PSAT Math - How to find the degree of a polynomial

Which of these polynomials has the greatest degree?

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

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

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

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

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

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:

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

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

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

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.

Give the degree of the polynomial

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

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.

Find the degree of the polynomial

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

$\small 5$

$\small 4$

$\small 3$

None of the other answers

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 $\small 5$ .

Give the degree of the polynomial

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

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:

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

Give the degree of the polynomial

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

Explanation

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

Give the degree of the polynomial

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

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.

Which of the following monomials has degree 999?

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