Question

identify the zeros of the graphed function.

a) \\(x = -1, 2, 4\\)
b) \\(x = -4, -2, 0, 1\\)
c) \\(x = -4, -2, 1\\)
d) \\(x = -1, 0, 2, 4\\)

question 4 (5 points)
use synthetic division to find the remainder if polynomial \\(p(x) = x^3 - 28x - 48\\) is divided by \\(x + 4\\).

a) 1
b) 3
c) 0
d) 2

Explanation:

Identify x-intercepts from the graph

To find the zeros of the graphed function, we look for the \(x\)-coordinates where the graph intersects or touches the \(x\)-axis (where \(y = 0\)). Looking closely at the grid:

• The curve crosses the \(x\)-axis on the left at \(x = -4\).
• The curve crosses the \(x\)-axis in the middle at \(x = -2\).
• The curve crosses the \(x\)-axis on the right at \(x = 1\).

Thus, the zeros of the graphed function are \(x = -4, -2, 1\). This matches option C.

Set up synthetic division

For the second question, we want to divide the polynomial \(P(x) = x^3 - 28x - 48\) by \(x + 4\).

• The divisor is \(x + 4\), so the test value for synthetic division is \(c = -4\).
• Write down the coefficients of \(P(x)\). Note that there is no \(x^2\) term, so its coefficient is \(0\).
• The coefficients are: \(1\) (for \(x^3\)), \(0\) (for \(x^2\)), \(-28\) (for \(x\)), and \(-48\) (constant).

Perform synthetic division calculation

We set up the synthetic division table:

-4 |   1    0   -28   -48
   |       -4    16    48
   ----------------------
       1   -4   -12     0

Let's calculate each column:

1. Bring down the first coefficient: \(1\).
2. Multiply \(1\) by \(-4\) to get \(-4\). Add to \(0\): \(0 + (-4) = -4\).
3. Multiply \(-4\) by \(-4\) to get \(16\). Add to \(-28\): \(-28 + 16 = -12\).
4. Multiply \(-12\) by \(-4\) to get \(48\). Add to \(-48\): \(-48 + 48 = 0\).

Determine the remainder

The last value in the bottom row of our synthetic division is the remainder.

• The remainder is \(0\).
• This corresponds to option C.

Answer:

Question 1

• A) \(x = -1, 2, 4\)
• B) \(x = -4, -2, 0, 1\)
• C) \(x = -4, -2, 1\) (Correct answer)
• D) \(x = -1, 0, 2, 4\)

Question 2

• A) 1
• B) 3
• C) 0 (Correct answer)
• D) 2