Multiply monomials by polynomials: Area model | Algebra 1 | Khan Academy

Main Ideas

The video explains how to multiply a monomial by a polynomial using an area model. A rectangle’s total area can be found either by multiplying its full height by its full width or by adding the areas of its smaller sections. These two approaches represent the distributive property.

Method

1. Identify the rectangle’s height and total width.
2. Split the width into the polynomial’s individual terms, creating smaller rectangular sections.
3. Find each section’s area by multiplying its width by the rectangle’s height.
4. Add the section areas to express the total area as an expanded polynomial.
5. Check that the result matches distributing the height across every term in the width.

Examples

Height (5), Width (3x^2 - x + 2)

• (5(3x^2) = 15x^2)
• (5(-x) = -5x)
• (5(2) = 10)

Total area: (\boxed{15x^2 - 5x + 10})

Height (2x), Width (3x + 4)

• ((2x)(3x) = 6x^2)
• ((2x)(4) = 8x)

Total area: (\boxed{6x^2 + 8x})

Notes on the Diagrams

The diagrams are conceptual and are not necessarily drawn to scale. A negative term such as (-x) can be represented abstractly; the drawing does not establish the actual widths or their relative sizes.

Speaker or Source

Khan Academy narrator/instructor — explains both examples using the area model and the distributive property.