Polynomial Operations - PSAT Math

Card 0 of 17

Question

Add the polynomials.

$(x^{2}+5x+12) + (3x^{3}+3x+8)$

Answer

We can add together each of the terms of the polynomial which have the same degree for our variable. $(3x^{3}) + (x^{2}) + (5x+3x) + (12+8) = 3x^{3}+x^{2}+8x+20$

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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

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

What is the remainder when the polynomial $x^{4} - 5x^{3} + 3x - 17$ is divided by ?

Answer

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 $x^{4} - 5x^{3} + 3x - 17$ by is the dividend evaluated at , which is

$x^{4} - 5x^{3} + 3x - 17$

$= 6^{4} - 5 \cdot 6^{3} + 3 \cdot 6 - 17$

$= 1,296 - 5 \cdot 216 + 18 - 17$

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Question

Find the degree of the polynomial

$\small - x^{2}+3x - 2x^{5} - x^{3} +4$

Answer

The degree of the polynomial is the largest degree of any one of it's individual terms.

$\small - x^{2}+3x - 2x^{5} - x^{3} +4$

The degree of $\small -x^2$ is $\small 2$

The degree of $\small 3x$ is $\small 1$

The degree of $\small -2x^5$ is $\small 5$

The degree of $\small -x^3$ is $\small 3$

The degree of $\small 4$ is $\small 0$

$\small 5$ is the largest degree of any one of the terms of the polynomial, and so the degree of the polynomial is $\small 5$ .

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Question

Give the degree of the polynomial

$777a^{12}b^{14}c^{16}d^{18}$

Answer

The polynomial has one term, so its degree is the sum of the exponents of the variables:

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Question

Give the degree of the polynomial

$x^{50}y^{30} - x^{70}y^{25}+ x^{45}y ^{45} - x^{30}y ^{70}$

Answer

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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Question

Give the degree of the polynomial

$y^{44} + y^{20} - y^{10} + y^{100}$

Answer

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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Question

Give the degree of the polynomial

$400x^{10}- 300x^{20} + 200x^{30}-100 x^{40}$

Answer

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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Question

Which of these polynomials has the greatest degree?

Answer

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:

$\underline{6x^{3}y^{2} + 4 xy^{3} - 100}$ :

$6x^{3}y^{2}: 3 + 2 = 5$

$4 xy^{3}: 1 + 3 = 4$

A constant term has degree 0.

The degree of this polynomial is 5.

$\underline{7xy^{4} - 4x^{2}+ y^{3}}$

$7xy^{4} : 1 + 4 = 5$

$4x^{2} : 2$

$y^{3}: 3$

The degree of this polynomial is 5.

$\underline{8x ^{4}y - 4 y^{2} + 9x^{3}y}$

$8x ^{4}y: 4 + 1= 5$

$4 y^{2} : 2$

$9x^{3}y: 3 + 1= 4$

The degree of this polynomial is 5.

$\underline{-5 x^{2}y^{2}+ 7x^{3}y^{2}+ 8x}$

$-5 x^{2}y^{2} : 2 + 2 = 4$

$7x^{3}y^{2} : 3 + 2 = 5$

The degree of this polynomial is 5.

All four polynomials have the same degree.

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Question

Which of the following monomials has degree 999?

Answer

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:

$444x^{999}yz$ :

$333x^{999}y^{999}z^{999}$ :

$999x^{4}y^{5}z^{6}$ : (note - 999 is the coefficient)

$100x^{333}y^{333}z^{333}$ :

$100x^{333}y^{333}z^{333}$ is the correct choice.

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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

Multiply:

$\left (100Y^{2}- 70Y + 49 \right )(10Y+7)$

Answer

$\left (100Y^{2}- 70Y + 49 \right )(10Y+7)$

$= \left [\left (10Y \right )^{2}- 10Y\cdot 7 +7^{2} \right ](10Y+7)$

This product fits the sum of cubes pattern, where :

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

So

$\left [\left (10Y \right )^{2}- 10Y\cdot 7 +7^{2} \right ](10Y+7)$

$= (10Y)^{3}+7^{3} = 1,000Y^{3} + 343$

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Question

Answer

Step 1: Distribute the negative to the second polynomial:

Step 2: Combine like terms:

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