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Algebra II : Intermediate Single-Variable Algebra

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Example Questions

Example Question #6 : How To Add Polynomials

Add:

$\frac{x}{x^{2}-2x-3} + \frac{1}{x^{2}-2x -3}$

Possible Answers:

$\frac{1}{x-3}$

$\frac{x-3}{x+1}$

$\frac{1}{x+1}$

$\frac{x+1}{x^{2}-2x-3}$

Correct answer:

$\frac{1}{x-3}$

Explanation:

First factor the denominators which gives us the following:

$\frac{x}{\left ( x+1 \right )\left ( x-3 \right )} + \frac{1}{\left ( x+1 \right )\left ( x-3 \right )}$

The two rational fractions have a common denominator hence they are like "like fractions". Hence we get:

$\frac{x+1}{\left ( x+1 \right )\left ( x-3 \right )}$

Simplifying gives us

$\frac{1}{x-3}$

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Example Question #571 : Intermediate Single Variable Algebra

Subtract:

$\frac{2x}{x-1} - \frac{3}{x^{2}-1}$

Possible Answers:

$\frac{2x^{2}-3}{\left ( x+1 \right )\left ( x-1 \right )}$

$\frac{2x-3}{x-1}$

$\frac{2}{x-1}$

$\frac{2x^{2}+2x-3}{\left ( x+1 \right )\left ( x-1 \right )}$

$\frac{2x-3}{x+1}$

Correct answer:

$\frac{2x^{2}+2x-3}{\left ( x+1 \right )\left ( x-1 \right )}$

Explanation:

First let us find a common denominator as follows:

$\frac{2x\left ( x+1 \right )}{\left ( x+1 \right )\left ( x-1 \right )} - \frac{3}{\left ( x+1 \right )\left ( x-1 \right )}$

Now we can subtract the numerators which gives us : $2x^{2}+2x-3$

So the final answer is $\frac{2x^{2}+2x-3}{\left ( x+1 \right )\left ( x-1 \right )}$

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Example Question #4 : How To Find The Solution To A Rational Equation With Lcd

Solve the rational equation:

$\small \frac{x+2}{2}-\frac{3}{2x-4}=3$

Possible Answers:

$\small x=-2$ or $\small x=2$

$\small x=\frac{1}{5}$

$\small x=1$ or $\small x=5$

$\small x=0$

no solution

Correct answer:

$\small x=1$ or $\small x=5$

Explanation:

With rational equations we must first note the domain, which is all real numbers except $\small \tiny x=2$ . (Recall, the denominator cannot equal zero. Thus, to find the domain set each denominator equal to zero and solve for what the variable cannot be.)

The least common denominator or $\small 2$ and $\small 2x-4$ is $\small 2(x-2)$ . Multiply every term by the LCD to cancel out the denominators. The equation reduces to $\small \small (x-2)(x+2)-3=6(x-2)$ . We can FOIL to expand the equation to $\small \small x^2 -4-3=6x-12$ . Combine like terms and solve: $\small x^2-6x+5=0$ . Factor the quadratic and set each factor equal to zero to obtain the solution, which is $\small x=5$ or $\small x=1$ . These answers are valid because they are in the domain.

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Example Question #11 : Solving Rational Expressions

Simplify: 3x-4xy-xy(6x+3)

Possible Answers:

3x+7xy-6x^2y

3x-7xy-6x^2y

3x-12xy-30x^2y

3x-xy-6x^2y

3x+xy-6x^2y

Correct answer:

3x-7xy-6x^2y

Explanation:

First, simplify the expression before attempting to combine like-terms.

3x-4xy-xy(6x+3)

3x-4xy-6x^2y-3xy

Combine like-terms.

3x-7xy-6x^2y

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Example Question #11 : Adding And Subtracting Rational Expressions

Simplify. 2x^2y+2xy-2xy^2+2x^2y^2

Possible Answers:

4x^2y^2

4x^2y^2-2xy

The question is simplified as is.

4x^2y+4xy-4xy^2+4x^2y^2

Correct answer:

The question is simplified as is.

Explanation:

The values can only be added or subtracted if there are like-terms in the expression. Since there are no like-terms in the question, the question is already simplified as is. All the other answers given are incorrect.

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Example Question #571 : Intermediate Single Variable Algebra

Simplify the rational expression: -(-2x+3)^2

Possible Answers:

-4x^2-9

-4x^2-12x+9

-4x^2-12x-9

-4x^2+12x-9

4x^2-12x+9

Correct answer:

-4x^2+12x-9

Explanation:

There are multiple operations required in this problem. The exponent must be eliminated before distributing the negative sign. Use the FOIL method which means to mulitply the first terms together, then multiply the outer terms together, then multiply the inner terms togethers, and lastly, mulitply the last terms together.

