Introduction to Linear Polynomials - Exploring linear patterns

Find the zero of the linear polynomial $p(x) = 5x - 15$.

To find the zero, we set $p(x) = 0$: $5x - 15 = 0$ $5x = 15$ $x = \frac{15}{5}$ $x = 3$

The zero of the polynomial is the value of $x$ that makes the expression equal to zero. For $ax + b$, it is always $-\frac{b}{a}$.

Consider the sequence $4, 7, 10, 13, \dots$. Express this pattern as a linear polynomial in terms of $n$, where $n$ is the position of the term.

1. Find the common difference: $7 - 4 = 3$, $10 - 7 = 3$. So, $d = 3$.
2. The general form is $T_n = dn + c$.
3. Substitute $n = 1, T_1 = 4$: $4 = 3(1) + c$ $4 = 3 + c$ $c = 1$
4. The linear expression is $T_n = 3n + 1$.

A sequence with a constant difference represents a linear pattern. The coefficient of $n$ is the common difference.

Which of the following are linear polynomials? (i) $2x + 3$ (ii) $x^2 - 5$ (iii) $\sqrt{x} + 1$ (iv) $4y$

(i) $2x + 3$ is linear (degree $1$). (ii) $x^2 - 5$ is not linear (degree $2$). (iii) $\sqrt{x} + 1$ is not a polynomial because the power of $x$ is $\frac{1}{2}$. (iv) $4y$ is linear (degree $1$).

A linear polynomial must have a variable with a power of exactly $1$ and must satisfy the definition of a polynomial (whole number exponents).