To read an expression the way algebra does, you need names for its parts. You have used these words before when combining like terms; now we make them exact, because the rules you will learn later (for factoring, for polynomials) talk about terms and factors all the time.
- A term is a piece that is added. In $4x^2 - x + 7$ the terms are $4x^2$, $-x$ and $7$.
- A factor is a piece that is multiplied. In $4x^2 = 4 \cdot x \cdot x$ the factors are $4$, $x$ and $x$.
- A coefficient is the number factor of a term, sign included. The coefficient of $x^2$ above is $4$, and the coefficient of $x$ is $-1$.
- A constant term has no variable: here it is $7$.
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Idea clave
Worked examples
Ejemplo resuelto: Naming every part
- Rewrite subtraction as adding the opposite: $4x^2 + (-x) + 7 + (-2x^3)$.
- Terms: $4x^2$, $-x$, $7$, $-2x^3$. That is four terms.
- Coefficients: $4$ (of $x^2$), $-1$ (of $x$), $-2$ (of $x^3$).
- The constant term is $7$.
Respuesta: Four terms; coefficients $4$, $-1$, $-2$; constant $7$.
Ejemplo resuelto: Seeing a block as one thing
- Treat $(x + 2)$ as a single block, as if it were one letter $u$: $3u^2 - 5u + 1$.
- Now the terms are easy to see: $3u^2$, $-5u$ and $1$.
Respuesta: Three terms. The coefficient of the block $(x + 2)^2$ is $3$.
Seeing structure tells you what a formula means. In $A = P(1 + r)^t$, the expression is a product of two factors, $P$ and $(1 + r)^t$. The starting amount $P$ is multiplied by the same factor $1 + r$ once for each of the $t$ years.
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