Algebra II

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Questions 1 - 10

Evaluate:

Explanation

Rewrite the problem as separate groups of binomials.

Use the FOIL method to expand the first two terms.

Simplify the right side.

Recall that since $i=\sqrt{-1}$ , the value of $i^2 = i\cdot i =-1$ .

Multiply this value with the third binomial.

Simplify the terms.

The answer is:

What is the center of the circular function ?

Explanation

Remember that the "shifts" involved with circular functions are sort of like those found in parabolas. When you shift a parabola left or right, you have to think "oppositely". A right shift requires you to subtract from the x-component, and a left one requires you to add. Hence, this circle has no horizontal shift, but does shift 6 upward for the vertical component.

You can also remember the general formula for a circle with center at and a radius of .

Comparing this to the given equation, we can determine the center point.

$h=0,\ k=6,\ r=5$

The center point is at (0,6) and the circle has a radius of 5.

Multiply the expressions:

$\small (x^{2} - 2x + 7) (x^{2} - 2x - 7)$

$\small x^{4}-4x^{3}+4x^2-49$

$\small \small x^{4}-4x^{3}+4x^2+49$

$\small \small \small x^{4}-4x^{3}-4x^2-49$

$\small \small \small \small \small x^{4}+4x^{3}-4x^2+49$

$\small \small \small \small x^{4}+4x^{3}-4x^2-49$

Explanation

You can look at this as the sum of two expressions multiplied by the difference of the same two expressions. Use the pattern

$\small (A+B)(A-B) = (A^{2}-B^{2})$ ,

where $\small A = x^{2} - 2x$ and $\small B = 7$ .

$\small \small \small (x^{2} - 2x + 7) (x^{2} - 2x - 7) = (x^{2} - 2x )^{2} - 7^{2} = (x^{2} - 2x )^{2} - 49$

To find $\small (x^{2} - 2x )^{2}$ , you use the formula for perfect squares:

$\small (A-B)^{2} = (A^{2}-2AB+B^{2})$ ,

where $\small \small A = x^{2}$ and $\small B = 2x$ .

$\small \small \small (x^{2}-2x)^{2} = ((x^{2})^{2}-2x^{2}(2x)+(2x)^{2}) = x^{4}-4x^{3}+4x^2$

Substituting above, the final answer is $\small x^{4}-4x^{3}+4x^2-49$ .

Solve the absolute value equation: $\left | 9x-9\right | = -1$

$\textup{No solution.}$

$\frac{8}{9},\frac{10}{9}$

$-\frac{8}{9},-\frac{10}{9}$

$-\frac{8}{9},\frac{10}{9}$

$\frac{8}{9},-\frac{10}{9}$

Explanation

Recall that the absolute value sign will convert any value to a positive sign. There will be no occurrences of that will evaluate into a negative one as a final solution.

There are no solutions for this equation.

The answer is: $\textup{No solution.}$

Solve: $2log_{10}(100^3)$

$\textup{Cannot be determined.}$

Explanation

Change the base of the inner term or log to base ten.

$2log_{10}(100^3)= 2log_{10}(10^{2(3)})= 2log_{10}(10^6)$

According to the log property:

$log_{b}(b^x) = x$

The log based ten and the ten to the power of will cancel, leaving just the power.

$2log_{10}(10^6) = 2(6) =12$

The answer is:

What are the coordinates of the center of a circle with the equation ?

Explanation

The equation of a circle is , in which (h, k) is the center of the circle. To derive the center of a circle from its equation, identify the constants immediately following x and y, and flip their signs. In the given equation, x is followed by -1 and y is followed by -6, so the coordinates of the center must be (1, 6).

varies directly with $x^{2}$ . If , what is if ?

Explanation

1. Since varies directly with $x^{2}$ :

$y=Kx^{2}$ with K being some constant.

2. Solve for K using the x and y values given:

$75=K(5^{2})$

3. Use the equation you solved for to find the value of y:

$y=3x^{2}$

$y=3(10^{2})$

Factor the following polynomial: .

Explanation

Because the term has a coefficient, you begin by multiplying the and the terms () together: $2\times-3=-6$ .

Find the factors of that when added together equal the second coefficient (the term) of the polynomial: .

There are four factors of : , and only two of those factors, , can be manipulated to equal when added together and manipulated to equal when multiplied together:

$1+-6=-5, 1\times-6=-6$

Determine the sum, if possible: $4-\frac{2}{3}+\frac{1}{9}-...$

$\frac{24}{7}$

$\textup{The sum cannot be determined.}$

$\frac{71}{21}$

$\frac{17}{5}$

$\frac{64}{19}$

Explanation

Determine the common ratio of the infinite series by dividing the second term with the first term and the third term with the second term. The common ratios should be similar.

$\frac{-\frac{2}{3}}{4} = -\frac{2}{3} \div 4= -\frac{2}{3}\times\frac{1}{4} = -\frac{1}{6}$