Precalculus Midterm

Divide by using long division: (5x^2 - 13x + 8) ÷ (x - 3).

5x + 2 + 14/x - 3. To solve this equation, put (x-3) on the outside of the division symbol, and (5x^2 - 13x + 8) on the inside. After solving the equation using long division, you should receive 5x + 2 with a remainder of 14.

Find the product (5x - 3)(x^3 -5x +2).

5x^4 - 3x^3 - 25x^2 +25x - 6. To solve this equation, just FOIL.

Divide by using long division: (5x + 6x^3 - 8) ÷ (x - 2).

6x^2 + 12x + 29 + 50/(x-2). To solve this equation, put (x-2) on the outside of the division symbol, and (5x + 6x^3 - 8) on the inside. After solving the equation using long division, you should receive 6x^2 + 12x + 29 with a remainder of 50.

What do you do when you are trying to determine what quartic function the graph represents?

When trying to determine what quartic function a graph represents, determine the zeros on the graph, and then choose the factors.

Solve the equation x^2 = 15 - 2x.

x = 3 or x = -5. To solve this equation, you must move 15 - 2x to the left side → x^2 + 2x - 15. Factor → (x+5)(x-3) And solve → x = -5 or x = 3.

Find the roots of the equation 28x - 98 = 2x^2.

x = 7. To solve this equation, you must first move 2x^2 to the other side of the equation → -2x^2 + 28x - 98. You can then factor this equation, and take a -2 out → -2(x^2 - 14x + 49). Factor → -2(x-7)(x-7). And solve → x = 7.

Solve the equation 3x^2 + 108 = 0.

x = ±6i. To solve this equation, you can use two methods. First, the quadratic formula: x=0±√0^2−4(3)(108)/2(3). Solve → 0±√-1296/6 = ±36i/6 = ±6. Second method: 3x^2/3 -108/3 = √x^2 = √-36. x = ±6i.

Use Pascal's Triangle to expand the expression (2x - 2)^4.

16x^4 - 64x^3 + 96x^2 - 64x + 16. To solve this equation, first expand the expression into 4 parts → (2x-2)(2x-2)(2x-2)(2x-2) Solve (2x-2)(2x-2) → 4x^2 - 4x - 4x + 4 = 4x^2 - 8x + 4. You can then split ( 4x^2 - 8x + 4) into 2 parts → ( 4x^2 - 8x + 4)( 4x^2 - 8x + 4). And solve → 16x^4 - 64x^3 + 96x^2 - 64x + 16.

Express 2√-81 in terms of i.

18i. To solve this equation, you must first take out the perfect square 9, remove the negative, and add an i → 2 x 9i. And solve → 18i.

Determine whether the binomial (x - 2) is a factor of the polynomial P(x) = x^3 +3x^2 - 4x - 9.

(x - 2) is not a factor of the polynomial P(x) = x^3 + 3x^2 - 4x - 9. To solve this equation, you have to divide (x^3 +3x^2) by (x - 2). After solving the equation using long division, you should receive x^2 + 5x + 6 with a remainder of 3.

Identify all of the real roots of 4x^4 + 31x^3 - 4x^2 - 89x + 22 = 0.

-2, 1/4, -3 + 2√5, -3 - 2√5. To solve this equation, you must first establish the degree of the polynomial. Since the degree is 4, there can only be 4 roots. To find these roots, you have to divide (4x^4 + 31x^3 - 4x^2 - 89x + 22) by (x + 2) and solve → 4x^3 + 23x^2 - 50x + 11. Since (4x^3 + 23x^2 - 50x + 11) is not factorable yet, you must divide it by x - 1/4 and solve → 4x^2 + 24x - 44. Using the quadratic formula, solve the equation 4x^2 + 24x - 44 → x=−24±√24^2−4(4)(-44)/2(4) = -24±√1280/8 = -24±16√5/8 = -3±2√5.

Rewrite the polynomial -6x^2 - 5x + 5x^4 - 10x^3 - 4x^5 + 10 in standard form. Then, identify the leading coefficient, degree, and number of terms. Name the polynomial.

-4x^5 + 5x^4 - 10x^3 - 6x^2 - 5x + 10. Leading coefficient: -4; degree: 5; number of terms: 6; name: quintic polynomial.

Find the product 2a^4b^2(5a^3b^3 - 3b^3).

10a^7b^5(5a^3b^3 - 3b^3). To solve this equation, just distribute.

Write the simplest polynomial function with zeros 2, 3, and -1/2.

P(x) = x^3 - 9/2x^2 + 7/2x + 3. To solve this equation, establish an equation with all of the factors → (x - 2)(x - 3)(x + 1/2). FOIl (x - 2)(x -3) → x^2 - 3x - 2x + 6 = x^2 - 5x + 6. Rewrite the new equation → (x^2 - 5x + 6)(x + 1/2). FOIL the entire equation → x^3 + 1/2x^2 - 5x^2 - 5/2x + 6x + 3. Simplify and establish the new function → x^3 - 9/2x^2 + 7/2 + 3.

Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. Graph: Down/down.

The degree is even, and the leading coefficient is negative.

Identify the leading coefficient, degree, and end behavior of the function P(x) = -3x^5 + 6x^3 + 6.

The leading coefficient is -3. The degree is 5. As x → -∞, P(x) → +∞ and as x → +∞, P(x) → -∞.

Find the minimum or maximum value of f(x) = x^2 +4x + 3.

The minimum value is -1. To solve equations like this, you have to establish if the leading coefficient is positive or negative. If the coefficient is positive, there is a minimum value. If the coefficient is negative, there is a maximum value. To solve this specific problem, you have to first find the axis of symmetry using the equation -b/2a. Therefore, -4/2(1) = -2. Then, you have to plug the axis of symmetry answer back into the equation. So, f(-2) = (-2)^2 + 4(-2) + 3 = 4 - 8 + 3 = - 4 + 3 = -1.

Solve the polynomial equation 4x^5 - 28x^4 + 40x^3 = 0 by factoring.

The roots are 0, 5, and 2. To solve this equation, you first must take 4x^3 out of the equation → 4x^3(x^2 - 7x + 10). Factor (x^2 - 7x + 10) → 4x^3(x - 5)(x - 2) And solve → x = 0, x = 5, x = 2.

Computer graphics programs often employ a method called cubic splines regression to smooth hand-drawn curves. This method involves splitting a hand-drawn curve into regions that can be modeled by cubic polynomials. A region of a hand-drawn curve is modeled by the function f(x) = -x^3 +3x^2 - 4. Use the graph of f(x) = -x^3 +3x^2 - 4 to identify the values of x for which f(x) = 0 and to factor f(x).

x = -1; x = 2; f(x) = -(x + 1)(x - 2)^2 To solve this equation, you simply look at the graph. The zeros of the graph are x = -1 or x = 2. Since the graph bounces at the value 2, there is a multiplicity of 2 for the factor (x - 2). Since the graph is going up/ down, the leading coefficient is negative.

Find the zeros of the function h(x) = x^2 - 19x + 60.

x = 15 or x = 4. To solve this equation, use the quadratic formula (x=−b±√b^2−4ac/2a). Plug in the values, x = -(-19)±√(19)^2−4(1)(60)/2(1). And solve, 19±11/2 → 19-11/2 = 8/2 = 4 → 19+11/2 = 30/2 = 15 Solutions: 15;4.