maths.free › Precalculus › Polynomial division and the remainder theorem

Polynomial division and the remainder theorem

Long division, synthetic division, and why f(a) is the remainder on dividing by x − a.

Dividing polynomials works like long division of integers: divide the leading terms, multiply back, subtract, repeat. The remainder theorem says dividing f(x) by x − a leaves remainder f(a), so a is a root exactly when the division is clean. Picture it: the quotient's graph is the original with the root's factor removed. Think it: this is the Euclidean algorithm again, over polynomials instead of integers.

Àwọn ààtò àwọn ìṣàmúlò-ètò · simplify (x^3 - 1)/(x - 1)

Algebra · Simplifying

Simplify (x^3 - 1)/(x - 1)

\frac{x^{3} - 1}{x - 1}

Àwọn ìṣàmúlò-ètò

\frac{x^{3} - 1}{x - 1}
Start from the expression.
x^{2} + x + 1
Factor numerator and denominator and cancel common factors.

Fi àwọn àgbèwọlé hàn

x^{2} + x + 1

Nígbà yìí Yan àwọn àkóónú kan, tàbí tẹ̀ àtí kọ́ àwọn àkóónú rẹ̀. Àwọn ìṣàmúlò-ètò gbogbo, àwòrán, àwọn à

Fi iṣẹ́ rẹ̀ pamọ́

A free account adds notes on every lesson, a record of what you have finished, your solved problems in one place, and a tutor you can ask about this page. The maths itself is open to everyone, signed in or not.

Fikún Àwọn Ìṣàmúlò-ètò

Àwọn àmì-ìwé tí a lo níbẹ̀

Tẹ àwọn àmì-ìwé fún ìṣàfarawé fún àwọn ìṣàmúlò-ètò, àwòrán, àti àwọn àmì-ìwé àwọn àmì-ìwé.

Bii o ṣe le: Polynomial division and the remainder theorem

Start from the expression.
Factor numerator and denominator and cancel common factors.

Àwọn Àtòjọ-ẹ̀yàn

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

Diẹ̀ nínú Precalculus

Complex numbers Polynomial functions Rational functions Sequences and series The binomial theorem Conic sections Vectors Exponential and logarithmic functions Parametric equations and polar coordinates