Polynomials

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Questions 1 - 10

1

Simplify the following polynomial:

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

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

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

$\frac{-y^3}{m^4n^2}$

$\frac{y^3}{m^4n^2}$

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

Explanation

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:

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

2

Simplify the following polynomial:

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

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

$\frac{a^2}{b}+b-\frac{a^4}{b^3}$

$\frac{a^3}{b}+b^2-\frac{a^4}{b^3}$

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

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

Explanation

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:

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

3

Factor the following polynomial:

Explanation

Begin by separating into like terms. You do this by multiplying and , then finding factors which sum to

Now, extract like terms:

Simplify:

4

Factor the following polynomial:

Explanation

Begin by separating into like terms. You do this by multiplying and , then finding factors which sum to

Now, extract like terms:

Simplify:

5

Simplify the following polynomial:

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

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

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

$\frac{-y^3}{m^4n^2}$

$\frac{y^3}{m^4n^2}$

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

Explanation

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:

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

6

Simplify the following polynomial:

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

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

$\frac{a^2}{b}+b-\frac{a^4}{b^3}$

$\frac{a^3}{b}+b^2-\frac{a^4}{b^3}$

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

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

Explanation

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:

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

7

Factor the following polynomial:

Explanation

To begin, distribute the squares:

Now, combine like terms:

8

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

$x^{2}-7x+12$

$x^{2}-3$

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

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

$x^{2}-x-3$

Explanation

is a composite function solved by substituting into :

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

9

Factor the following polynomial:

Explanation

To begin, distribute the squares:

Now, combine like terms:

10

Simplify the following polynomial:

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

$2n^{x+7}$

$n^{x+7}$

$3n^{x+7}$

$2n^{x+9}$

$n^{x+9}$

Explanation

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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