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Algebra II : Expressions

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Example Question #1 : Simplifying Expressions

Find the product:

$\small 7n(8n-2)$

Possible Answers:

$\small n^2+14$

$\small n-14$

$\small n^2-14n$

$\small 56n^2-14n$

Correct answer:

$\small 56n^2-14n$

Explanation:

First, mulitply the mononomial by the first term of the polynomial:

$\small 7n\times8n\ = 56n^2$

Second, multiply the monomial by the second term of the polynomial:

$\small 7n\times (-2)\ = -14n$

Add the terms together:

$\small 56n^2\ +\ (-14n)\ = 56n^2-14n$

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Example Question #1 : Simplifying Expressions

Multiply, expressing the product in simplest form:

$\small \frac{15x^{5}}{14y^{3}} \cdot \frac{21y^{2} }{25x^{3}}$

Possible Answers:

$\small \small \small \small \frac{10 x^{2}}{9y }$

$\small \small \small \frac{9 x^{2}y}{10 }$

$\small \small \small \small \frac{9 y}{10x^{2} }$

$\small \small \small \small \small \frac{10 y}{9x^{2} }$

$\small \small \frac{9 x^{2}}{10 y}$

Correct answer:

$\small \small \frac{9 x^{2}}{10 y}$

Explanation:

Cross-cancel the coefficients by dividing both 15 and 25 by 5, and both 14 and 21 by 7:

$\small \small \small \small \frac{15x^{5}}{14y^{3}} \cdot \frac{21y^{2} }{25x^{3}} = \small \small \small \small \frac{3x^{5}}{2y^{3}} \cdot \frac{3y^{2} }{5x^{3}} = \frac{3x^{5} \cdot 3y^{2}}{5x^{3} \cdot 2y^{3}} = \frac{3 \cdot 3 \cdot x^{5} y^{2}}{5\cdot 2\cdot x^{3} y^{3}} = \frac{9 x^{5} y^{2}}{10 x^{3} y^{3}}$

Now use the quotient rule on the variables by subtracting exponents:

$\small \frac{9 x^{5} y^{2}}{10 x^{3} y^{3}} = \frac{9 x^{5-3}}{10 y^{3-2}} = \frac{9 x^{2}}{10 y^{1}} = \frac{9 x^{2}}{10 y}$

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Example Question #5 : Simplifying Expressions

Simplify the following:

$\frac{2xy^2}{3yz^2} \times \frac{15y^2z}{14x^2y}$

Possible Answers:

$\frac{7y^2}{5x^2z}$

$\frac{5y^2}{7xz}$

$\frac{5xz}{7y^2}$

$\frac{5xy^2}{7z}$

Correct answer:

$\frac{5y^2}{7xz}$

Explanation:

In this problem, you have two fractions being multiplied. You can first simplify the coefficients in the numerators and denominators. You can divide and cancel the 2 and 14 each by 2, and the 3 and 15 each by 3:

$\frac{2xy^2}{3yz^2} \times \frac{15y^2z}{14x^2y} = \frac{xy^2}{yz^2} \times \frac{5y^2z}{7x^2y}$

You can multiply the two numerators and two denominators, keeping in mind that when multiplying like variables with exponents, you simplify by adding the exponents together:

$\frac{xy^2}{yz^2} \times \frac{5y^2z}{7x^2y} = \frac{5xy^4z}{7x^2y^2z^2}$

Any variables that are both in the numerator and denominator can be simplified by subtracting the numerator's exponent by the denominator's exponent. If you end up with a negative exponent in the numerator, you can move the variable to the denominator to keep the exponent positive:

$\frac{5xy^4z}{7x^2y^2z^2} = \frac{5y^2}{7xz}$

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Example Question #3 : How To Factor A Variable

Factor the expression:

$\small \small 2xy^3-14xy^2+9xy$

Possible Answers:

$\small xy^3(2-14xy^2+9xy)$

$\small 2xy(y^2-7xy+9)$

$\small x(2y^3-14y^2+9y)$

$\small 2xy^3(-14xy^2+9xy)$

$\small xy(2y^2-14y+9)$

Correct answer:

$\small xy(2y^2-14y+9)$

Explanation:

To find the greatest common factor, we need to break each term into its prime factors:

$\small 2xy^3=2 \cdot x \cdot y \cdot y \cdot y$

$\small 14xy^2 = 2 \cdot 7 \cdot x \cdot y \cdot y$

$\small 9xy = 3 \cdot 3 \cdot x \cdot y$

Looking at which terms all three expressions have in common; thus, the GCF is $\small xy$ . We then factor this out: $\small \small xy(2y^2 -14y +9)$ .

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Example Question #2 : Monomials

Expand: 8x(3x+7)

Possible Answers:

11x^2 + 15x

24x^2 + 56x

24x^2 + 56

24x + 56

11x + 15

Correct answer:

24x^2 + 56x

Explanation:

To expand, multiply 8x by both terms in the expression (3x + 7).

