Simplifying Polynomials - High School Math

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Question

Simplify.

$x^{-6}x^3x^9$

Answer

Put the negative exponent on the bottom so that you have $\frac{x^{12}}{x^{6}}$ which simplifies further to $x^{6}$ .

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Question

Simplify the following expression:

$3x^9y^4 \cdot (2y)^3$ .

Answer

First, multiply out the second expression so that you get .

Then, multiply your like terms, taking care to remember that when multiplying terms that have the same base, you add the exponents. Thus, you get $24x^9y^{12}$ .

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Question

Simplify:

$\frac{13x^2y^5z^9}{169xy^5z^4}$

Answer

Focus on each pair of like terms. The completely cancel out, there is one left on top, and five left on the bottom.

$\frac{13}{169}$ reduces to $\frac{1}{13}$ .

Put that all together to get $\frac{xz^5}{13}$ .

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Question

Expand this expression:

Answer

Use the FOIL method (First, Outer, Inner, Last):

Put all of these terms together:

Combine like terms:

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Question

Simplify the following polynomial:

$ab^{-2}(a^2b+a^{-1}b^3-a^3b^{-1})$

Answer

To simplify the polynomial, begin by multiplying the first binomial by every term within the parentheses:

$ab^{-2}(a^2b+a^{-1}b^3-a^3b^{-1})$

$aa^2b^{-2}b+aa^{-1}b^{-2}b^3-aa^3b^{-2}b^{-1}$

Now, combine like terms:

$a^3b^{-1}+b^1-a^4b^{-3}$

Convert the polynomial into fraction form:

$\frac{a^3}{b}+b-\frac{a^4}{b^3}$

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Question

Simplify the following polynomial:

$\frac{-18a^{-2}b^3c^{-4}}{12ab^{-2}c^{-3}}$

Answer

To simplify the polynomial, begin by rearranging the terms to have positive exponents:

$\frac{-18a^{-2}b^3c^{-4}}{12ab^{-2}c^{-3}}$

$\frac{-18b^3b^2c^{3}}{12aa^2c^{4}}$

Now, combine like terms:

$\frac{-18b^5}{12a^3c^{1}}$

Simplify the integers:

$\frac{-3b^5}{2a^3c}$

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Question

Simplify the following polynomial:

$(5n^{3x}-2m^{x+2})^2$

Answer

Squaring the polynomial is equivalent to:

$(5n^{3x}-2m^{x+2})^2$

$(5n^{3x}-2m^{x+2})(5n^{3x}-2m^{x+2})$

Use the FOIL method to multiply the terms:

F - First

O - Outer

I - Inner

L - Last

$25n^{6x}-10n^{3x}m^{x+2}-10n^{3x}m^{x+2}+4m^{2x+4}$

Combine like terms:

$25n^{6x}-20n^{3x}m^{x+2}+4m^{2x+4}$

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Question

Simplify the following polynomial:

$(\frac{-a^{-2}b^3c^{-1}}{3a^{-4}b^{-1}c^3})^{-2}$

Answer

To simplify the polynomial, begin by rearranging the terms to have positive exponents:

$(\frac{-a^{-2}b^3c^{-1}}{3a^{-4}b^{-1}c^3})^{-2}$

$(\frac{-a^{4}b^3b^1}{3a^{2}c^3c^1})^{-2}$

Rearrange the terms once again so that the outer exponent is positive. Also, combine like terms:

$(\frac{-3c^4}{a^{2}b^4})^{2}$

Now, square the polynomial:

$\frac{9c^8}{a^{4}b^8}$

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Question

Simplify the following polynomial:

Answer

To simplify the polynomial, begin by combining the terms within the parentheses and multiplying the integers:

Now, add together like terms:

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Question

Simplify the following polynomial:

$\frac{-9x^{-2}y^3z^{-1}}{3x^{-4}yz^2}$

Answer

To simplify the polynomial, begin by rearranging the terms to have positive exponents:

$\frac{-9x^{-2}y^3z^{-1}}{3x^{-4}yz^2}$

$\frac{-9x^{4}y^3}{3x^{2}yz^2z^1}$

Now, combine like terms:

$\frac{-9x^{2}y^2}{3z^3}$

Simplify the integers:

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

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Question

Simplify the following polynomial:

$(\frac{2y^{-2}}{-2})^{-1}(\frac{m^2n}{y})^{-2}$

Answer

Begin by reversing the numerator and denominator so that the exponents are positive:

$(\frac{2y^{-2}}{-2})^{-1}(\frac{m^2n}{y})^{-2}$

$(\frac{-2}{2y^{-2}})^{1}(\frac{y}{m^2n})^{2}$

$(\frac{-2y^{2}}{2})^{1}(\frac{y}{m^2n})^{2}$

Square the right side of the expression and multiply:

$(\frac{-2y^{2}}{2})(\frac{y^2}{m^4n^2})$

$\frac{-2y^{4}}{2m^4n^2}$

Simplify:

$\frac{-y^{4}}{m^4n^2}$

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Question

Simplify the following polynomial:

$\frac{6n^{2x+3}}{3n^{x-4}}$

Answer

Begin by simplifying the integers:

$\frac{6n^{2x+3}}{3n^{x-4}}$

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

Subtract the exponent in the denominator from the exponent in the numerator:

$2n^{x+7}$

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Question

Simplify the following polynomial:

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

Answer

Begin by multiplying the terms:

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

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

Convert into fraction form:

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

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Question

Simplify the following polynomial:

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

Answer

Use the FOIL method to multiply the terms: F (first) O (outer) I (inner) L (last)

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

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

$16x^{2a+4}-24x^{a+2}3y^{3a}+9y^{6a}$

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Question

Simplify the following polynomial:

$(x^{n+1}+y^{2n-1})(x^{n+1}-y^{2n-1})$

Answer

Use the FOIL method to multiply the terms: F (first) O (outer) I (inner) L (last)

$(x^{n+1}+y^{2n-1})(x^{n+1}-y^{2n-1})$

$(x^{2n+2}+x^{n+1}y^{2n-1}-x^{n+1}y^{2n-1}-y^{4n-2})$

Combine like terms:

$x^{2n+2}-y^{4n-2}$

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Question

If and $g(x)=x^{2}-x$ , what is ?

Answer

is a composite function solved by substituting into :

$(x-3)^{2}-(x-3)=x^{2}-6x+9-x+3=x^{2}-7x+12$

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