PRE CALCULUS Pre-Calculus Polynomial Functions 3

Question 1

Using synthetic division, what is the quotient when x³ + 2x² - 5x - 6 is divided by (x - 2)?

Accepted Answer

x² + 4x + 3

Answer

Synthetic division with c = 2 yields coefficients 1, 4, 3, remainder 0, giving x² + 4x + 3.

Question 2

According to the Rational Zero Theorem, which is a possible rational zero of f(x) = 2x³ - 3x + 6?

Accepted Answer

±3/2

Answer

Possible rational zeros are factors of the constant (6) over factors of the leading coefficient (2), giving ±3/2 among others.

Question 3

How many positive real zeros can f(x) = x⁴ - x³ + x² - x + 1 have according to Descartes' Rule?

Accepted Answer

0 or 2 or 4

Answer

There are 4 sign changes, so Descartes' Rule gives 4, 2, or 0 positive real zeros.

Question 4

If f(x) has degree 6 with real coefficients and 2 + 3i is a zero, what other zero must exist?

Accepted Answer

2 - 3i

Answer

Complex zeros of polynomials with real coefficients come in conjugate pairs, so 2 - 3i must also be a zero.

Question 5

What is the remainder when f(x) = x⁴ - 3x² + 2 is divided by (x + 1)?

Accepted Answer

0

Answer

By the Remainder Theorem, f(-1) = 1 - 3 + 2 = 0.

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