8x multiplied by 3x is 24x².

8x multiplied by 7 is 56x.

Therefore, 8x(3x + 7) = 24x² + 56x.

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Example Question #4501 : Algebra 1

Simplify:

-5(x+15)-3x

Possible Answers:

-8x-75

2x+15

2x-75

-8x+15

None of the other answers are correct.

Correct answer:

-8x-75

Explanation:

First, distribute –5 through the parentheses by multiplying both terms by –5.

-5(x+15)-3x=-5x-75-3x

Then, combine the like-termed variables (–5x and –3x).

-5x-75-3x=-8x-75

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Example Question #1 : How To Multiply Binomials With The Distributive Property

Expand:

$\frac{1}{3}(2x^2-6)(4x-\frac{2}{3})$

Possible Answers:

$-2\frac{2}{3}x^3-\frac{4}{9}x^2-8x+1\frac{1}{3}$

$2\frac{2}{3}x^3+\frac{4}{9}x^2-8x+1\frac{1}{3}$

$2\frac{2}{3}x^3-\frac{4}{9}x^2-8x-1\frac{1}{3}$

$2\frac{2}{3}x^3-\frac{4}{9}x^2+8x+1\frac{1}{3}$

$2\frac{2}{3}x^3-\frac{4}{9}x^2-8x+1\frac{1}{3}$

Correct answer:

$2\frac{2}{3}x^3-\frac{4}{9}x^2-8x+1\frac{1}{3}$

Explanation:

First, FOIL:

$\frac{1}{3}(8x^3-\frac{4}{3}x^2-24x+\frac{12}{3})$

Simplify:

$\frac{1}{3}(8x^3-\frac{4}{3}x^2-24x+4)$

Distribute the $\frac{1}{3}$ through the parentheses:

$\frac{8}{3}x^3-\frac{4}{9}x^2-\frac{24}{3}x+\frac{4}{3}$

Rewrite to make the expression look like one of the answer choices:

$2\frac{2}{3}x^3-\frac{4}{9}x^2-8x+1\frac{1}{3}$

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Example Question #11 : Simplifying Expressions

Simplify the expression.

$15x^{3}y^{2}+8x^{2}+3x^{3}y^{2}-4x^{2}$

Possible Answers:

$18x^{3}y^{2}-4x^{2}$

$18x^{3}y^{2}+4x^{2}$

$22x^{6}y^{4}$

$22x^3y^{2}x^{2}$

None of the other answers are correct.

Correct answer:

$18x^{3}y^{2}+4x^{2}$

Explanation:

When simplifying polynomials, only combine the variables with like terms.

$15x^{3}y^{2}$ can be added to $3x^{3}y^{2}$ , giving $18x^{3}y^{2}$ .

$4x^{2}$ can be subtracted from $8x^{2}$ to give $4x^{2}$ .

Combine both of the terms into one expression to find the answer: $18x^{3}y^{2}+4x^{2}$

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Example Question #12 : Simplifying Expressions

Simplify the following expression.

$(5c^{2}+5)+(-5c-5)$

Possible Answers:

$5c^{2}-5c-25$

$5c^{2}-5c+10$

$5c^{2}-5c$

Correct answer:

$5c^{2}-5c$

Explanation:

$(5c^{2}+5)+(-5c-5)$

This is not a FOIL problem, as we are adding rather than multiplying the terms in parentheses.

Add like terms to solve.

$5c^{2}$ and -5c have no like terms and cannot be combined with anything.

5 and -5 can be combined however:

5-5=0

This leaves us with $5c^{2}-5c$ .

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Example Question #1 : How To Subtract Polynomials

Simplify the following:

4x^3-2x(x-2)(x+3)

Possible Answers:

4x^3-2x(x^2+x-6)

4x^3-2x^3-2x^2+12x

2x^3-2x^2+12x

8x^3-2x+16

None of the other answers are correct.

Correct answer:

2x^3-2x^2+12x

Explanation:

4x^3-2x(x-2)(x+3)

First, FOIL the two binomials:

4x^3-2x(x^2+x-6)

Then distribute the through the terms in parentheses:

4x^3-2x^3-2x^2+12x

Combine like terms:

2x^3-2x^2+12x

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Algebra II : Expressions

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Simplifying Expressions

Example Question #1 : Simplifying Expressions

Example Question #5 : Simplifying Expressions

Example Question #3 : How To Factor A Variable

Example Question #2 : Monomials

Example Question #4501 : Algebra 1

Example Question #1 : How To Multiply Binomials With The Distributive Property

Example Question #11 : Simplifying Expressions

Example Question #12 : Simplifying Expressions

Example Question #1 : How To Subtract Polynomials

All Algebra II Resources

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