CRM 3.2 - Synthetic and Long Division Review

Front

What is synthetic division?

Back

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

Front

What is long division in the context of polynomials?

Back

Long division for polynomials is a method used to divide a polynomial by another polynomial, similar to numerical long division. It involves dividing the leading terms, multiplying, and subtracting until the remainder is found.

Front

How do you determine if (x - c) is a factor of a polynomial f(x)?

Back

To determine if (x - c) is a factor of f(x), use the Factor Theorem: if f(c) = 0, then (x - c) is a factor of f(x).

Front

What is the Remainder Theorem?

Back

The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder of this division is f(c).

Front

If f(x) = x^3 - 4x + 1, what is f(2)?

Back

f(2) = 2^3 - 4(2) + 1 = 8 - 8 + 1 = 1.

Front

What is the result of dividing (x^3 - 3x^2 + 4) by (x - 1)?

Back

The result is x^2 - 2x + 2 with a remainder of 2.

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What does it mean if the remainder is zero when dividing a polynomial?

Back

If the remainder is zero, it means that the divisor is a factor of the polynomial.

Front

What is the first step in synthetic division?

Back

The first step in synthetic division is to write down the coefficients of the polynomial in descending order.

Front

How do you find the remainder when dividing by (x + 1)?

Back

To find the remainder when dividing by (x + 1), evaluate the polynomial at x = -1.

Front

What is the significance of the leading coefficient in polynomial division?

Back

The leading coefficient affects the degree of the quotient and the behavior of the polynomial as x approaches infinity.