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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #1 : Simplifying Polynomials

Rewrite the expression in simplest terms.

$5x^{^{3}} - 14x^{2} - 10x + 3 - (4x+3) * x^{2} - (8-10x)$

Possible Answers:

5x^3 -17x^2 - 4x - 5

x^3 - 11x^2 - 5

5x^3 -11x^2 - 24x - 5

$x^{3}-17x^{2}-5$

x^3 - 11x^2 - 20x - 5

Correct answer:

$x^{3}-17x^{2}-5$

Explanation:

In simplifying this expression, be mindful of the order of operations (parenthical, division/multiplication, addition/subtraction).

$5x^{^{3}} - 14x^{2} - 10x + 3 - (4x+3) * x^{2} - (8-10x)$

Since operations invlovling parentheses occur first, distribute the factors into the parenthetical binomials. Note that the x^2 outside the first parenthetical binomial is treated as -x^2 since the parenthetical has a negative (minus) sign in front of it. Similarly, multiply the members of the expression in the second parenthetical by because of the negative (minus) sign in front of it. Distributing these factors results in the following polynomial.

5x^3 - 14x^2 - 10x + 3 - 4x^3 - 3x^2 - 8 + 10x

Now like terms can be added and subtracted. Arranging the members of the polynomial into groups of like terms can help with this. Be sure to retain any negative signs when rearranging the terms.

5x^3 - 4x^3 -14x^2 - 3x^2 - 10x + 10x + 3 - 8

Adding and subtracting these terms results in the simplified expression below.

x^3 - 17x^2 -5

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Example Question #1 : Simplifying Polynomials

Multiply:

$\left (x^{2} + 2x + 7 \right ) \left (x^{2} + x - 6 \right )$

Possible Answers:

$x^{4} + 3 x^{3} -x^{2} - 5x - 42$

$x^{4} + 3 x^{3} -15x^{2} - 19x - 42$

$x^{4} + 3 x^{3} + 19x^{2} - 5x - 42$

$x^{4} + 3 x^{3} +3x^{2} - 5x - 42$

$x^{4} + 3 x^{3} -x^{2} - 19x - 42$

Correct answer:

$x^{4} + 3 x^{3} +3x^{2} - 5x - 42$

Explanation:

$\left (x^{2} + 2x + 7 \right ) \left (x^{2} + x - 6 \right )$

$= x^{2}\left (x^{2} + x - 6 \right ) + 2x \left (x^{2} + x - 6 \right ) + 7 \left (x^{2} + x - 6 \right )$

$= x^{2} \cdot x^{2} + x^{2} \cdot x - x^{2} \cdot 6 + 2x \cdot x^{2} + 2x \cdot x - 2x \cdot 6 + 7 \cdot x^{2} + 7 \cdot x - 7 \cdot 6$

$= x^{4} + x^{3} - 6x^{2} + 2x^{3} + 2x^{2} - 12x + 7x^{2} +7x - 42$

$= x^{4} + x^{3} + 2x^{3} - 6x^{2} + 2x^{2} + 7x^{2} - 12x +7x - 42$

$= x^{4} + 3 x^{3} +3x^{2} - 5x - 42$

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Example Question #1 : Simplifying Polynomials

Simplify the expression.

$\small (3x+2)-(x+4)(x+1)$

Possible Answers:

$\small -x^2-2x-2$

$\small -x^2+8x+6$

$\small x^2+8x+6$

$\small -x^2-8x-6$

Correct answer:

$\small -x^2-2x-2$

Explanation:

$\small (3x+2)-(x+4)(x+1)$

Use FOIL to expand the monomials.

(x+4)(x+1)

x^2+x+4x+4=x^2+5x+4

Return this expansion to the original expression.

$\small (3x+2)-(x^2+5x+4)$

Distribute negative sign.

$\small (3x+2)-x^2-5x-4$

Combine like terms.

$\small -x^2+3x-5x+2-4$

$\small -x^2-2x-2$

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Example Question #72 : Variables

Divide the trinomial below by .

$18x^{2}+ 9x -3$

Possible Answers:

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

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

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

-x+1

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

Correct answer:

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

Explanation:

$18x^{2}+ 9x -3$

We can accomplish this division by re-writing the problem as a fraction.

$\frac{18x^2+9x-3}{9x}$

The denominator will distribute, allowing us to address each element separately.

$\frac{18x^{2}+ 9x -3}{9x}=\frac{18x^{2}}{9x}+\frac{ 9x}{9x}-\frac{3}{9x}$

Now we can cancel common factors to find our answer.

$(\frac{18}{9}* \frac{x^{2}}{x})+1-(\frac{3}{9}* \frac{1}{x})$

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

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Example Question #1 : Simplifying Polynomials

Evaluate the following:

$(2x^2+\frac{3}{4}x-5) - (x^2+x-10)$

Possible Answers:

$3x^2-\frac{1}{4}x + 5$

$x^2-\frac{7}{4}x + 10$

$x^2-\frac{1}{4}x + 5$

$x^2-\frac{1}{4}x -15$

Correct answer:

$x^2-\frac{1}{4}x + 5$

Explanation:

To subtract these two trinomials, you first need to flip the sign on every term in the second trinomial, since it is being subtrated:

$(2x^2+\frac{3}{4}x-5) - (x^2+x-10)$

$2x^2+\frac{3}{4}x-5 - x^2-x+10$

Next you can combine like terms. You have two terms with x^2 , two terms with , and two terms with no variable:

$x^2-\frac{1}{4}x+5$

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Example Question #1 : Simplifying Polynomials

Subtract the expressions below.

