \div x = x^2$; subtract $x^2(x - 2) = x^3 - 2x^2$, leaving $x - 5$.

$x \div x = 1$; subtract $1(x - 2)$, leaving $-3$.

$Q(x) = x^2 + 1$, remainder $-3$.

Flashcards

Degree

Highest power of $x$

Leading coefficient

Coefficient of the highest power

Division statement

$P(x) = (x - a)Q(x) + R$

Practice

Remainder when $x^3 - 2x^2 + x - 5$ is divided by $x - 2$?
Quotient for the same division?
Degree of $2x^3 - 5x^2 + x - 7$?
Leading coefficient of $2x^3 - 5x^2 + x - 7$?
$P(x) = x^3 - 2x^2 + x - 5$. Find $P(2)$.
Constant term of $2x^3 - 5x^2 + x - 7$?
Divide $x^3 + 4x^2 + 5x + 2$ by $x + 1$. Quotient?
Degree of $(x^2 + 1)(x^3 - x)$?