Remainder and Factor Theorems.

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Remainder and Factor Theorems

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Presentation transcript:

Remainder and Factor Theorems

Some basics A polynomial of degree zero is a constant A polynomial of degree 1 is called linear e.g. A polynomial of degree 2 is called quadratic Degree 3 – cubic, 4 – quartic, 5 – quintic etc. “Degree” is also sometimes called “order” descending order ascending order unordered

Identities and Equations An equation tells you that something is true in a particular situation, not always! Now look at this equation: Always true Certainly! The LHS and RHS are identically equal This is called an “identity”

Equating coefficients This may be obvious, but it’s very useful! Example: Identical polynomials have identical coefficients. Note: an identity is an equality relation. See page 83 text book.

Long Division of Polynomials

Long Division of Polynomials Divide 6x2 – 26x + 12 by x – 4. The dividend is 6x2 – 26x + 12 and the divisor is x – 4. We begin by arranging them as follows:

Long Division of Polynomials The last line then contains the remainder. The top line contains the quotient.

Long Division of Polynomials The result of the division can be interpreted as.

Long Division of Polynomials

Long Division of Polynomials We summarize the long division process in the following theorem.

Long Division of Polynomials

Long Division of Polynomials

Long Division of Polynomials

Long Division of Polynomials

Factor Theorem

Factor Theorem

Factor Theorem

Synthetic Division

Synthetic division is a quick method of dividing polynomials. It can be used when the divisor is of the form x – c. In synthetic division, we write only the essential parts of the long division.

Long Division vs. Synthetic Division Compare the following long and synthetic divisions, in which we divide 2x3 – 7x2 + 5 by x – 3.

Long Division vs. Synthetic Division

Thus, 2x3 – 7x2 + 5 = (x – 3)(2x2 – x – 3) – 4. Synthetic Division From the last line, we see that the quotient is 2x2 – x – 3 and the remainder is –4. Thus, 2x3 – 7x2 + 5 = (x – 3)(2x2 – x – 3) – 4.