How to add exponential variables

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ISEE Upper Level Quantitative Reasoning › How to add exponential variables

Questions 1 - 10

1

Rewrite the polynomial in standard form:

$9x^{4} - 3x ^{2} + 8x^{5} -2x + 7 - 4x^{3}$

$8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x + 7$

$7-2x- 3x ^{2}- 4x^{3}+ 9x^{4} + 8x^{5}$

$9x^{4} + 8x^{5} + 7- 4x^{3} - 3x ^{2} -2x$

$-2x - 3x ^{2}- 4x^{3}+ 7+ 8x^{5} + 9x^{4}$

$9x^{4} + 8x^{5} + 7-2x - 3x ^{2} - 4x^{3}$

Explanation

The degree of a term of a polynomial with one variable is the exponent of that variable. The terms of a polynomial in standard form are written in descending order of degree. Therefore, we rearrange the terms by their exponent, from 5 down to 0, noting that we can rewrite the and constant terms with exponents 1 and 0, respectively:

$9x^{4} - 3x ^{2} + 8x^{5} -2x + 7 - 4x^{3}$

$= 9x^{4} - 3x ^{2} + 8x^{5} -2x ^{1} + 7x ^{0} - 4x^{3}$

$=8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x ^{1}+ 7x^{0}$

$=8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x+ 7$

2

Simplify:

$\frac{(x^{2}+y^{2})^{2} - (x^{2} - y^{2})^{2}}{x^{4}y^{4}}$

$\frac{4}{x^{2}y^{2}}$

$\frac{4}{x^{4}y^{4}}$

$\frac{4}{xy}$

$\frac{4}{x^{3}y^{3}}$

Explanation

$\frac{(x^{2}+y^{2})^{2} - (x^{2} - y^{2})^{2}}{x^{4}y^{4}}$

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

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

$=\frac{4x^{2}y^{2}}{x^{4}y^{4}}$

$=\frac{4}{x^{2}y^{2}}$

3

Add:

Explanation

4

Define

What is $g(x ^{2} + 3x)$ ?

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

$2 x ^{2} + 6x - 10$

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

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

Explanation

Substitute $x ^{2} + 3x$ for in the definition:

$g(x ^{2} + 3x ) = 2 \left ( x ^{2} + 3x \right ) - 5$

$g(x ^{2} + 3x ) = 2 \left ( x ^{2}\right ) + 2 \left ( 3x \right ) - 5$

$g(x ^{2} + 3x ) = 2 x ^{2} + 6x - 5$

5

Simplify:

Explanation

Expand each term by using FOIL:

Rearrange to group like-terms together.

Simplify by combining like-terms.

6

Assume that . Simplify:

$\frac{(x^{2}+y^{2})^{2} +(x^{2} - y^{2})^{2}}{x^{4}y^{4}}$

$\frac{2x^{4}+2y^{4} }{x^{4}y^{4}}$

$\frac{2x^{4}-2y^{4} }{x^{4}y^{4}}$

$\frac{4}{x^{2}y^{2}}$

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

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

Explanation

$\frac{(x^{2}+y^{2})^{2} + (x^{2} + y^{2})^{2}}{x^{4}y^{4}}$

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

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

$=\frac{2x^{4}+2y^{4} }{x^{4}y^{4}}$

7

If , simplify:

$\frac{(x-y)^2+4xy}{x+y}$

Explanation

$\frac{(x-y)^2+4xy}{x+y}=\frac{(x^2-2xy+y^2)+4xy}{x+y}$

$=\frac{(x^2-2xy+4xy+y^2)}{x+y}$

$=\frac{(x^2+2xy+y^2)}{x+y}$

$=\frac{(x+y)^2}{x+y}=x+y$

8

Simplify:

The expression can not be simplified further

Explanation

Start by reordering the expression to group like-terms together.

Combine like-terms to simplify.

9

Simplify:

Explanation

We can expand the first term using FOIL:

Reorder the expression to group like-terms together.

Simplify by combining like-terms.

10

If , simplify:

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

Explanation

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

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

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

$=\frac{2x^3+\frac{6}{x}}{\frac{2}{x}}$

$=\frac{\frac{2x^4+6}{x}}{\frac{2}{x}}$

$=\frac{2x^4+6}{2}$

$=\frac{2(x^4+3)}{2}$

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