Polynomial Operations - SAT Math

Card 0 of 19

Question

Given a♦b = (a+b)/(a-b) and b♦a = (b+a)/(b-a), which of the following statement(s) is(are) true:

I. a♦b = -(b♦a)

II. (a♦b)(b♦a) = (a♦b)2

III. a♦b + b♦a = 0

Answer

Notice that - (a-b) = b-a, so statement I & III are true after substituting the expression. Substitute the expression for statement II gives ((a+b)/(a-b))((a+b)/(b-a))=((a+b)(b+a))/((-1)(a-b)(a-b))=-1 〖(a+b)〗2/〖(a-b)〗2 =-((a+b)/(a-b))2 = -(a♦b)2 ≠ (a♦b)2

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Question

If a positive integer a is divided by 7, the remainder is 4. What is the remainder if 3_a_ + 5 is divided by 3?

Answer

The best way to solve this problem is to plug in an appropriate value for a. For example, plug-in 11 for a because 11 divided by 7 will give us a remainder of 4.

Then 3_a + 5_, where a = 11, gives us 38. Then 38 divided by 3 gives a remainder of 2.

The algebra method is as follows:

a divided by 7 gives us some positive integer b, with a remainder of 4.

Thus,

a / 7 = b 4/7

a / 7 = (7_b +_ 4) / 7

a = (7_b_ + 4)

then 3_a + 5 =_ 3 (7_b_ + 4) + 5

(3_a_+5)/3 = \[3(7_b_ + 4) + 5\] / 3

= (7_b_ + 4) + 5/3

The first half of this expression (7_b_ + 4) is a positive integer, but the second half of this expression (5/3) gives us a remainder of 2.

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Question

Answer

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Question

Simplify: $\frac{6x^7y^3z^9}{3x^6y^3z}$

Answer

Cancel by subtracting the exponents of like terms:

$\frac{6x^7y^3z^9}{3x^6y^3z} = 2x^{7-6}y^{3-3}z^{9-1}=2xy^0z^8=2xz^8$

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Question

If 3 less than 15 is equal to 2x, then 24/x must be greater than

Answer

Set up an equation for the sentence: 15 – 3 = 2x and solve for x. X equals 6. If you plug in 6 for x in the expression 24/x, you get24/6 = 4. 4 is only choice greater than a.

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Question

By what expression can be multiplied to yield the product $6x^{3} +20x^{2} +5x-21$ ?

Answer

Divide $6x^{3} +20x^{2} +5x-21$ by by setting up a long division.

Divide the lead term of the dividend, $6x^{3}$ , by that of the divisor, ; the result is $\frac{6x^{3}}{3x} = 2 x^{2}$

Enter that as the first term of the quotient. Multiply this by the divisor:

$2 x^{2} (3x+7) = 2 x^{2} (3x)+2 x^{2} ( 7) = 2x^{3} + 14x^{2}$

Subtract this from the dividend. This is shown in the figure below.

Repeat the process with the new difference:

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

$2 x (3x+7) = 2 x (3x)+2 x ( 7) = 2x^{2} + 14x$

Repeating:

$\frac{-9x}{3x} = -3$

The quotient - and the correct response - is $2x^{2 }+ 2x - 3$ .

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Question

Divide $x^{3} + 125y ^{3}$ by .

Answer

It is not necessary to work a long division if you recognize $x^{3} + 125y ^{3}$ as the sum of two perfect cube expressions:

$x^{3} + 125y ^{3} = x^{3} + 5^{3} y ^{3} = x^{3} + (5y )^{3}$

A sum of cubes can be factored according to the pattern

$A^{3}+ B^{3} = (A+B) (A^{2}+ AB +B^{2})$ ,

so, setting ,

$x^{3} + (5y )^{3} =(x+5y) [x ^{2}+ x \cdot 5y + (5y)^{2}]$

$=(x+5y)(x ^{2}+ 5xy + 25y^{2})$

Therefore,

$\frac{x^{3} + 125y ^{3}}{x+ 5y} = \frac{(x+5y)(x ^{2}+ 5xy + 25y^{2})}{x+ 5y} = x ^{2}+ 5xy + 25y^{2}$

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Question

Find the degree of the polynomial:

$x^{3}-3x^{5}+7$

Answer

To find the degree of a polynomial we must find the largest exponent in the function.

The degree of the polynomial $x^{3}-3x^{5}+7$ is 5, as the largest exponent of is 5 in the second term.

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Question

What is the degree of the polynomial $5x^{2}y^{2}+4y^{3}+3x+10$ ?

Answer

When a polynomial has more than one variable, we need to find the degree by adding the exponents of each variable in each term.

$5x^{2}y^{2}$ has a degree of 4 (since both exponents add up to 4), so the polynomial has a degree of 4 as this term has the highest degree.

