UPSKILL MATH PLUS

Learn Mathematics through our AI based learning portal with the support of our Academic Experts!

### Theory:

Theorem
If a polynomial $$p(x)$$ of degree greater than or equal to one is divided by a linear polynomial $$(x - a)$$ then the remainder is $$p(a)$$, where $$a$$ is any real number.
Explanation:
If a polynomial $$p(x)$$ is divided by $$(x-a)$$, then $$p(a)$$ is the remainder.
The theorem states that, any polynomial $$p(x)$$ when divided by an expression of the form $$(x-a)$$ leaves a remainder $$p(a)$$.
Example:
1. Find the remainder when $$x^{2} + 3x + 2$$ is divided by $$x + 2$$.

Given:

The polynomial $$p(x) = x^{2} + 3x + 2$$.

To find:

The remainder when $$p(x) = x^{2} + 3x + 2$$ is divided by $$x + 2$$.

Theorem used:

If a polynomial $$p(x)$$ is divided by $$(x-a)$$, then $$p(a)$$ is the remainder.

Solution:

Step 1: Find the zero of the polynomial $$x = a$$.

Equate $$x + 2$$ to zero and solve for $$x$$.

$$x$$ $$+$$ $$2$$ $$=$$ $$0$$

$$x$$ $$=$$ $$-2$$

Step 2: Find the remainder $$p(-2)$$.

Substitute $$x = -2$$ in $$p(x)$$.

$$p(-2) = (-2)^{2} + 3(-2) + 2$$

$$= 4 - 6 + 2$$

$$= 0$$

2. Find the remainder when $$x^{2} + 3x - 2$$ is divided by $$x + 1$$.

Given:

The polynomial $$p(x) = x^{2} + 3x - 2$$.

To find:

The remainder when $$p(x) = x^{2} + 3x - 2$$ is divided by $$x + 1$$.

Theorem used:

If a polynomial $$p(x)$$ is divided by $$(x-a)$$, then $$p(a)$$ is the remainder.

Solution:

Step 1: Find the zero of the polynomial $$x = a$$.

Equate $$x + 1$$ to zero and solve for $$x$$.

$$x$$ $$+$$ $$1$$ $$=$$ $$0$$

$$x$$ $$=$$ $$-1$$

Step 2: Find the remainder $$p(-1)$$.

Substitute $$x = -1$$ in $$p(x)$$.

$$p(-1) = (-1)^{2} + 3(-1) - 2$$

$$= 1 - 3 - 2$$

$$= -4$$
Important!
In Example (1), $$x + 2$$ is the factor of the polynomial $$x^{2} + 3x + 2$$ as it satisfies the equation  $$p(x) = 0$$.

In Example (2), $$x + 1$$ is not the factor of the polynomial $$x^{2} + 3x - 2$$ as it does not satisfies the equation  $$p(x) = 0$$.