Polynomials - PSAT Math
Card 1 of 343
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?
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?
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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.
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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Simplify: 
Simplify:
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Cancel by subtracting the exponents of like terms:
Cancel by subtracting the exponents of like terms:
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Add the polynomials.
Add the polynomials.
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We can add together each of the terms of the polynomial which have the same degree for our variable. 
We can add together each of the terms of the polynomial which have the same degree for our variable.
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If 3 less than 15 is equal to 2x, then 24/x must be greater than
If 3 less than 15 is equal to 2x, then 24/x must be greater than
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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.
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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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
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
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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
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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What is the remainder when the polynomial 
is divided by 
?
What is the remainder when the polynomial
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By the remainder theorem, if a polynomial 
is divided by the linear binomial 
, the remainder is 
- that is, the polynomial evaluated at 
. The remainder of dividing 
by 
is the dividend evaluated at 
, which is
By the remainder theorem, if a polynomial
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Find the degree of the polynomial
Find the degree of the polynomial
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The degree of the polynomial is the largest degree of any one of it's individual terms.
The degree of 
is 
The degree of 
is 
The degree of 
is 
The degree of 
is 
The degree of 
is 
is the largest degree of any one of the terms of the polynomial, and so the degree of the polynomial is 
.
The degree of the polynomial is the largest degree of any one of it's individual terms.
The degree of
The degree of
The degree of
The degree of
The degree of
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Give the degree of the polynomial
Give the degree of the polynomial
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The polynomial has one term, so its degree is the sum of the exponents of the variables:
The polynomial has one term, so its degree is the sum of the exponents of the variables:
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Give the degree of the polynomial
Give the degree of the polynomial
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The degree of a polynomial in more than one variable is the greatest degree of any of the terms; the degree of a term is the sum of the exponents. The degrees of the terms in the given polynomial are:
The degree of the polynomial is the greatest of these degrees, 100.
The degree of a polynomial in more than one variable is the greatest degree of any of the terms; the degree of a term is the sum of the exponents. The degrees of the terms in the given polynomial are:
The degree of the polynomial is the greatest of these degrees, 100.
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Give the degree of the polynomial
Give the degree of the polynomial
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The degree of a polynomial in one variable is the greatest exponent of any of the powers of the variable. The terms have as their exponents, in order, 44, 20, 10, and 100; the greatest of these is 100, which is the degree.
The degree of a polynomial in one variable is the greatest exponent of any of the powers of the variable. The terms have as their exponents, in order, 44, 20, 10, and 100; the greatest of these is 100, which is the degree.
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Give the degree of the polynomial
Give the degree of the polynomial
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The degree of a polynomial in one variable is the greatest exponent of any of the powers of the variable. The terms have as their exponents, in order, 10, 20, 30, and 40; 40 is the greatest of them and is the degree of the polynomial.
The degree of a polynomial in one variable is the greatest exponent of any of the powers of the variable. The terms have as their exponents, in order, 10, 20, 30, and 40; 40 is the greatest of them and is the degree of the polynomial.
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Which of these polynomials has the greatest degree?
Which of these polynomials has the greatest degree?
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The degree of a polynomial is the highest degree of any term; the degree of a term is the exponent of its variable or the sum of the exponents of its variables, with unwritten exponents being equal to 1. For each term in a polynomial, write the exponent or add the exponents; the greatest number is its degree. We do this with all four choices:
:
A constant term has degree 0.
The degree of this polynomial is 5.
The degree of this polynomial is 5.
The degree of this polynomial is 5.
The degree of this polynomial is 5.
All four polynomials have the same degree.
The degree of a polynomial is the highest degree of any term; the degree of a term is the exponent of its variable or the sum of the exponents of its variables, with unwritten exponents being equal to 1. For each term in a polynomial, write the exponent or add the exponents; the greatest number is its degree. We do this with all four choices:
The degree of this polynomial is 5.
The degree of this polynomial is 5.
The degree of this polynomial is 5.
The degree of this polynomial is 5.
All four polynomials have the same degree.
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Which of the following monomials has degree 999?
Which of the following monomials has degree 999?
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The degree of a monomial term is the sum of the exponents of its variables, with the default being 1.
For each monomial, this sum - and the degree - is as follows:
: 
: 
: 
(note - 999 is the coefficient)
: 
is the correct choice.
The degree of a monomial term is the sum of the exponents of its variables, with the default being 1.
For each monomial, this sum - and the degree - is as follows:
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and
What is 
?
and
What is
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so we multiply the two function to get the answer. We use 
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Multiply:
Multiply:
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This product fits the sum of cubes pattern, where 
:
So
This product fits the sum of cubes pattern, where
So
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Step 1: Distribute the negative to the second polynomial:
Step 2: Combine like terms:
Step 1: Distribute the negative to the second polynomial:
Step 2: Combine like terms:
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Solve for 
.
Solve for
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Factor the expression
numerator: find two numbers that add to 2 and multiply to -8 \[use 4,-2\]
denominator: find two numbers that add to 5 and multiply to -14 \[use 7,-2\]
new expression:
Cancel the 
and cross multiply.
Factor the expression
numerator: find two numbers that add to 2 and multiply to -8 \[use 4,-2\]
denominator: find two numbers that add to 5 and multiply to -14 \[use 7,-2\]
new expression:
Cancel the
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Multiply the binomial.
Multiply the binomial.
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By multiplying with the foil method, we multiply our first values giving 
, our outside values giving 
. our inside values which gives 
, and out last values giving 
.
By multiplying with the foil method, we multiply our first values giving
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Give the coefficient of 
in the binomial expansion of 
.
Give the coefficient of
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If the expression 
is expanded, then by the binomial theorem, the 
term is
or, equivalently, the coefficient of 
is
Therefore, the 
coefficient can be determined by setting
If the expression
or, equivalently, the coefficient of
Therefore, the
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