Expand and simplify: (9 - r)(2

Expand and simplify: (9 - r)(2 - 3r)

Understand the Problem

The question is asking to expand the product of two binomials involving the variable 'r'. This involves applying the distributive property (often remembered with the acronym FOIL) to multiply each term in the first binomial by each term in the second binomial, and then simplifying by combining like terms.

Answer

$3r^2 - 29r + 18$

Steps to Solve

Apply the distributive property (FOIL)

Multiply each term in the first binomial $(9 - r)$ by each term in the second binomial $(2 - 3r)$.

$(9 - r)(2 - 3r) = 9(2) + 9(-3r) - r(2) - r(-3r)$

Perform the multiplications

Calculate each of the products:

$9(2) = 18$ $9(-3r) = -27r$ $-r(2) = -2r$ $-r(-3r) = 3r^2$

So, we have:

$18 - 27r - 2r + 3r^2$

Combine like terms

Combine the '$r$' terms: $-27r - 2r = -29r$.

The expression now becomes:

$18 - 29r + 3r^2$

Rearrange terms

Rearrange the terms in descending order of the exponent of $r$:

$3r^2 - 29r + 18$

More Information

The expanded form of $(9 - r)(2 - 3r)$ is $3r^2 - 29r + 18$. This is a quadratic expression in the variable 'r'.

Tips

A common mistake is incorrectly applying the distributive property or making sign errors when multiplying the terms. For example, forgetting that $-r \times -3r = 3r^2$, not $-3r^2$. Another common mistake is not combining like terms correctly.

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