Adding and Subtracting Polynomials: Practice

Drop the parentheses and combine each power.

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

Distribute the minus sign to all three terms, then combine.

$6x^2 + 2x - 1 - 3x^2 + 4x - 5 = 3x^2 + 6x - 6$

The $x$ terms give $2x + 4x = 6x$ and the constants give $-1 - 5 = -6$.

Neither polynomial has an $x^2$ term. Combine the $x^3$, $x$, and constant terms.

$(2x^3 + x^3) + (-x + 5x) - 4 = 3x^3 + 4x - 4$

Only one polynomial has an $x^4$ term, so nothing cancels the top power.

$x^4 + 3x^2 + (2x - x) = x^4 + 3x^2 + x$

The highest power is $4$, so the degree is $4$.

Combine each power. The $x$ terms and the constants each cancel.

$(x^2 + x^2) + (x - x) + (1 - 1) = 2x^2$

Distribute the minus sign; the $x^3$ terms cancel, the $x^2$ terms add, and the $x$ terms add.

$x^2 + x^2 = 2x^2, \qquad -2x - 2x = -4x$

The result is $2x^2 - 4x$.

Add the three side lengths, combining the $x$ terms and the constants.

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

Distribute the minus sign; the $x^2$ terms and the constants cancel.

$6x - 2x = 4x$

Everything else cancels, leaving $4x$.

Add all three polynomials at once, one power at a time.

$(3x^2 + x^2) + (-4x + 2x) + (2 - 5) = 4x^2 - 2x - 3$

A rectangle has two lengths and two widths. Add all four side lengths.

$(3x + 2) + (3x + 2) + (x + 4) + (x + 4) = 8x + 12$

The $x$ terms give $8x$ and the constants give $2 + 2 + 4 + 4 = 12$.

Distribute the minus sign; the constant $7$ cancels.

$2x^2 + 7 - x^2 - 7 = x^2$

Combine each power. The $x$ terms cancel and the constants cancel.

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