Basic operations with polynomials

A polynomial is a sum of terms. A term is a number multiplied by letters with powers, such as $3x^2$, $-5x$, or $7$. The number in front is called the coefficient.

Like terms have the same letters raised to the same powers. So $2x$ and $3x$ are alike, and $3mn$ and $5mn$ are alike, but $x^2$ and $x$ are not. To combine like terms, keep the letter part and add the numbers in front. For example $2x + 3x = 5x$, and $3mn - 5mn = (3 - 5)mn = -2mn$. Terms that are not alike are simply left as they are.

To remove a bracket, multiply every term inside it by the term outside, one at a time. Take $m(3x + 4y^2)$: first $m \times 3x = 3mx$, then $m \times 4y^2 = 4my^2$, which gives $3mx + 4my^2$.

Be careful when a minus sign sits in front of a bracket, because it multiplies every inside term by a negative. Take $-x(-x + 2y)$: $-x \times -x = x^2$ (negative times negative is positive), and $-x \times 2y = -2xy$, giving $x^2 - 2xy$.

To write an expression in the form $ax^2 + bx + c$, first multiply out every bracket, then gather the like terms: all the $x^2$ terms together, then all the $x$ terms, then the plain numbers.

Combine $4x + 5x$.
They are like terms, so add the numbers in front:
$4x + 5x = 9x$.
In the same way, $8y - 3y = 5y$.

Combine $3mn - 5mn$.
The letter part $mn$ is the same, so add the numbers:
$(3 - 5)mn = -2mn$.

Expand $2(x + 3)$.
Multiply each inside term by $2$:
$2 \times x = 2x$ and $2 \times 3 = 6$,
so $2(x + 3) = 2x + 6$.

Expand $n(2a + 5b^2)$.
Multiply each inside term by $n$:
$n \times 2a = 2an$ and $n \times 5b^2 = 5nb^2$,
so the result is $2an + 5nb^2$.

Expand $-y(-y + 3x - 2x^2)$.
Multiply each inside term by $-y$:
$-y \times -y = y^2$,
$-y \times 3x = -3xy$,
$-y \times -2x^2 = 2x^2 y$.
So the result is $y^2 - 3xy + 2x^2 y$.

Write $-x(3 + 4x) + 2(x + 10 + \frac{3x^2}{2})$ in the form $ax^2 + bx + c$.
First multiply out: $-3x - 4x^2 + 2x + 20 + 3x^2$.
Then gather like terms:
$x^2$ terms give $-4x^2 + 3x^2 = -x^2$,
$x$ terms give $-3x + 2x = -x$,
leaving $-x^2 - x + 20$.