Simplifying Expressions - Math

Card 0 of 484

Question

$\small $$\frac{x^2$-5x-6}{x+1}$$

For all values $x\neq -1$ , which of the following is equivalent to the expression above?

Answer

First, factor the numerator. We need factors that multiply to $\small -6$ and add to $\small -5$ .

$\small 1*-6=-6\ $\text{and}$\ 1+(-6)=-5$

$\small $x^2$-5x-6=(x-6)(x+1)$

We can plug the factored terms into the original expression.

$\small $$\frac{x^2$-5x-6}{x+1}$=\frac{(x-6)(x+1)}{(x+1)}$$

Note that $\small (x+1)$ appears in both the numerator and the denominator. This allows us to cancel the terms.

$$\frac{(x-6)(x+1)}{(x+1)}$=(x-6)$

This is our final answer.

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Question

Simplify the expression.

$\small $$\frac{(x^2y$^$2)(x^3y$^$4z)}{x^4y$^$3z^4$}$$

Answer

$\small $$\frac{(x^2y$^$2)(x^3y$^$4z)}{x^4y$^$3z^4$}$$

Because we are only multiplying terms in the numerator, we can disregard the parentheses.

$\small $$\frac{(x^2y$^$2)(x^3y$^$4z)}{x^4y$^$3z^4$$}$=\frac{x^2y$^$2x^3y$^$4z}{x^4y$^$3z^4$}$$

To combine like terms in the numerator, we add their exponents.

$\small $$\frac{x^5y$^$6z}{x^4y$^$3z^4$}$$

To combine like terms between the numerator and denominator, subtract the denominator exponent from the numerator exponent.

Remember that any negative exponents stay in the denominator.

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Question

Give the value of that makes the polynomial the square of a linear binomial.

Answer

A quadratic trinomial is a perfect square if and only if takes the form

for some values of and .

, so

and .

For to be a perfect square, it must hold that

$Nx = 2AB = 2 \cdot 3x \cdot 10 = 60x$ ,

so . This is the correct choice.

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Question

Exponentiate:

$\left ( 9R - 10\right $)^{3}$$

Answer

The difference of two terms can be cubed using the pattern

Where :

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Question

How many of the following are prime factors of ?

I)

II)

III)

IV)

Answer

Factor all the way to its prime factorization.

can be factored as the difference of two perfect square terms as follows:

$=\left ( $9y^{2}$ \right $)^{2}$ - $1^{2}$$

$=\left (9 $y^{2}$ + 1 \right )\left ( $9y^{2}$ - 1 \right )$

is a factor, and, as the sum of squares, it is a prime. is also a factor, but it is not a prime factor - it can be factored as the difference of two perfect square terms. We continue:

$\left (9 $y^{2}$ + 1 \right )\left ( $9y^{2}$ - 1 \right )$

$=\left ( $9y^{2}$ + 1 \right )\left [ \left (3y \right ) ^{2} - $1^{2}$ \right ]$

$=\left ( $9y^{2}$ + 1 \right )\left (3y+1 \right )\left (3 y-1\right )$

Therefore, of the given four choices, only is not a factor, so the correct response is three.

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Question

Factor:

Answer

$=\left (3Q \right $)^{3}$- 3 \cdot $90Q^{2}$+3\cdot 300Q $-10^{3}$$

$=\left (3Q \right $)^{3}$- 3 \cdot $9Q^{2}$\cdot 10+3\cdot 3Q \cdot $10^{2}$ $-10^{3}$$

$=\left (3Q \right $)^{3}$- 3 \cdot \left (3Q \right $)^{2}$\cdot 10+3\cdot 3Q \cdot $10^{2}$ $-10^{3}$$

This can be factored out as the cube of a difference, where :

Therefore,

$=\left (3Q \right $)^{3}$- 3 \cdot \left (3Q \right $)^{2}$\cdot 10+3\cdot 3Q \cdot $10^{2}$ $-10^{3}$$

$= (3Q - $10)^{3}$$

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Question

Simplify:

Answer

To simplify, we begin by simplifying the numerator. When muliplying like bases with different exponents, their exponents are added.

For x: $\small 4+3=7$

For y: $\small 2+1=3$

For z: $\small 3+1=4$

The numerator is now .

When dividing like bases, their exponents are subtracted.

For x: $\small 7-4=3$

For y: $\small 3-(-3)=6$

For z: $\small 4-3=1$

Thus, our answer is .

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Question

Which of the following linear binomials is a factor of the polynomial ?

Answer

By the factor theorem, a polynomial is divisible by the linear binomial if and only if . We can use this fact to test each of the binomials by evaluating the dividend for the appropriate value of .

: Evaluate the polynomial at :

: Evaluate the polynomial at :

: Evaluate the polynomial at :

: Evaluate the polynomial at :

: Evaluate the polynomial at :

The dividend assumes the value of 0 at , so of the choices given, is the factor.

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Question

What is increased by 40%?

