Draw a model to represent the polynomial x^2+2x+4
1. A large square is shown. To the right of the large square is a rectangle whose height is the same as the large square but whose width is much more narrow. To the right of the rectangle are four congruent small squares. The squares are arranged in two columns. Each column has two squares with the top square slightly above the bottom square.
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3. Two large congruent squares are shown in a row. To the right of the large squares are four congruent small squares. The small squares are arranged in two columns. Each column has two squares with the top square slightly above the bottom square.
4. none of these
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x^2+2x+4
This polynomial can be represented by a quadratic function that forms a U-shape when graphed. The vertex of this U-shaped curve is located at the point (-1, 3), and the curve opens upwards.
Graphically, the model can be represented as follows:
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Alternatively, using algebra, we can expand the polynomial as follows:
x^2 + 2x + 4 = (x + 1)^2 + 3
This form of the polynomial shows that it is the sum of a perfect square (x+1)^2 and a constant term 3. We can use this form to graph the polynomial as a shifted parabola, with the vertex at (-1, 3), as shown below:
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Draw a model to represent the polynomial x2 + 2x + 4. (1 point)
model aA large square is shown. To the right of the large square is a rectangle whose height is the same as the large square but whose width is much more narrow. To the right of the rectangle are four congruent small squares. The squares are arranged in two columns. Each column has two squares with the top square slightly above the bottom square.
model b
model cTwo large congruent squares are shown in a row. To the right of the large squares are four congruent small squares. The small squares are arranged in two columns. Each column has two squares with the top square slightly above the bottom square.
none of these
The correct model is model b.
Simplify the polynomial.
–3f2 + 4f – 3 + 8f2 + 7f + 1
(1 point)
5f2 – 11f + 2
11f2 + 11f + 2
5f2 + 11f – 2
–5f2 + 11f – 2
Combine like terms to simplify the polynomial:
-3f^2 + 4f - 3 + 8f^2 + 7f + 1 = 5f^2 + 11f - 2
Therefore, the simplified polynomial is 5f^2 + 11f - 2.
The correct answer is: 5f2 + 11f – 2.
Add or subtract.
(2x2 + 6x + 1) + (–7x2 + 2x – 3) (1 point)
5x2 – 4x – 2
–5x2 + 8x – 2
5x2 – 8x + 2
–9x2 – 8x + 2
Add the like terms to simplify the expression:
(2x^2 + 6x + 1) + (-7x^2 + 2x - 3) = -5x^2 + 8x - 2
Therefore, the simplified expression is -5x^2 + 8x - 2.
The correct answer is: -5x2 + 8x – 2.
(3x2 – 7x – 4) – (6x2 – 6x + 1) (1 point)
–3x2 – x –5
–3x2 –13x + 5
9x2 – x + 5
3x2 – 13x – 5
Distribute the negative sign in the second set of parentheses and then combine like terms to simplify the expression:
(3x^2 - 7x - 4) - (6x^2 - 6x + 1) = 3x^2 - 7x - 4 - 6x^2 + 6x - 1
= -3x^2 - x - 5
Therefore, the simplified expression is -3x^2 - x - 5.
The correct answer is: –3x2 – x –5.
Write the perimeter of the figure as a polynomial. Simplify.
A trapezoid is shown. The top side is labeled 2 x. The right side is labeled 4 x plus 1. The bottom side is labeled 3 x. The left side is labeled 4 x plus 1. (1 point)
13x + 2
13x + 1
9x + 1
9x + 2
The perimeter of the trapezoid is the sum of the lengths of all four sides, so we can write it as follows:
Perimeter = 2x + 4x + 1 + 3x + 4x + 1
Perimeter = 13x + 2
Therefore, the perimeter of the trapezoid as a polynomial is 13x + 2.
The correct answer is: 13x + 2.
The area of room A is (5x2 – 7x – 6) ft2. Room B has an area of (8x2 + 6x – 1) ft2. If room A is the larger room, how much greater is the area of room A than the area of room B? (1 point)
–3x2 – x – 7
13x2 – 13x – 7
–3x2 – 13x – 5
3x2 – x – 5
To find the difference in area between Room A and Room B, we need to subtract the area of Room B from the area of Room A:
Area of Room A - Area of Room B
= (5x^2 - 7x - 6) - (8x^2 + 6x - 1)
= 5x^2 - 7x - 6 - 8x^2 - 6x + 1
= -3x^2 - 13x - 5
Therefore, the difference in area between Room A and Room B is -3x^2 - 13x - 5.
The correct answer is: –3x2 – 13x – 5.
In the expression –7x – 5x2 + 5, what is the coefficient of x? (1 point)
7
5
–5
–7
The coefficient of x is -7.
The correct answer is: –7
Write the expression using a single exponent.
22 • 28
(1 point)
410
210
416
216
We can simplify the expression using the exponent rule that states "when two exponential expressions with the same base are multiplied, we add their exponents." Therefore:
22 · 28 = 2^(2) · 2^(8) = 2^(2 + 8) = 2^(10)
So, the expression can be written as 2^(10).
Therefore, the correct answer is: 2^(10).
Write the expression using scientific notation.
(2.5 • 104)(4 • 103) (1 point)
10 • 107
10 • 108
1 • 108
10 • 1012
To write the expression in scientific notation, we need to multiply the two numbers and then adjust the decimal point so that there is one digit to the left of the decimal:
(2.5 × 10^4) × (4 × 10^3) = (2.5 × 4) × 10^(4 + 3) = 10 × 10^7 = 1 × 10^8
Therefore, the expression in scientific notation is 1 × 10^8.
The correct answer is: 1 • 108.