Question

1. divide $4x^3 + 2x^2 + 3x + 4$ by $x + 4$ using long or synthetic division.

a. $4x^2 - 14x + 59$
b. $4x^2 + 18x - 53$, r 240
c. $4x^2 - 14x + 59$, r -232
d. $4x^2 + 18x - 53$
what are the zeros of the function? what are their multiplicities?

1. $f(x) = 4x^3 - 12x^2 - 16x$

a. the numbers 1, -4, and 0 are zeros of multiplicity 2
b. the numbers -1, 4, and 0 are zeros of multiplicity 2
c. the numbers -1, 4, and 0 are zeros of multiplicity 1
d. the numbers 1, -4, and 0 are zeros of multiplicity 1

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

a. the numbers -1 and -3 are zeros of multiplicity 2; the number 0 is a zero of multiplicity 1
b. the number 0 is a zero of multiplicity 2; the numbers 1 and 3 are zeros of multiplicity 1
c. the numbers 0 and 1 are zeros of multiplicity 2; the number 3 is a zero of multiplicity 1
d. the number 0 is a zero of multiplicity 2; the numbers -1 and -3 are zeros of multiplicity 1
what are the vertical asymptotes for the graph of the rational function?

1. $y = \frac{x + 8}{x^2 - 9x + 14}$

a. $x = 2, x = -7$
b. $x = 2, x = 7$
c. $x = -8$
d. $x = 2, x = 7$

1. find the horizontal asymptote of the graph of $y = \frac{3x^6 - 7x + 9}{7x^2 + 7x + 9}$.

a. $y = 3$
b. $y = \frac{3}{7}$
c. $y = 0$
d. no horizontal asymptote

1. what is an equivalent expression of $\frac{2x^2 + 26x + 80}{x^2 - 2x - 35}$? specify the domain.

a. $\frac{2(x + 8)}{x - 7}; x \
eq -8, x \
eq 7$
b. $\frac{x + 8}{x - 7}; x \
eq -8, x \
eq 7$
c. $\frac{2(x + 8)}{x - 7}; x \
eq -5, x \
eq 7$
d. $\frac{x + 8}{x - 7}; x \
eq 5, x \
eq 7$

Explanation:

Question 51 (Polynomial Division)

Step 1: Set up synthetic division

For dividing \(4x^3 + 2x^2 + 3x + 4\) by \(x + 4\), use root \(r=-4\) for synthetic division. Coefficients: \(4, 2, 3, 4\).

Step 2: Perform synthetic division

Bring down \(4\). Multiply by \(-4\): \(4\times(-4)=-16\). Add to next coefficient: \(2 + (-16)=-14\). Multiply \(-14\times(-4)=56\). Add to next coefficient: \(3 + 56 = 59\). Multiply \(59\times(-4)=-236\). Add to last coefficient: \(4 + (-236)=-232\).
So quotient is \(4x^2 - 14x + 59\), remainder \(-232\).

Step 1: Factor the function

\(f(x)=4x^3 - 12x^2 - 16x = 4x(x^2 - 3x - 4)\). Factor quadratic: \(x^2 - 3x - 4=(x - 4)(x + 1)\). So \(f(x)=4x(x - 4)(x + 1)\).

Step 2: Identify zeros and multiplicities

Zeros are \(x = 0\), \(x = 4\), \(x=-1\), each with multiplicity \(1\) (since each factor is to the first power).

Step 1: Factor the function

\(f(x)=x^4 - 4x^3 + 3x^2 = x^2(x^2 - 4x + 3)\). Factor quadratic: \(x^2 - 4x + 3=(x - 1)(x - 3)\). So \(f(x)=x^2(x - 1)(x - 3)\).

Step 2: Identify zeros and multiplicities

Zero \(x = 0\) has multiplicity \(2\) (from \(x^2\)), zeros \(x = 1\) and \(x = 3\) have multiplicity \(1\) (from linear factors).

Answer:

c. \(4x^2 - 14x + 59\), R \(-232\)

Question 52 (Zeros and Multiplicities)