$\left (\frac{1}{3}x^{2}+\frac{1}{6}xy-\frac{5}{4}y^{2} \right )-\left (\frac{1}{9}x^{2}-\frac{4}{3}xy+y^{2} \right )$

Possible Answers:

None of the other answers are correct.

$\frac{2}{9}x^{2}+\frac{3}{2}xy-\frac{5}{4}y^{2}$

$\frac{1}{27}x^{2}-\frac{2}{9}xy-\frac{5}{4}y^{2}$

$\frac{2}{9}x^{2}+\frac{3}{2}xy-\frac{9}{4}y^{2}$

$\frac{2}{9}x^{2}+\frac{7}{6}xy-\frac{1}{4}y^{2}$

Correct answer:

$\frac{2}{9}x^{2}+\frac{3}{2}xy-\frac{9}{4}y^{2}$

Explanation:

$\left (\frac{1}{3}x^{2}+\frac{1}{6}xy-\frac{5}{4}y^{2} \right )-\left (\frac{1}{9}x^{2}-\frac{4}{3}xy+y^{2} \right )$

Since we are only adding and subtracting (there is no multiplication or division), we can remove the parentheses.

$\frac{1}{3}x^{2}+\frac{1}{6}xy-\frac{5}{4}y^{2}-\left \frac{1}{9}x^{2}+\frac{4}{3}xy-y^{2} \right$

Regroup the expression so that like variables are together. Remember to carry positive and negative signs.

$(\frac{1}{3}x^{2}-\left \frac{1}{9}x^{2})+(\frac{1}{6}xy+\frac{4}{3}xy)-(\frac{5}{4}y^{2}-y^{2} \right)$

For all fractional terms, find the least common multiple in order to add and subtract the fractions.

$(\frac{3}{9}x^{2}-\left \frac{1}{9}x^{2})+(\frac{1}{6}xy+\frac{8}{6}xy)-(\frac{5}{4}y^{2}-\frac{4}{4}y^{2} \right)$

Combine like terms and simplify.

$\frac{2}{9}x^{2}+\frac{9}{6}xy-\frac{9}{4}y^{2}$

$\frac{2}{9}x^{2}+\frac{3}{2}xy-\frac{9}{4}y^{2}$

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Example Question #1 : Simplifying Polynomials

Multiply:

(4x^2-5x+9)(3x-4)

Possible Answers:

12x^3-31x^2+47x-36

31x^3-12x^2+47x-36

33x^2+27x-36

16x^2+20x-36

Correct answer:

12x^3-31x^2+47x-36

Explanation:

Set up this problem vertically like you would a normal multiplication problem without variables. Then, multiply the term to each term in the trinomial. Next, multiply the term to each term in the trinomial (keep in mind your placeholder!).

Then combine the two, which yields:

12x^3-31x^2+47x-36

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Example Question #1 : Simplifying Polynomials

Evaluate the following:

$\frac{1}{2}(4x^2-5x+10)+\frac{1}{4}(7x^2-5x+12)$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

$\frac{1}{2}(4x^2-5x+10)+\frac{1}{4}(7x^2-5x+12)$

First distribute the $\frac{1}{2}$ :

$2x^2-\frac{5}{2}x+5+\frac{1}{4}(7x^2-5x+12)$

Then distribute the $\frac{1}{4}$ :

$2x^2-\frac{5}{2}x+5+\frac{7}{4}x^2-\frac{5}{4}x+\frac{12}{4}$

Finally combine like terms:

$2x^2-\frac{10}{4}x+5+1\frac{3}{4}x^2-\frac{5}{4}x+3$

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

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

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Example Question #1 : How To Multiply Trinomials

Multiply the expressions:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

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

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Example Question #1141 : Algebra Ii

Simplify:

$y=\frac{(x^2-16)(x+7)}{x+4}+(x-1)^2+13$

Possible Answers:

y=2x^2+x-14

$y=\frac{(x^2-16)(x+7)}{x+4}+x^2-2x+14$

y=x^2-x-14

$y=\frac{x^3+7x^2-16x-112}{x+4}+(x-1)^2+13$

y=2(x^2+x-7)

Correct answer:

y=2x^2+x-14

Explanation:

$y=\frac{(x^2-16)(x+7)}{x+4}+(x-1)^2+13$

First, factor the numerator of the quotient term by recognizing the difference of squares:

$y=\frac{(x-4)(x+4)(x+7)}{x+4}+(x-1)^2+13$

Cancel out the common term from the numerator and denominator:

y=(x-4)(x+7)+(x-1)^2+13

FOIL (First Outer Inner Last) the first two terms of the equation:

y=x^2+7x-4x-28+x^2-2x+1+13

Combine like terms:

y=2x^2+x-14

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Simplifying Polynomials

Example Question #1 : Simplifying Polynomials

Example Question #1 : Simplifying Polynomials

Example Question #72 : Variables

Example Question #1 : Simplifying Polynomials

Example Question #1 : Simplifying Polynomials

Example Question #1 : Simplifying Polynomials

Example Question #1 : Simplifying Polynomials

Example Question #1 : How To Multiply Trinomials

Example Question #1141 : Algebra Ii

All Algebra II Resources

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