Expanding expressions where one bracket is linear and the other is quadratic means multiplying a bracket with two terms by a bracket with three terms. The goal is to remove the brackets and simplify the result.

 

The method is the same as before: multiply every term in the first bracket by every term in the second bracket, then collect like terms.

 

 
Understanding the Structure

A linear expression has a highest power of \( x \) equal to 1, for example \( x + 2 \).

 

A quadratic expression has a highest power of \( x \) equal to 2, for example \( x^2 + 3x + 1 \).

 

When these are multiplied together, the result will usually be a cubic expression, with a highest power of \( x^3 \).

 

 

Expanding a Linear Bracket by a Quadratic Bracket

Consider the expression:

$$
(x + 2)(x^2 + 3x + 1)
$$

 

Multiply \( x \) by every term in the quadratic bracket:

$$
x \times x^2 + x \times 3x + x \times 1
$$

$$
= x^3 + 3x^2 + x
$$

 

Now multiply \( 2 \) by every term in the quadratic bracket:

$$
2 \times x^2 + 2 \times 3x + 2 \times 1
$$

$$
= 2x^2 + 6x + 2
$$

 

Add the results together:

$$
x^3 + 3x^2 + x + 2x^2 + 6x + 2
$$

 

Collect like terms:

$$
x^3 + 5x^2 + 7x + 2
$$

 

Another Example

Consider:

$$
(2x - 1)(x^2 - 4x + 3)
$$

 

Multiply \( 2x \) by each term:

$$
2x \times x^2 - 2x \times 4x + 2x \times 3
$$

$$
= 2x^3 - 8x^2 + 6x
$$

 

Multiply \( -1 \) by each term:

$$
-x^2 + 4x - 3
$$

 

Combine all terms:

$$
2x^3 - 8x^2 + 6x - x^2 + 4x - 3
$$

 

Collect like terms:

$$
2x^3 - 9x^2 + 10x - 3
$$

 

 
Key Points to Remember

Multiply every term in the linear expression by every term in the quadratic expression.
Work systematically to avoid missing terms.
Collect like terms carefully at the end.

 

Expanding linear and quadratic expressions correctly is an important skill for simplifying algebraic expressions and solving higher level equations.