SAT Math - Polynomial Operations

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

$x ^{2}+ 5xy + 25y^{2}$

$x ^{2}-5xy + 25y^{2}$

$x ^{2}-10xy + 25y^{2}$

$x ^{2}+10xy + 25y^{2}$

$x ^{2} + 25y^{2}$

Explanation

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,

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

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

$x ^{2}+ 5xy + 25y^{2}$

$x ^{2}-5xy + 25y^{2}$

$x ^{2}-10xy + 25y^{2}$

$x ^{2}+10xy + 25y^{2}$

$x ^{2} + 25y^{2}$

Explanation

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,

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

Find the degree of the polynomial:

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

Explanation

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.

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?

Explanation

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.

Find the degree of the polynomial:

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

Explanation

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.

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?

Explanation

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.

Find the degree of the following polynomial: