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Algebra II : Intermediate Single-Variable Algebra

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Example Questions

Example Question #21 : Polynomials

Multiply the expressions:

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

Possible Answers:

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

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

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

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

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

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 #11 : Intermediate Single Variable Algebra

Simplify:

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

Possible Answers:

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

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

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

y=x^2-x-14

y=2x^2+x-14

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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Example Question #11 : Simplifying Polynomials

$(3x^{2}-4xy+2y^{2})-(x^{2}+2xy-5y^{2})=?$

Possible Answers:

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

$2x^{2}-6xy-7y^{2}$

$2x^{2}-6xy+7y^{2}$

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

$2x^{2}+6xy-7y^{2}$

Correct answer:

$2x^{2}-6xy+7y^{2}$

Explanation:

$(3x^{2}-4xy+2y^{2})-(x^{2}+2xy-5y^{2})$

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

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

$=2x^{2}-6xy+7y^{2}$

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Example Question #13 : Simplifying Polynomials

Expand:

$\tiny \tiny \tiny \tiny \small (2x-3y)(4x^{2}+6xy+9y^{2})$

Possible Answers:

$(2x-3y)^{2}$

$(2x-3y)(2x+3y)^{2}$

$8x^{3}-27y^{3}$

$(2x+3y)^{2}$

$8x^{3}+27x^{3}$

Correct answer:

$8x^{3}-27y^{3}$

Explanation:

$(2x-3y)(4x^{2}+6xy+9y^{2})$

$= 8x^{3}+12x^{2}y+18xy^{2}-12x^{2}y-18xy^{2}-27y^{2}$

$=8x^{3}-27y^{3}$

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Example Question #13 : Simplifying Polynomials

Simplify

$\frac{2x^2-4x-30}{5-x}$

Possible Answers:

x+5

x+3

2x+6

-2x-6

2x+10

Correct answer:

-2x-6

Explanation:

The polynominal breaks down to 2(x-5)(x+3) , and once you factor out a from the denomintator and cancel, you are left with -2(x+3) .

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Example Question #14 : Simplifying Polynomials

Simplify

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

Possible Answers:

x^3+2x^2+x+2

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

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

x+1

Correct answer:

x+1

Explanation:

First, you FOIL the binomial in the numerator. Then combine like terms, which simplifies the numerator to x^3+3x^2+3x+1 , which can be further simplified to (x+1)^3 . The coefficients for this can be found in Pascal's Triangle, so they do not have to be memorized. Once the numerator and denominator are in the same base, the exponents cancel out, leaving you with the correct answer.

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Example Question #12 : Polynomials

Simplify the following expression

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

Possible Answers:

$6y^{2}+6x^{2}$

$7x^{2}-x+9y-3y^{2}$

Expresion cannot be simplified any further

$9y + 7x^{2} + 3xy^{2}$

$3y^{2}+x+9y+7x^{2}$

Correct answer:

$7x^{2}-x+9y-3y^{2}$

Explanation:

Simplify a polynomial expression by combining like terms. Like terms are those that have exactly the same variable and degree (exponent). This one is reasonably simple with only addition and subtraction of polynomials involved. The like terms are...

8y + y = 9y

$4x^{2}+3x^{2}=7x^{2}$

The remaining terms are $-3y^{2}$ and

Combine everything together as done below. As long as the signs are right, order of operations lets you arrange the final answer various ways, but you should know this be able to identify the correct answer even if yours was arranged differently.

$9y+7x^{2}-3y^{2}-x$

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Example Question #4 : How To Add Polynomials

Simplify

$(2x^{2}-5) + (4x^{3}-3x^{2}+2x-1)$

Possible Answers:

$5x^{2}+2x-6$

$4x^{3}-x^{2}+2x-6$

$5x^{2}-6$

$4x^{3}-6x^{2}+1x-6$

Correct answer:

$4x^{3}-x^{2}+2x-6$

Explanation:

$(2x^{2}-5) + (4x^{3}-3x^{2}+2x-1)$

To simplify you combind like terms:

$(4x^{3}\boldsymbol{-3x^{2}+2x^{2}} +2x\boldsymbol{-1-5)}$

Answer:

$4x^{3}-x^{2}+2x-6$

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Example Question #11 : Intermediate Single Variable Algebra

Simplify.

$(4a^{2}b+2ab) - (3a^{2}b+4ab+2)$

Possible Answers:

$a^{2}b+6ab+2$

$a^{2}b-2ab-2$

$a^{2}b+2ab-2$

$a^{2}b-6ab-2$

Correct answer:

$a^{2}b-2ab-2$

Explanation:

Simplify

$(4a^{2}b+2ab) - (3a^{2}b+4ab+2)$

Distribute the negative: $\boldsymbol{-}3a^{2}b\boldsymbol{-}4ab\boldsymbol{-}2$

Then combinde like terms

$\mathbf{\boldsymbol{-}3a^{2}b+4a^{2}b}(\mathbf{\boldsymbol{-}4ab+2ab})\boldsymbol{-}2$

Answer:

$a^{2}b-2ab-2$

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Example Question #4 : How To Subtract Polynomials

Subtract:

$(7.5x^{2} + 1.4x - 4.6) - (x^{2} + 6.2x - 3.5)$

Possible Answers:

$6.5x^{2} - 4.8x - 1.1$

$7.5x^{2} - 4.8x - 1.1$

$7.5x^{2} +7.2x - 1.1$

$6.5x^{2} +7.2x - 1.1$

$6.5x^{2} - 4.8x - 8.1$

Correct answer:

$6.5x^{2} - 4.8x - 1.1$

Explanation:

$(7.5x^{2} + 1.4x - 4.6) - (x^{2} + 6.2x - 3.5)$

$= 7.5x^{2} + 1.4x - 4.6 - x^{2} - 6.2x + 3.5$

$= 7.5x^{2} - 1x^{2} + 1.4x- 6.2x - 4.6 + 3.5$

$= 6.5x^{2} - 4.8x - 1.1$

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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 #21 : Polynomials

Example Question #11 : Intermediate Single Variable Algebra

Example Question #11 : Simplifying Polynomials

Example Question #13 : Simplifying Polynomials

Example Question #13 : Simplifying Polynomials

Example Question #14 : Simplifying Polynomials

Example Question #12 : Polynomials

Example Question #4 : How To Add Polynomials

Example Question #11 : Intermediate Single Variable Algebra

Example Question #4 : How To Subtract Polynomials

All Algebra II Resources

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