-(-2x+3)^2

= -[(-2x+3)(-2x+3)]

=-[(-2x)(-2x)+(-2x)(3)+(3)(-2x)+(3)(3)]

=-[4x^2-6x-6x+9]

=-[4x^2-12x+9]

The negative sign can now be distributed.

=-4x^2+12x-9

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Example Question #12 : Adding And Subtracting Rational Expressions

$\frac{2x-1}{xy}+\frac{4y+3}{y^2}$

Possible Answers:

$\frac{3xy+3x-y}{xy^2}$

$\frac{6xy+3x-y}{xy}$

$\frac{6xy+3x-y}{xy^2}$

$\frac{3}{xy^2}$

$\frac{6xy+3x-y}{y^2}$

Correct answer:

$\frac{6xy+3x-y}{xy^2}$

Explanation:

First, find the common denominator, which is xy^2 . Then, make sure to offset each numerator. Multiply $\frac{2x-1}{xy}$ by y to get $\frac{(2x-1)y}{xy^2}$ . Multiply $\frac{4y+3}{y^2}$ by x to get $\frac{(4y+3)x}{xy^2}$ . Then, combine numerators to get (2x-1)(y)+(4y+3)(x)=2xy-y+4xy+3x=6x+3x-y . Then, put the numerator over the denominator to get your answer: $\frac{6xy+3x-y}{xy^2}$ .

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Example Question #11 : Adding And Subtracting Rational Expressions

Subtract the expressions: $\frac{8x}{x-3}-\frac{2}{x-7}$

Possible Answers:

$\frac{8x^2+58x+6}{(x-3)(x-7)}$

The answer is not present.

$\frac{8x^2-58x-6}{(x-3)(x-7)}$

$\frac{8x^2+58x-6}{(x-3)(x-7)}$

$\frac{8x^2-58x+6}{(x-3)(x-7)}$

Correct answer:

$\frac{8x^2-58x+6}{(x-3)(x-7)}$

Explanation:

$\frac{8x}{x-3}-\frac{2}{x-7}$

Make a common denominator:

$\frac{x-7}{x-7}*\frac{8x}{x-3}-\frac{2}{x-7}*\frac{x-3}{x-3}$

$\frac{(x-7)8x}{x-3(x-7)}-\frac{2(x-3)}{x-7(x-3)}$

Combine and subtract numerators:

$\frac{(x-7)8x-2(x-3)}{(x-7)(x-3)}$

$\frac{8x^2-58x+6}{(x-3)(x-7)}$

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Example Question #12 : Adding And Subtracting Rational Expressions

$\frac{4}{x-1}+\frac{3}{x+1}$

Possible Answers:

$\frac{7x-1}{x^2-1}$

$\frac{7x+1}{x-1}$

$\frac{7x+1}{x^2-1}$

$\frac{7x+1}{x^2+1}$

$\frac{1}{x^2-1}$

Correct answer:

$\frac{7x+1}{x^2-1}$

Explanation:

To start this problem, you must first identify the common denominator, which in this case is teh two denominators multiplied together: (x-1)(x+1) . Next, offset the numerators with the newly changed denominators: $\frac{4(x+1)}{(x-1)(x+1)}+\frac{3(x-1)}{(x-1)(x+1)}$ . Then, add numerators to get: 4x+4+3x-3=7x+1 . Put the numerator over your denominator and check to make sure it can't be simplified anymore (it can't). Your answer is: $\frac{7x+1}{x^2-1}$ .

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Example Question #13 : Adding And Subtracting Rational Expressions

$\frac{2-x}{x^2}+\frac{4}{x}$

Possible Answers:

$-\frac{3x+2}{x^2}$

$\frac{3x+2}{x^2}$

$\frac{3x+2}{x}$

$\frac{3x-2}{x^2}$

$\frac{3}{x^2}$

Correct answer:

$\frac{3x+2}{x^2}$

Explanation:

To combine these rational expressions, first find the common denominator. In this case, it is x^2 . Then, offset the second equation so that you get the correct denominator: $\frac{4}{x}=\frac{4x}{x^2}$ . Then, combine the numerators: 2-x+4x=3x+2 . Put your numerator over the denominator for your answer: $\frac{3x+2}{x^2}$ .

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Algebra II : Intermediate Single-Variable Algebra

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #6 : How To Add Polynomials

Example Question #571 : Intermediate Single Variable Algebra

Example Question #4 : How To Find The Solution To A Rational Equation With Lcd

Example Question #11 : Solving Rational Expressions

Example Question #11 : Adding And Subtracting Rational Expressions

Example Question #571 : Intermediate Single Variable Algebra

Example Question #12 : Adding And Subtracting Rational Expressions

Example Question #11 : Adding And Subtracting Rational Expressions

Example Question #12 : Adding And Subtracting Rational Expressions

Example Question #13 : Adding And Subtracting Rational Expressions

All Algebra II Resources

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