Question

1. which system of inequalities is graphed below?

graph of a system of inequalities

a. \\(\

$$\begin{cases} y > x^2 + 7x - 8 \\\\ y < x + 7 \\end{cases}$$

\\)
b. \\(\

$$\begin{cases} y < x^2 + 7x - 8 \\\\ y > x + 7 \\end{cases}$$

\\)
c. \\(\

$$\begin{cases} y > x^2 - 7x - 8 \\\\ y < -x + 7 \\end{cases}$$

\\)
d. \\(\

$$\begin{cases} y < x^2 - 7x - 8 \\\\ y > -x + 7 \\end{cases}$$

\\)

1. classify \\(-2x^4 - x^3 + 8x^2 + 12\\) by degree.

a. quartic
b. quintic
c. quadratic
d. cubic

1. classify \\(8x^4 + 7x^3 + 5x^2 + 8\\) by number of terms.

a. trinomial
b. binomial
c. polynomial of 5 terms
d. polynomial of 4 terms

Explanation:

Question 37:

Step1: Analyze the parabola (quadratic inequality)

The parabola is a dashed curve opening upwards (since the coefficient of \(x^2\) is positive). The shaded region is above the parabola, so the inequality for the parabola should be \(y > ax^2 + bx + c\). Let's find the equation of the parabola. The roots seem to be around \(x = -8\) and \(x = 1\)? Wait, no, looking at the options, let's check the quadratic terms. Option c and d have \(x^2 - 7x - 8\). Let's factor \(x^2 - 7x - 8=(x - 8)(x + 1)\), so roots at \(x = 8\) and \(x = -1\), which matches the graph (the parabola crosses x-axis at around -1 and 8). So the quadratic is \(y = x^2 - 7x - 8\). The shaded region is above the parabola, so \(y > x^2 - 7x - 8\)? Wait no, wait the parabola is dashed, and the shaded area is inside the parabola? Wait no, the graph shows the shaded region is between the parabola and the line. Wait, no, the parabola is opening upwards, and the shaded region is above the parabola? Wait no, looking at the options, let's check the line. The line has a negative slope (since it's going down from left to right), so its equation should be \(y = -x + 7\) (since when x=0, y=7; when y=0, x=7? Wait no, the line in the graph seems to cross y-axis at 7 and x-axis at 7? Wait no, the line in the options: option c has \(y < -x + 7\), which has slope -1, y-intercept 7. Let's check the shaded region: the region is below the line (since the line is dashed and the shaded area is below it) and above the parabola? Wait no, the parabola is opening upwards, and the shaded region is inside the parabola (above the parabola's vertex). Wait, let's check the options:

Option c: \(

$$\begin{cases} y > x^2 - 7x - 8 \\ y < -x + 7 \end{cases}$$

\)

Option d: \(

$$\begin{cases} y < x^2 - 7x - 8 \\ y > -x + 7 \end{cases}$$

\)

Wait, the parabola is dashed, and the shaded region is above the parabola (since the parabola opens upwards, the inside would be above the vertex, but the inequality for the parabola: if the shaded region is above the parabola, then \(y > \) the parabola. The line: the shaded region is below the line (since the line is dashed and the shaded area is below it), so \(y < \) the line. The line has negative slope, so equation \(y = -x + 7\) (slope -1, y-intercept 7). So the line inequality is \(y < -x + 7\), and the parabola inequality: since the shaded region is above the parabola (inside the parabola, as the parabola opens upwards, the region above the vertex is inside), so \(y > x^2 - 7x - 8\) (since the parabola is \(y = x^2 - 7x - 8\), and the shaded area is above the parabola, i.e., \(y\) is greater than the parabola's \(y\) for a given \(x\)). Wait, but let's check the options: option c is \(y > x^2 - 7x - 8\) and \(y < -x + 7\), which matches: the shaded region is above the parabola (so \(y >\) parabola) and below the line (so \(y <\) line). The line has slope -1, so \(y = -x + 7\), so \(y < -x + 7\) (shaded below the line). The parabola: \(y = x^2 - 7x - 8\), shaded above it (so \(y > x^2 - 7x - 8\)). So option c.

Step2: Analyze the line

The line has a negative slope, so its equation is \(y = -x + 7\) (slope -1, y-intercept 7). The shaded region is below this line, so the inequality is \(y < -x + 7\).

Step3: Combine the two inequalities

The parabola is \(y = x^2 - 7x - 8\) (opening upwards, roots at x=-1 and x=8), and the shaded region is above the parabola (so \(y > x^2 - 7x - 8\)) and below the line \(y = -x + 7\) (so \(y < -x + 7\)). This matches option c.

Step1: Recall the degree of a polynomial

The degree of a polynomial is the highest power of \(x\) with a non-zero coefficient.

Step2: Identify the highest power in \(-2x^4 - x^3 + 8x^2 + 12\)

The terms are \(-2x^4\) (degree 4), \(-x^3\) (degree 3), \(8x^2\) (degree 2), and \(12\) (degree 0). The highest degree is 4.

Step3: Classify by degree

A polynomial with degree 4 is called a quartic polynomial.

Step1: Recall the number of terms in a polynomial

A term is a single monomial (a product of a coefficient and variables, or a constant).

Step2: Count the terms in \(8x^4 + 7x^3 + 5x^2 + 8\)

The terms are \(8x^4\), \(7x^3\), \(5x^2\), and \(8\). That's 4 terms.

Step3: Classify by number of terms

A polynomial with 4 terms is called a polynomial of 4 terms.

Answer:

c. \(

$$\begin{cases} y > x^2 - 7x - 8 \\ y < -x + 7 \end{cases}$$

\)

Question 38: