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PSAT Math : Polynomials

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Example Questions

Example Question #1 : Polynomial Operations

Give the degree of the polynomial

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

Possible Answers:

837

777

Correct answer:

Explanation:

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

12+14 + 16+18 =60

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Example Question #1 : Polynomials

Give the degree of the polynomial

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

Possible Answers:

100

365

Correct answer:

100

Explanation:

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:

50 + 30 = 80

70 + 25= 95

45+45 = 90

30 + 70 = 100

The degree of the polynomial is the greatest of these degrees, 100.

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Example Question #1 : Polynomial Operations

Give the degree of the polynomial

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

Possible Answers:

174

100

154

Correct answer:

100

Explanation:

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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Example Question #1 : Polynomial Operations

Give the degree of the polynomial

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

Possible Answers:

400

200

100

Correct answer:

Explanation:

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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Example Question #17 : Algebra

Which of these polynomials has the greatest degree?

Possible Answers:

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

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

All of the polynomials given in the other responses have the same degree.

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

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

Correct answer:

All of the polynomials given in the other responses have the same degree.

Explanation:

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:

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

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

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

100: 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.

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

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

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

8x: 1

The degree of this polynomial is 5.

All four polynomials have the same degree.

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Example Question #21 : Variables

Which of the following monomials has degree 999?

Possible Answers:

$444x^{999}yz$

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

$999x^{4}y^{5}z^{6}$

None of the other responses is correct.

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

Correct answer:

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

Explanation:

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:

$444x^{999}yz$ : 999 + 1 + 1 = 1,001

$333x^{999}y^{999}z^{999}$ : 999+999+999 = 2,997

$999x^{4}y^{5}z^{6}$ : 4+5+6 = 15 (note - 999 is the coefficient)

$100x^{333}y^{333}z^{333}$ : 333+333+333 = 999

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

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Example Question #22 : Variables

Find the degree of the polynomial

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

Possible Answers:

$\small 3$

None of the other answers

$\small 4$

$\small 5$

Correct answer:

$\small 5$

Explanation:

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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Example Question #1 : Polynomials

Add the polynomials.

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

Possible Answers:

$3x^{3}+x^{2}+6x+17$

$4x^{2}+8x+20$

$3x^{3}+x^{2}+8x+20$

$4x^{3}+2x^{2}+4x+14$

Correct answer:

$3x^{3}+x^{2}+8x+20$

Explanation:

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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Example Question #21 : Variables

(x^4+7x^2-5x+4)-(-4x^4+5x^3-x^2+3)

Possible Answers:

5x^4-5x^3+8x^2-5x+1

5x^4+5x^3+6x^2-5x+7

x^4+7x^2-10x+1

x^4-5x^3+11x^2-5x+1

-3x^4+5x^3+6x^2-5x+7

Correct answer:

5x^4-5x^3+8x^2-5x+1

Explanation:

Step 1: Distribute the negative to the second polynomial:

(x^4+7x^2-5x+4)-(-4x^4+5x^3-x^2+3)

x^4+7x^2-5x+4+4x^2-5x^3+x^2-3

Step 2: Combine like terms:

x^4+4x^4-5x^3+7x^2+x^2-5x+4-3

5x^4-5x^3+8x^2-5x+1

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Example Question #2 : Polynomials

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

and

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

What is FG(x) ?

Possible Answers:

$(FG)(x) = x^{3} - x - 3$

$(FG)(x) = x^{5} + x^{4} - x^{3} + 2x^{2} - 5x -10$

$(FG)(x) = x^{5} + x^{4} - x - 2$

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

$(FG)(x) = x^{5} + x^{4} +4x^{3} + 7x^{2} - 5x +10$

Correct answer:

$(FG)(x) = x^{5} + x^{4} +4x^{3} + 7x^{2} - 5x +10$

Explanation:

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

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PSAT Math : Polynomials

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : Polynomial Operations

Example Question #1 : Polynomials

Example Question #1 : Polynomial Operations

Example Question #1 : Polynomial Operations

Example Question #17 : Algebra

Example Question #21 : Variables

Example Question #22 : Variables

Example Question #1 : Polynomials

Example Question #21 : Variables

Example Question #2 : Polynomials

All PSAT Math Resources

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