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Question

Find the degree of the following polynomial:

$7xy^{2}z^{3}+8y^{5}-3x^{4}+4$

Answer

When a polynomial has more than one variable, we need to find the degree by adding the exponents of each variable in each term.

Even though $8y^{5}$ has a degree of 5, it is not the highest degree in the polynomial -

$7xy^{2}z^{3}$ has a degree of 6 (with exponents 1, 2, and 3). Therefore, the degree of the polynomial is 6.

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Question

Solve each problem and decide which is the best of the choices given.

What is the degree of the following polynomial?

Answer

The degree is defined as the largest exponent in the polynomial. In this case, it is .

$3x^4-9x^2+12x^{\color{Red} 7}-9x$

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Question

What is the degree of this polynomial?

Answer

When an exponent with a power is raised to another power, the value of the power are multiplied.

$x^2(x^{2*3})=x^2(x^6)$

When multiplying exponents you add the powers together

$x^{6+2}=x^8$

The degree of a polynomial is the determined by the highest power. In this problem the highest power is 8.

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Question

Find the degree of the following polynomial:

Answer

The degree of a polynomial is the largest exponent on one of its variables (for a single variable), or the largest sum of exponents on variables in a single term (for multiple variables).

Here, the term with the largest exponent is , so the degree of the whole polynomial is 6.

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Question

$F(x) = x^{3} + x^{2} - x + 2$

and

$G(x) = x^{2} + 5$

What is ?

Answer

$(FG)(x) = F(x)G(x)$ so we multiply the two function to get the answer. We use $x^{m}x^{n} = x^{m+n}$

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Question

Find the product:

Answer

Find the product:

Step 1: Use the distributive property.

Step 2: Combine like terms.

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Question

and represent positive quantities.

$x^{2} = 20$

$y^{2} = 5$

Evaluate $(x-y) (x^{2}+xy+y^{2})$ .

Answer

$(x-y) (x^{2}+xy+y^{2})$ can be recognized as the pattern conforming to that of the difference of two perfect cubes:

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

Additionally,

$x^{2} = 20$ and is positive, so

$x = \sqrt{20}$

Using the product of radicals property, we see that

$x = \sqrt{4 \cdot 5} = \sqrt{4 } \cdot \sqrt{ 5} = 2 \sqrt{ 5}$

and

$x ^{3 } = x^{2} \cdot x = 20 \cdot 2 \sqrt{5} = 40 \sqrt{5}$

$y^{2} = 5$ and is positive, so

$y = \sqrt{5}$ ,

and

$y^{3}= y^{2} \cdot y = 5 \sqrt{5}$

Substituting for $x ^{3 }$ and $y^{3}$ , then collecting the like radicals,

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

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

$= 40 \sqrt{5}-5 \sqrt{5}$

$=( 40 -5 )\sqrt{5}$

$=35 \sqrt{5}$ .

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Question

represents a positive quantity; represents a negative quantity.

$x^{4} = 116$

$y^{2} = 11$

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

Answer

The first two binomials are the difference and the sum of the same two expressions, which, when multiplied, yield the difference of their squares:

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

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

Again, a sum is multiplied by a difference to yield a difference of squares, which by the Power of a Power Property, is equal to:

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

$= x^{2\cdot 2} - y^{2\cdot 2}$

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

$y^{2} = 11$ , so by the Power of a Power Property,

$(y^{2} )^{2}= 11^{2}$

$y^{2\cdot 2} = 121$

$y^{4} = 121$

Also, $x^{4} = 116$ , so we can now substitute accordingly:

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

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

Note that the signs of and are actually irrelevant to the problem.

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Question

represents a positive quantity; represents a negative quantity.

$x^{6} = 625$

$y ^{6} = 64$

Evaluate $(x-y) (x^{2}+xy+y^{2})$ .

Answer

$(x-y) (x^{2}+xy+y^{2})$ can be recognized as the pattern conforming to that of the difference of two perfect cubes:

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

Additionally, by way of the Power of a Power Property,

$(x^{3})^{ 2 } = x^{3 \cdot 2} = x^{6 }$ , making $x^{3}$ a square root of $x^{6}$ , or 625; since is positive, so is $x^{3}$ , so

$x^{3} = \sqrt{625} = 25$ .

Similarly, $y ^{3}$ is a square root of $y ^{6}$ , or 64; since is negative, so is $y^{3}$ (as an odd power of a negative number is negative), so

$y^{3} = - \sqrt{64} = -8$ .

Therefore, substituting:

$(x-y) (x^{2}+xy+y^{2}) = x^{3} - y^{3} = 25 - (-8) = 25+8= 33$ .

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Question

Simplify the following expression:

$(2q^{2}+4q+7)-(3q^{2}-5)$

Answer

This is not a FOIL problem, as we are adding rather than multiplying the terms in parentheses.

Add like terms together:

$2q^{2}-3q^{2}=-1q^{2}=-q^2$

has no like terms.

Combine these terms into one expression to find the answer:

$-q^{2}+4q+12$

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