Answer

A number increased by 40% is equivalent to 100% of the number plus 40% of the number. This is taking 140% of the number, or, equivalently, multiplying it by 1.4.

Therefore, increased by 40% is 1.4 times this, or

$1.4 \left (3t+ 7 \right ) = 1.4 \cdot 3t + 1.4 \cdot 7 = 4.2 t + 9.8$

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Question

Simplify the expression:

Answer

To solve this problem, we first need to factor the numerator. We are looking for two numbers that multiply to equal -8 and sum to equal 2.

Now, we can write out our expression in fraction form.

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

Since we have the like term in the numerator and denominator, we can cancel them out of our expression.

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

Thus, our answer is .

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Question

Which of the following is a prime factor of ?

Answer

is the difference of two squares:

As such, it can be factored as follows:

The first factor is the sum of cubes and the second is the difference of cubes; each can be factored further:

Therefore,

Of the choices, appears in the prime factorization and is therefore the correct choice.

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Question

Decrease by 20%. Which of the following will this be equal to?

Answer

A number decreased by 20% is equivalent to 100% of the number minus 20% of the number. This is taking 80% of the number, or, equivalently, multiplying it by 0.8.

Therefore, decreased by 20% is 0.8 times this, or

$0.8 \left (9x + 4y \right ) = 0.8 \cdot 9x + 0.8 \cdot 4y = 7.2x + 3.2y$

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Question

Divide:

$\left ( 12 x ^{4} + $16x^{3}$+ $64x^{2}$+8 x \right ) \div $8x^{2}$$

Answer

Divide termwise:

$\left ( 12 x ^{4} + $16x^{3}$+ $64x^{2}$+8 x \right ) \div $8x^{2}$$

$=\frac{ 12 x ^{4}$ + $16x^{3}$+ $64x^{2}$+8 x }{ $8x^{2}$}$

$=\frac{ 12 x ^{4}$ }{ $8x^{2}$} + $\frac{ $16x^{3}$$ }{ $8x^{2}$} + $\frac{ $64x^{2}$$ }{ $8x^{2}$} +\frac{ 8 x }{ $8x^{2}$$}$

$=\frac{ 12 }{ 8 }$x ^{4-2} + $\frac{ 16 }{ $8}$x^{3-2}$ + $\frac{ 64 }{ 8 }$ +\frac{ 8 }{ 8}$ \cdot $$\frac{1}{x^{2-1}$$}$

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

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Question

Decrease by 30%. Which of the following will this be equal to?

Answer

A number decreased by 30% is equivalent to 100% of the number minus 30% of the number. This is taking 70% of the number, or, equivalently, multiplying it by 0.7.

Therefore, decreased by 30% is 0.7 times this, or

$0.7 \left (6y + 17 \right ) = 0.7 \cdot 6y +0.7 \cdot 17 = 4.2y + 11.9$

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Question

The polynomial is divisible by the linear binomial . Evaluate .

Answer

By the factor theorem, a polynomial is divisible by the linear binomial if and only if . Therefore, we want the value of that makes the polynomial equal to 0 when evaluated at .

$1,296- 27 \cdot 36+ 30- A = 0$

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Question

Factor:

Answer

can be rewritten as and is therefore the sum of two cubes. As such, it can be factored using the pattern

where .

$= $(5T+9)[(5T)^{2}$-5T \cdot 9 $+9^{2}$]$

$= $(5T+9)(25T^{2}$-45T +81)$

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Question

Exponentiate:

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

Answer

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

Vertical multiplication is perhaps the easiest way to multiply trinomials.

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

$$\underline{x ^{4 }$ + $x^{3 }$ + $7x^{2}$ ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; }$

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Question

Factor completely:

Answer

The grouping technique works here:

$=\left $(t^{3}$ $+4t^{2}$ \right )-\left (9t +36 \right )$

$=\left (t ^{2} \cdot t $+t^{2}$ \cdot 4} \right )-\left (9 \cdot t +9 \cdot 4 \right )$

$=t ^{2} \left (t + 4} \right )-9 \left ( t + 4 \right )$

$=\left (t ^{2}-9 \right ) \left (t + 4} \right )$

The first factor is the difference of squares and can be factored further accordingly:

$\left (t ^{2}-9 \right ) \left (t + 4} \right )$

$=\left (t ^{2}- $3^{2}$ \right ) \left (t + 4} \right )$

$=\left (t -3 \right ) \left (t +3 \right ) \left (t + 4} \right )$

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Question

Factor completely:

Answer

Since the first term is a perfect cube, the factoring pattern we are looking to take advantage of is the difference of cubes pattern. However, 243 is not a perfect cube of an integer , so the factoring pattern cannot be applied. No other pattern fits, so the polynomial is a prime.

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Question

Simplify the following expression:

Answer

When simplifying an equation,you must find a common factor for all values in the equation, including both sides.

and, can all be divided by so divide them all at once

$\left ( $\frac{3}{3}$=1, $\frac{18}{3}$=6, $\frac{81}{3}$=27\right )$ .

This leaves you with

.

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