Question

1. use synthetic division to evaluate the polynomial $y=-11 + x^2 + x^3 - 9x$ at $x=3$ then mark the answer as a coordinate on the graph.

1. you use synthetic division to evaluate the polynomial f(x) at $x=-4$ and find that you get a remainder of 9. which of the following are true? select all that apply.

a) the (-4, 9) is a zero of the polynomial.
b) $f(-4)=9$
c) the coordinate (-4, 9) lies on the graph of the polynomial.
d) the polynomial has a factor of $x-4$.

find all zeros using synthetic division.

1. $f(x)=3x^3 - 16x^2 + 16x$
2. $f(x)=4x^3 + 16x^2 - x - 4$ given that $x+4$ is a factor.

Explanation:

Question 11

Step1: Rearrange polynomial by degree

$y = x^3 + x^2 - 9x - 11$

Step2: Set up synthetic division for $x=3$

$$\begin{array}{r|rrrr} 3 & 1 & 1 & -9 & -11 \\ & & 3 & 12 & 9 \\ \hline & 1 & 4 & 3 & -2 \end{array}$$

Step3: Identify remainder as $y$-value

The remainder is $-2$, so $y=-2$ when $x=3$.

• A zero of a polynomial means $f(x)=0$, but the remainder is 9, so A is false.
• The Remainder Theorem states the remainder equals $f(-4)$, so $f(-4)=9$, making B true.
• If $f(-4)=9$, the point $(-4,9)$ lies on the polynomial's graph, so C is true.
• A factor $x-a$ requires remainder 0; here remainder is 9, so D is false.

Step1: Factor out common $x$ first

$f(x) = x(3x^2 - 16x + 16)$

Step2: Use synthetic division on quadratic for $x=4$

$$\begin{array}{r|rrr} 4 & 3 & -16 & 16 \\ & & 12 & -16 \\ \hline & 3 & -4 & 0 \end{array}$$

Step3: Factor resulting linear term

$3x^2 -16x +16=(x-4)(3x-4)$

Step4: Set all factors to zero

$x=0$, $x-4=0$, $3x-4=0$

Answer:

The coordinate is $(3, -2)$

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Question 12