Adding and Subtracting Polynomials: Practice

The degree of a polynomial is the largest exponent on the variable. The powers present are $3$, $1$, and $0$.

$\max(3, 1, 0) = 3$

So the degree is $3$, from the $5x^3$ term.

A binomial has exactly two terms. Count the terms in each choice.

$x^2 - 9 \;\text{has } 2 \text{ terms}$

The first choice is a trinomial (three terms), $5x$ is a monomial (one term), and the last has four terms.

The terms $6x$ and $3x$ share the power $x^1$, so add their coefficients and keep the power.

$6x + 3x = (6 + 3)x = 9x$

The exponent stays $1$, so the answer is $9x$, not $9x^2$.

Both terms carry $x^2$, so subtract the coefficients and keep the power.

$8x^2 - 5x^2 = (8 - 5)x^2 = 3x^2$

The power does not change, so it is $3x^2$, not $3x^4$.

The constant term is the term with no variable, the $x^0$ term.

$4x^2 - 7x + \mathbf{10}$

The constant term is $10$.

Drop the parentheses, then combine the $x$ terms and the constants.

$2x + 3 + 5x + 1 = (2x + 5x) + (3 + 1) = 7x + 4$

Distribute the minus sign to both terms of the second polynomial, so $-5$ becomes $+5$.

$9x - 2 - 4x + 5 = 5x + 3$

The constant is $-2 + 5 = 3$.

Standard form lists the terms from the highest power down to the constant.

$2x^3 - x^2 + 5$

The other choices have the same terms in a scrambled order.

Count the terms: $x^2$ and $-4$, which is two terms.

$x^2 - 4 \;\text{has } 2 \text{ terms}$

Two terms make a binomial.

The two polynomials have no matching powers except the constants, so only the constants combine.

$x^2 + 2x + (3 + 5) = x^2 + 2x + 8$

The $x^2$ and the $2x$ have no like partner and carry down unchanged.

Distribute the minus sign; the $-3$ becomes $+3$ and cancels the first $-3$.

$6x^2 - 3 - 2x^2 + 3 = 4x^2$

Group the $x$ terms together and the constants together.

$(5x + 3x) + (2 + 4) = 8x + 6$